Special
DiracDelta
- class diofant.functions.special.delta_functions.DiracDelta(arg, k=Integer(0))[source]
The DiracDelta function and its derivatives.
DiracDelta function has the following properties:
diff(Heaviside(x),x) = DiracDelta(x)integrate(DiracDelta(x-a)*f(x),(x,-oo,oo)) = f(a)andintegrate(DiracDelta(x-a)*f(x),(x,a-e,a+e)) = f(a)DiracDelta(x) = 0for allx != 0DiracDelta(g(x)) = Sum_i(DiracDelta(x-x_i)/abs(g'(x_i)))Wherex_i-s are the roots ofg
Derivatives of
k-th order of DiracDelta have the following property:DiracDelta(x,k) = 0, for allx != 0
See also
diofant.functions.special.delta_functions.Heaviside,diofant.simplify.simplify.simplify,diofant.functions.special.delta_functions.DiracDelta.is_simple,diofant.functions.special.tensor_functions.KroneckerDeltaReferences
- is_simple(self, x)[source]
Tells whether the argument(args[0]) of DiracDelta is a linear expression in x.
x can be:
a symbol
Examples
>>> DiracDelta(x*y).is_simple(x) True >>> DiracDelta(x*y).is_simple(y) True
>>> DiracDelta(x**2+x-2).is_simple(x) False
>>> DiracDelta(cos(x)).is_simple(x) False
- simplify(self, x)[source]
Compute a simplified representation of the function using property number 4.
x can be:
a symbol
Examples
>>> DiracDelta(x*y).simplify(x) DiracDelta(x)/Abs(y) >>> DiracDelta(x*y).simplify(y) DiracDelta(y)/Abs(x)
>>> DiracDelta(x**2 + x - 2).simplify(x) DiracDelta(x - 1)/3 + DiracDelta(x + 2)/3
Heaviside
Gamma, Beta and related Functions
- class diofant.functions.special.gamma_functions.gamma(arg)[source]
The gamma function
\[\Gamma(x) := \int^{\infty}_{0} t^{x-1} e^{t} \mathrm{d}t.\]The
gammafunction implements the function which passes through the values of the factorial function, i.e. \(\Gamma(n) = (n - 1)!\) when n is an integer. More general, \(\Gamma(z)\) is defined in the whole complex plane except at the negative integers where there are simple poles.Examples
Several special values are known:
>>> gamma(1) 1 >>> gamma(4) 6 >>> gamma(Rational(3, 2)) sqrt(pi)/2
The Gamma function obeys the mirror symmetry:
>>> conjugate(gamma(x)) gamma(conjugate(x))
Differentiation with respect to x is supported:
>>> diff(gamma(x), x) gamma(x)*polygamma(0, x)
Series expansion is also supported:
>>> gamma(x).series(x, 0, 3) 1/x - EulerGamma + x*(EulerGamma**2/2 + pi**2/12) + x**2*(-EulerGamma*pi**2/12 + polygamma(2, 1)/6 - EulerGamma**3/6) + O(x**3)
We can numerically evaluate the gamma function to arbitrary precision on the whole complex plane:
>>> gamma(pi).evalf(40) 2.288037795340032417959588909060233922890 >>> gamma(1+I).evalf(20) 0.49801566811835604271 - 0.15494982830181068512*I
See also
lowergammaLower incomplete gamma function.
uppergammaUpper incomplete gamma function.
polygammaPolygamma function.
loggammaLog Gamma function.
digammaDigamma function.
trigammaTrigamma function.
diofant.functions.special.beta_functions.betaEuler Beta function.
References
- class diofant.functions.special.gamma_functions.loggamma(z)[source]
The
loggammafunction implements the logarithm of the gamma function i.e, \(\log\Gamma(x)\).Examples
Several special values are known. For numerical integral arguments we have:
>>> loggamma(-2) oo >>> loggamma(0) oo >>> loggamma(1) 0 >>> loggamma(2) 0 >>> loggamma(3) log(2)
and for symbolic values:
>>> n = Symbol('n', integer=True, positive=True) >>> loggamma(n) log(gamma(n)) >>> loggamma(-n) oo
for half-integral values:
>>> loggamma(Rational(5, 2)) log(3*sqrt(pi)/4) >>> loggamma(n/2) log(2**(-n + 1)*sqrt(pi)*gamma(n)/gamma(n/2 + 1/2))
and general rational arguments:
>>> L = loggamma(Rational(16, 3)) >>> expand_func(L).doit() -5*log(3) + loggamma(1/3) + log(4) + log(7) + log(10) + log(13) >>> L = loggamma(Rational(19, 4)) >>> expand_func(L).doit() -4*log(4) + loggamma(3/4) + log(3) + log(7) + log(11) + log(15) >>> L = loggamma(Rational(23, 7)) >>> expand_func(L).doit() -3*log(7) + log(2) + loggamma(2/7) + log(9) + log(16)
The loggamma function has the following limits towards infinity:
>>> loggamma(oo) oo >>> loggamma(-oo) zoo
The loggamma function obeys the mirror symmetry if \(x \in \mathbb{C} \setminus \{-\infty, 0\}\):
>>> c = Symbol('c', complex=True, real=False) >>> conjugate(loggamma(c)) loggamma(conjugate(c))
Differentiation with respect to x is supported:
>>> diff(loggamma(x), x) polygamma(0, x)
Series expansion is also supported:
>>> loggamma(x).series(x, 0, 4) -log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + O(x**4)
We can numerically evaluate the gamma function to arbitrary precision on the whole complex plane:
>>> loggamma(5).evalf(30) 3.17805383034794561964694160130 >>> loggamma(I).evalf(20) -0.65092319930185633889 - 1.8724366472624298171*I
See also
gammaGamma function.
lowergammaLower incomplete gamma function.
uppergammaUpper incomplete gamma function.
polygammaPolygamma function.
digammaDigamma function.
trigammaTrigamma function.
diofant.functions.special.beta_functions.betaEuler Beta function.
References
- class diofant.functions.special.gamma_functions.polygamma(n, z)[source]
The function
polygamma(n, z)returnslog(gamma(z)).diff(n + 1).It is a meromorphic function on \(\mathbb{C}\) and defined as the (n+1)-th derivative of the logarithm of the gamma function:
\[\psi^{(n)} (z) := \frac{\mathrm{d}^{n+1}}{\mathrm{d} z^{n+1}} \log\Gamma(z).\]Examples
Several special values are known:
>>> polygamma(0, 1) -EulerGamma >>> polygamma(0, Rational(1, 2)) -2*log(2) - EulerGamma >>> polygamma(0, Rational(1, 3)) -3*log(3)/2 - sqrt(3)*pi/6 - EulerGamma >>> polygamma(0, Rational(1, 4)) -3*log(2) - pi/2 - EulerGamma >>> polygamma(0, 2) -EulerGamma + 1 >>> polygamma(0, 23) -EulerGamma + 19093197/5173168
>>> polygamma(0, oo) oo >>> polygamma(0, -oo) oo >>> polygamma(0, I*oo) oo >>> polygamma(0, -I*oo) oo
Differentiation with respect to x is supported:
>>> diff(polygamma(0, x), x) polygamma(1, x) >>> diff(polygamma(0, x), (x, 2)) polygamma(2, x) >>> diff(polygamma(0, x), (x, 3)) polygamma(3, x) >>> diff(polygamma(1, x), x) polygamma(2, x) >>> diff(polygamma(1, x), (x, 2)) polygamma(3, x) >>> diff(polygamma(2, x), x) polygamma(3, x) >>> diff(polygamma(2, x), (x, 2)) polygamma(4, x)
>>> diff(polygamma(n, x), x) polygamma(n + 1, x) >>> diff(polygamma(n, x), (x, 2)) polygamma(n + 2, x)
We can rewrite polygamma functions in terms of harmonic numbers:
>>> polygamma(0, x).rewrite(harmonic) harmonic(x - 1) - EulerGamma >>> polygamma(2, x).rewrite(harmonic) 2*harmonic(x - 1, 3) - 2*zeta(3) >>> ni = Symbol('n', integer=True) >>> polygamma(ni, x).rewrite(harmonic) (-1)**(n + 1)*(-harmonic(x - 1, n + 1) + zeta(n + 1))*factorial(n)
See also
gammaGamma function.
lowergammaLower incomplete gamma function.
uppergammaUpper incomplete gamma function.
loggammaLog Gamma function.
digammaDigamma function.
trigammaTrigamma function.
diofant.functions.special.beta_functions.betaEuler Beta function.
References
- diofant.functions.special.gamma_functions.digamma(x)[source]
The digamma function is the first derivative of the loggamma function i.e,
\[\psi(x) := \frac{\mathrm{d}}{\mathrm{d} z} \log\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z) }\]In this case,
digamma(z) = polygamma(0, z).See also
gammaGamma function.
lowergammaLower incomplete gamma function.
uppergammaUpper incomplete gamma function.
polygammaPolygamma function.
loggammaLog Gamma function.
trigammaTrigamma function.
diofant.functions.special.beta_functions.betaEuler Beta function.
References
- diofant.functions.special.gamma_functions.trigamma(x)[source]
The trigamma function is the second derivative of the loggamma function i.e,
\[\psi^{(1)}(z) := \frac{\mathrm{d}^{2}}{\mathrm{d} z^{2}} \log\Gamma(z).\]In this case,
trigamma(z) = polygamma(1, z).See also
gammaGamma function.
lowergammaLower incomplete gamma function.
uppergammaUpper incomplete gamma function.
polygammaPolygamma function.
loggammaLog Gamma function.
digammaDigamma function.
diofant.functions.special.beta_functions.betaEuler Beta function.
References
- class diofant.functions.special.gamma_functions.uppergamma(a, z)[source]
The upper incomplete gamma function.
It can be defined as the meromorphic continuation of
\[\Gamma(s, x) := \int_x^\infty t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \gamma(s, x).\]where \(\gamma(s, x)\) is the lower incomplete gamma function,
lowergamma. This can be shown to be the same as\[\Gamma(s, x) = \Gamma(s) - \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),\]where \({}_1F_1\) is the (confluent) hypergeometric function.
The upper incomplete gamma function is also essentially equivalent to the generalized exponential integral:
\[\operatorname{E}_{n}(x) = \int_{1}^{\infty}{\frac{e^{-xt}}{t^n} \, dt} = x^{n-1}\Gamma(1-n,x).\]Examples
>>> from diofant.abc import s >>> uppergamma(s, x) uppergamma(s, x) >>> uppergamma(3, x) E**(-x)*x**2 + 2*E**(-x)*x + 2*E**(-x) >>> uppergamma(-Rational(1, 2), x) -2*sqrt(pi)*(-erf(sqrt(x)) + 1) + 2*E**(-x)/sqrt(x) >>> uppergamma(-2, x) expint(3, x)/x**2
See also
gammaGamma function.
lowergammaLower incomplete gamma function.
polygammaPolygamma function.
loggammaLog Gamma function.
digammaDigamma function.
trigammaTrigamma function.
diofant.functions.special.beta_functions.betaEuler Beta function.
References
https://en.wikipedia.org/wiki/Incomplete_gamma_function#Upper_incomplete_Gamma_function
Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
https://en.wikipedia.org/wiki/Exponential_integral#Relation_with_other_functions
- class diofant.functions.special.gamma_functions.lowergamma(a, x)[source]
The lower incomplete gamma function.
It can be defined as the meromorphic continuation of
\[\gamma(s, x) := \int_0^x t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \Gamma(s, x).\]This can be shown to be the same as
\[\gamma(s, x) = \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),\]where \({}_1F_1\) is the (confluent) hypergeometric function.
Examples
>>> from diofant.abc import s >>> lowergamma(s, x) lowergamma(s, x) >>> lowergamma(3, x) 2 - E**(-x)*x**2 - 2*E**(-x)*x - 2*E**(-x) >>> lowergamma(-Rational(1, 2), x) -2*sqrt(pi)*erf(sqrt(x)) - 2*E**(-x)/sqrt(x)
See also
gammaGamma function.
uppergammaUpper incomplete gamma function.
polygammaPolygamma function.
loggammaLog Gamma function.
digammaDigamma function.
trigammaTrigamma function.
diofant.functions.special.beta_functions.betaEuler Beta function.
References
https://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_Gamma_function
Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
- class diofant.functions.special.beta_functions.beta(x, y)[source]
The beta integral is called the Eulerian integral of the first kind by Legendre:
\[\mathrm{B}(x,y) := \int^{1}_{0} t^{x-1} (1-t)^{y-1} \mathrm{d}t.\]Beta function or Euler’s first integral is closely associated with gamma function. The Beta function often used in probability theory and mathematical statistics. It satisfies properties like:
\[\begin{split}\mathrm{B}(a,1) = \frac{1}{a} \\ \mathrm{B}(a,b) = \mathrm{B}(b,a) \\ \mathrm{B}(a,b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\end{split}\]Therefore for integral values of a and b:
\[\mathrm{B} = \frac{(a-1)! (b-1)!}{(a+b-1)!}\]Examples
The Beta function obeys the mirror symmetry:
>>> conjugate(beta(x, y)) beta(conjugate(x), conjugate(y))
Differentiation with respect to both x and y is supported:
>>> diff(beta(x, y), x) (polygamma(0, x) - polygamma(0, x + y))*beta(x, y)
>>> diff(beta(x, y), y) (polygamma(0, y) - polygamma(0, x + y))*beta(x, y)
We can numerically evaluate the gamma function to arbitrary precision on the whole complex plane:
>>> beta(pi, pi).evalf(40) 0.02671848900111377452242355235388489324562
>>> beta(1 + I, 1 + I).evalf(20) -0.2112723729365330143 - 0.7655283165378005676*I
See also
diofant.functions.special.gamma_functions.gammaGamma function.
diofant.functions.special.gamma_functions.uppergammaUpper incomplete gamma function.
diofant.functions.special.gamma_functions.lowergammaLower incomplete gamma function.
diofant.functions.special.gamma_functions.polygammaPolygamma function.
diofant.functions.special.gamma_functions.loggammaLog Gamma function.
diofant.functions.special.gamma_functions.digammaDigamma function.
diofant.functions.special.gamma_functions.trigammaTrigamma function.
References
Error Functions and Fresnel Integrals
- class diofant.functions.special.error_functions.erf(arg)[source]
The Gauss error function. This function is defined as:
\[\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \mathrm{d}t.\]Examples
Several special values are known:
>>> erf(0) 0 >>> erf(oo) 1 >>> erf(-oo) -1 >>> erf(I*oo) oo*I >>> erf(-I*oo) -oo*I
In general one can pull out factors of -1 and I from the argument:
>>> erf(-z) -erf(z)
The error function obeys the mirror symmetry:
>>> conjugate(erf(z)) erf(conjugate(z))
Differentiation with respect to z is supported:
>>> diff(erf(z), z) 2*E**(-z**2)/sqrt(pi)
We can numerically evaluate the error function to arbitrary precision on the whole complex plane:
>>> erf(4).evalf(30) 0.999999984582742099719981147840
>>> erf(-4*I).evalf(30) -1296959.73071763923152794095062*I
See also
References
- class diofant.functions.special.error_functions.erfc(arg)[source]
Complementary Error Function. The function is defined as:
\[\mathrm{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \mathrm{d}t\]Examples
Several special values are known:
>>> erfc(0) 1 >>> erfc(oo) 0 >>> erfc(-oo) 2 >>> erfc(I*oo) -oo*I >>> erfc(-I*oo) oo*I
The error function obeys the mirror symmetry:
>>> conjugate(erfc(z)) erfc(conjugate(z))
Differentiation with respect to z is supported:
>>> diff(erfc(z), z) -2*E**(-z**2)/sqrt(pi)
It also follows
>>> erfc(-z) -erfc(z) + 2
We can numerically evaluate the complementary error function to arbitrary precision on the whole complex plane:
>>> erfc(4).evalf(30) 0.0000000154172579002800188521596734869
>>> erfc(4*I).evalf(30) 1.0 - 1296959.73071763923152794095062*I
See also
References
- class diofant.functions.special.error_functions.erfi(z)[source]
Imaginary error function. The function erfi is defined as:
\[\mathrm{erfi}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{t^2} \mathrm{d}t\]Examples
Several special values are known:
>>> erfi(0) 0 >>> erfi(oo) oo >>> erfi(-oo) -oo >>> erfi(I*oo) I >>> erfi(-I*oo) -I
In general one can pull out factors of -1 and I from the argument:
>>> erfi(-z) -erfi(z)
>>> conjugate(erfi(z)) erfi(conjugate(z))
Differentiation with respect to z is supported:
>>> diff(erfi(z), z) 2*E**(z**2)/sqrt(pi)
We can numerically evaluate the imaginary error function to arbitrary precision on the whole complex plane:
>>> erfi(2).evalf(30) 18.5648024145755525987042919132
>>> erfi(-2*I).evalf(30) -0.995322265018952734162069256367*I
See also
References
- class diofant.functions.special.error_functions.erf2(x, y)[source]
Two-argument error function. This function is defined as:
\[\mathrm{erf2}(x, y) = \frac{2}{\sqrt{\pi}} \int_x^y e^{-t^2} \mathrm{d}t\]Examples
Several special values are known:
>>> erf2(0, 0) 0 >>> erf2(x, x) 0 >>> erf2(x, oo) -erf(x) + 1 >>> erf2(x, -oo) -erf(x) - 1 >>> erf2(oo, y) erf(y) - 1 >>> erf2(-oo, y) erf(y) + 1
In general one can pull out factors of -1:
>>> erf2(-x, -y) -erf2(x, y)
The error function obeys the mirror symmetry:
>>> conjugate(erf2(x, y)) erf2(conjugate(x), conjugate(y))
Differentiation with respect to x, y is supported:
>>> diff(erf2(x, y), x) -2*E**(-x**2)/sqrt(pi) >>> diff(erf2(x, y), y) 2*E**(-y**2)/sqrt(pi)
See also
References
- class diofant.functions.special.error_functions.erfinv(z)[source]
Inverse Error Function. The erfinv function is defined as:
\[\mathrm{erf}(x) = y \quad \Rightarrow \quad \mathrm{erfinv}(y) = x\]Examples
Several special values are known:
>>> erfinv(0) 0 >>> erfinv(1) oo
Differentiation with respect to x is supported:
>>> diff(erfinv(x), x) E**(erfinv(x)**2)*sqrt(pi)/2
We can numerically evaluate the inverse error function to arbitrary precision on [-1, 1]:
>>> erfinv(0.2) 0.179143454621292
See also
References
- class diofant.functions.special.error_functions.erfcinv(z)[source]
Inverse Complementary Error Function. The erfcinv function is defined as:
\[\mathrm{erfc}(x) = y \quad \Rightarrow \quad \mathrm{erfcinv}(y) = x\]Examples
Several special values are known:
>>> erfcinv(1) 0 >>> erfcinv(0) oo
Differentiation with respect to x is supported:
>>> diff(erfcinv(x), x) -E**(erfcinv(x)**2)*sqrt(pi)/2
See also
References
- class diofant.functions.special.error_functions.erf2inv(x, y)[source]
Two-argument Inverse error function. The erf2inv function is defined as:
\[\mathrm{erf2}(x, w) = y \quad \Rightarrow \quad \mathrm{erf2inv}(x, y) = w\]Examples
Several special values are known:
>>> erf2inv(0, 0) 0 >>> erf2inv(1, 0) 1 >>> erf2inv(0, 1) oo >>> erf2inv(0, y) erfinv(y) >>> erf2inv(oo, y) erfcinv(-y)
Differentiation with respect to x and y is supported:
>>> diff(erf2inv(x, y), x) E**(-x**2 + erf2inv(x, y)**2) >>> diff(erf2inv(x, y), y) E**(erf2inv(x, y)**2)*sqrt(pi)/2
See also
References
- class diofant.functions.special.error_functions.FresnelIntegral(z)[source]
Base class for the Fresnel integrals.
- class diofant.functions.special.error_functions.fresnels(z)[source]
Fresnel integral S.
This function is defined by
\[\operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t.\]It is an entire function.
Examples
Several special values are known:
>>> fresnels(0) 0 >>> fresnels(oo) 1/2 >>> fresnels(-oo) -1/2 >>> fresnels(I*oo) -I/2 >>> fresnels(-I*oo) I/2
In general one can pull out factors of -1 and \(i\) from the argument:
>>> fresnels(-z) -fresnels(z) >>> fresnels(I*z) -I*fresnels(z)
The Fresnel S integral obeys the mirror symmetry \(\overline{S(z)} = S(\bar{z})\):
>>> conjugate(fresnels(z)) fresnels(conjugate(z))
Differentiation with respect to \(z\) is supported:
>>> diff(fresnels(z), z) sin(pi*z**2/2)
Defining the Fresnel functions via an integral
>>> integrate(sin(pi*z**2/2), z) 3*fresnels(z)*gamma(3/4)/(4*gamma(7/4)) >>> expand_func(integrate(sin(pi*z**2/2), z)) fresnels(z)
We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane:
>>> fresnels(2).evalf(30) 0.343415678363698242195300815958
>>> fresnels(-2*I).evalf(30) 0.343415678363698242195300815958*I
See also
fresnelcFresnel cosine integral.
References
The converging factors for the fresnel integrals by John W. Wrench Jr. and Vicki Alley
- class diofant.functions.special.error_functions.fresnelc(z)[source]
Fresnel integral C.
This function is defined by
\[\operatorname{C}(z) = \int_0^z \cos{\frac{\pi}{2} t^2} \mathrm{d}t.\]It is an entire function.
Examples
Several special values are known:
>>> fresnelc(0) 0 >>> fresnelc(oo) 1/2 >>> fresnelc(-oo) -1/2 >>> fresnelc(I*oo) I/2 >>> fresnelc(-I*oo) -I/2
In general one can pull out factors of -1 and \(i\) from the argument:
>>> fresnelc(-z) -fresnelc(z) >>> fresnelc(I*z) I*fresnelc(z)
The Fresnel C integral obeys the mirror symmetry \(\overline{C(z)} = C(\bar{z})\):
>>> conjugate(fresnelc(z)) fresnelc(conjugate(z))
Differentiation with respect to \(z\) is supported:
>>> diff(fresnelc(z), z) cos(pi*z**2/2)
Defining the Fresnel functions via an integral
>>> integrate(cos(pi*z**2/2), z) fresnelc(z)*gamma(1/4)/(4*gamma(5/4)) >>> expand_func(integrate(cos(pi*z**2/2), z)) fresnelc(z)
We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane:
>>> fresnelc(2).evalf(30) 0.488253406075340754500223503357
>>> fresnelc(-2*I).evalf(30) -0.488253406075340754500223503357*I
See also
fresnelsFresnel sine integral.
References
The converging factors for the fresnel integrals by John W. Wrench Jr. and Vicki Alley
Exponential, Logarithmic and Trigonometric Integrals
- class diofant.functions.special.error_functions.Ei(z)[source]
The classical exponential integral.
For use in Diofant, this function is defined as
\[\operatorname{Ei}(x) = \sum_{n=1}^\infty \frac{x^n}{n\, n!} + \log(x) + \gamma,\]where \(\gamma\) is the Euler-Mascheroni constant.
If \(x\) is a polar number, this defines an analytic function on the Riemann surface of the logarithm. Otherwise this defines an analytic function in the cut plane \(\mathbb{C} \setminus (-\infty, 0]\).
Background
The name exponential integral comes from the following statement:
\[\operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t\]If the integral is interpreted as a Cauchy principal value, this statement holds for \(x > 0\) and \(\operatorname{Ei}(x)\) as defined above.
Note that we carefully avoided defining \(\operatorname{Ei}(x)\) for negative real \(x\). This is because above integral formula does not hold for any polar lift of such \(x\), indeed all branches of \(\operatorname{Ei}(x)\) above the negative reals are imaginary.
However, the following statement holds for all \(x \in \mathbb{R}^*\):
\[\int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t = \frac{\operatorname{Ei}\left(|x|e^{i \arg(x)}\right) + \operatorname{Ei}\left(|x|e^{- i \arg(x)}\right)}{2},\]where the integral is again understood to be a principal value if \(x > 0\), and \(|x|e^{i \arg(x)}\), \(|x|e^{- i \arg(x)}\) denote two conjugate polar lifts of \(x\).
Examples
The exponential integral in Diofant is strictly undefined for negative values of the argument. For convenience, exponential integrals with negative arguments are immediately converted into an expression that agrees with the classical integral definition:
>>> Ei(-1) -I*pi + Ei(exp_polar(I*pi))
This yields a real value:
>>> Ei(-1).evalf(chop=True) -0.219383934395520
On the other hand the analytic continuation is not real:
>>> Ei(polar_lift(-1)).evalf(chop=True) -0.21938393439552 + 3.14159265358979*I
The exponential integral has a logarithmic branch point at the origin:
>>> Ei(x*exp_polar(2*I*pi)) Ei(x) + 2*I*pi
Differentiation is supported:
>>> Ei(x).diff(x) E**x/x
The exponential integral is related to many other special functions. For example:
>>> Ei(x).rewrite(expint) -expint(1, x*exp_polar(I*pi)) - I*pi >>> Ei(x).rewrite(Shi) Chi(x) + Shi(x)
See also
expintGeneralized exponential integral.
E1Special case of the generalized exponential integral.
liLogarithmic integral.
LiOffset logarithmic integral.
SiSine integral.
CiCosine integral.
ShiHyperbolic sine integral.
ChiHyperbolic cosine integral.
diofant.functions.special.gamma_functions.uppergammaUpper incomplete gamma function.
References
Abramowitz & Stegun, section 5: http://people.math.sfu.ca/~cbm/aands/page_228.htm
- class diofant.functions.special.error_functions.expint(nu, z)[source]
Generalized exponential integral.
This function is defined as
\[\operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z),\]where \(\Gamma(1 - \nu, z)\) is the upper incomplete gamma function (
uppergamma).Hence for \(z\) with positive real part we have
\[\operatorname{E}_\nu(z) = \int_1^\infty \frac{e^{-zt}}{t^\nu} \mathrm{d}t,\]which explains the name.
The representation as an incomplete gamma function provides an analytic continuation for \(\operatorname{E}_\nu(z)\). If \(\nu\) is a non-positive integer the exponential integral is thus an unbranched function of \(z\), otherwise there is a branch point at the origin. Refer to the incomplete gamma function documentation for details of the branching behavior.
Examples
>>> from diofant.abc import nu
Differentiation is supported. Differentiation with respect to z explains further the name: for integral orders, the exponential integral is an iterated integral of the exponential function.
>>> expint(nu, z).diff(z) -expint(nu - 1, z)
Differentiation with respect to nu has no classical expression:
>>> expint(nu, z).diff(nu) -z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, -nu + 1), ()), z)
At non-postive integer orders, the exponential integral reduces to the exponential function:
>>> expint(0, z) E**(-z)/z >>> expint(-1, z) E**(-z)/z + E**(-z)/z**2
At half-integers it reduces to error functions:
>>> expint(Rational(1, 2), z) -sqrt(pi)*erf(sqrt(z))/sqrt(z) + sqrt(pi)/sqrt(z)
At positive integer orders it can be rewritten in terms of exponentials and expint(1, z). Use expand_func() to do this:
>>> expand_func(expint(5, z)) z**4*expint(1, z)/24 + E**(-z)*(-z**3 + z**2 - 2*z + 6)/24
The generalized exponential integral is essentially equivalent to the incomplete gamma function:
>>> expint(nu, z).rewrite(uppergamma) z**(nu - 1)*uppergamma(-nu + 1, z)
As such it is branched at the origin:
>>> expint(4, z*exp_polar(2*pi*I)) I*pi*z**3/3 + expint(4, z) >>> expint(nu, z*exp_polar(2*pi*I)) z**(nu - 1)*(E**(2*I*pi*nu) - 1)*gamma(-nu + 1) + expint(nu, z)
See also
References
- diofant.functions.special.error_functions.E1(z)[source]
Classical case of the generalized exponential integral.
This is equivalent to
expint(1, z).
- class diofant.functions.special.error_functions.li(z)[source]
The classical logarithmic integral.
For the use in Diofant, this function is defined as
\[\operatorname{li}(x) = \int_0^x \frac{1}{\log(t)} \mathrm{d}t \,.\]Examples
Several special values are known:
>>> li(0) 0 >>> li(1) -oo >>> li(oo) oo
Differentiation with respect to z is supported:
>>> diff(li(z), z) 1/log(z)
Defining the \(li\) function via an integral:
The logarithmic integral can also be defined in terms of Ei:
>>> li(z).rewrite(Ei) Ei(log(z)) >>> diff(li(z).rewrite(Ei), z) 1/log(z)
We can numerically evaluate the logarithmic integral to arbitrary precision on the whole complex plane (except the singular points):
>>> li(2).evalf(30) 1.04516378011749278484458888919
>>> li(2*I).evalf(30) 1.0652795784357498247001125598 + 3.08346052231061726610939702133*I
We can even compute Soldner’s constant by the help of mpmath:
>>> from mpmath import findroot >>> print(findroot(li, 2)) 1.45136923488338
Further transformations include rewriting \(li\) in terms of the trigonometric integrals \(Si\), \(Ci\), \(Shi\) and \(Chi\):
>>> li(z).rewrite(Si) -log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)) >>> li(z).rewrite(Ci) -log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)) >>> li(z).rewrite(Shi) -log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z)) >>> li(z).rewrite(Chi) -log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))
See also
References
- class diofant.functions.special.error_functions.Li(z)[source]
The offset logarithmic integral.
For the use in Diofant, this function is defined as
\[\operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2)\]Examples
The following special value is known:
>>> Li(2) 0
Differentiation with respect to z is supported:
>>> diff(Li(z), z) 1/log(z)
The shifted logarithmic integral can be written in terms of \(li(z)\):
>>> Li(z).rewrite(li) li(z) - li(2)
We can numerically evaluate the logarithmic integral to arbitrary precision on the whole complex plane (except the singular points):
>>> Li(2).evalf(30) 0
>>> Li(4).evalf(30) 1.92242131492155809316615998938
See also
References
- class diofant.functions.special.error_functions.Si(z)[source]
Sine integral.
This function is defined by
\[\operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t.\]It is an entire function.
Examples
The sine integral is an antiderivative of sin(z)/z:
>>> Si(z).diff(z) sin(z)/z
It is unbranched:
>>> Si(z*exp_polar(2*I*pi)) Si(z)
Sine integral behaves much like ordinary sine under multiplication by
I:>>> Si(I*z) I*Shi(z) >>> Si(-z) -Si(z)
It can also be expressed in terms of exponential integrals, but beware that the latter is branched:
>>> Si(z).rewrite(expint) -I*(-expint(1, z*exp_polar(-I*pi/2))/2 + expint(1, z*exp_polar(I*pi/2))/2) + pi/2
See also
References
- class diofant.functions.special.error_functions.Ci(z)[source]
Cosine integral.
This function is defined for positive \(x\) by
\[\operatorname{Ci}(x) = \gamma + \log{x} + \int_0^x \frac{\cos{t} - 1}{t} \mathrm{d}t = -\int_x^\infty \frac{\cos{t}}{t} \mathrm{d}t,\]where \(\gamma\) is the Euler-Mascheroni constant.
We have
\[\operatorname{Ci}(z) = -\frac{\operatorname{E}_1\left(e^{i\pi/2} z\right) + \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2}\]which holds for all polar \(z\) and thus provides an analytic continuation to the Riemann surface of the logarithm.
The formula also holds as stated for \(z \in \mathbb{C}\) with \(\Re(z) > 0\). By lifting to the principal branch we obtain an analytic function on the cut complex plane.
Examples
The cosine integral is a primitive of \(\cos(z)/z\):
>>> Ci(z).diff(z) cos(z)/z
It has a logarithmic branch point at the origin:
>>> Ci(z*exp_polar(2*I*pi)) Ci(z) + 2*I*pi
The cosine integral behaves somewhat like ordinary \(\cos\) under multiplication by \(i\):
>>> Ci(polar_lift(I)*z) Chi(z) + I*pi/2 >>> Ci(polar_lift(-1)*z) Ci(z) + I*pi
It can also be expressed in terms of exponential integrals:
>>> Ci(z).rewrite(expint) -expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2
See also
References
- class diofant.functions.special.error_functions.Shi(z)[source]
Sinh integral.
This function is defined by
\[\operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t.\]It is an entire function.
Examples
The Sinh integral is a primitive of \(\sinh(z)/z\):
>>> Shi(z).diff(z) sinh(z)/z
It is unbranched:
>>> Shi(z*exp_polar(2*I*pi)) Shi(z)
The \(\sinh\) integral behaves much like ordinary \(\sinh\) under multiplication by \(i\):
>>> Shi(I*z) I*Si(z) >>> Shi(-z) -Shi(z)
It can also be expressed in terms of exponential integrals, but beware that the latter is branched:
>>> Shi(z).rewrite(expint) expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2
See also
References
- class diofant.functions.special.error_functions.Chi(z)[source]
Cosh integral.
This function is defined for positive \(x\) by
\[\operatorname{Chi}(x) = \gamma + \log{x} + \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t,\]where \(\gamma\) is the Euler-Mascheroni constant.
We have
\[\operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right) - i\frac{\pi}{2},\]which holds for all polar \(z\) and thus provides an analytic continuation to the Riemann surface of the logarithm. By lifting to the principal branch we obtain an analytic function on the cut complex plane.
Examples
The \(\cosh\) integral is a primitive of \(\cosh(z)/z\):
>>> Chi(z).diff(z) cosh(z)/z
It has a logarithmic branch point at the origin:
>>> Chi(z*exp_polar(2*I*pi)) Chi(z) + 2*I*pi
The \(\cosh\) integral behaves somewhat like ordinary \(\cosh\) under multiplication by \(i\):
>>> Chi(polar_lift(I)*z) Ci(z) + I*pi/2 >>> Chi(polar_lift(-1)*z) Chi(z) + I*pi
It can also be expressed in terms of exponential integrals:
>>> Chi(z).rewrite(expint) -expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2
See also
References
Bessel Type Functions
- class diofant.functions.special.bessel.BesselBase(nu, z)[source]
Abstract base class for bessel-type functions.
This class is meant to reduce code duplication. All Bessel type functions can 1) be differentiated, and the derivatives expressed in terms of similar functions and 2) be rewritten in terms of other bessel-type functions.
Here “bessel-type functions” are assumed to have one complex parameter.
To use this base class, define class attributes
_aand_bsuch that2*F_n' = -_a*F_{n+1} + b*F_{n-1}.- property argument
The argument of the bessel-type function.
- property order
The order of the bessel-type function.
- class diofant.functions.special.bessel.besselj(nu, z)[source]
Bessel function of the first kind.
The Bessel \(J\) function of order \(\nu\) is defined to be the function satisfying Bessel’s differential equation
\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu^2) w = 0,\]with Laurent expansion
\[J_\nu(z) = z^\nu \left(\frac{1}{\Gamma(\nu + 1) 2^\nu} + O(z^2) \right),\]if \(\nu\) is not a negative integer. If \(\nu=-n \in \mathbb{Z}_{<0}\) is a negative integer, then the definition is
\[J_{-n}(z) = (-1)^n J_n(z).\]Examples
Create a Bessel function object:
>>> b = besselj(n, z)
Differentiate it:
>>> b.diff(z) besselj(n - 1, z)/2 - besselj(n + 1, z)/2
Rewrite in terms of spherical Bessel functions:
>>> b.rewrite(jn) sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi)
Access the parameter and argument:
>>> b.order n >>> b.argument z
References
Abramowitz, Milton; Stegun, Irene A., eds. (1965), “Chapter 9”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1
- class diofant.functions.special.bessel.bessely(nu, z)[source]
Bessel function of the second kind.
The Bessel \(Y\) function of order \(\nu\) is defined as
\[Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu) - J_{-\mu}(z)}{\sin(\pi \mu)},\]where \(J_\mu(z)\) is the Bessel function of the first kind.
It is a solution to Bessel’s equation, and linearly independent from \(J_\nu\).
Examples
>>> b = bessely(n, z) >>> b.diff(z) bessely(n - 1, z)/2 - bessely(n + 1, z)/2 >>> b.rewrite(yn) sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi)
References
- class diofant.functions.special.bessel.besseli(nu, z)[source]
Modified Bessel function of the first kind.
The Bessel I function is a solution to the modified Bessel equation
\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 + \nu^2)^2 w = 0.\]It can be defined as
\[I_\nu(z) = i^{-\nu} J_\nu(iz),\]where \(J_\nu(z)\) is the Bessel function of the first kind.
Examples
>>> besseli(n, z).diff(z) besseli(n - 1, z)/2 + besseli(n + 1, z)/2
References
- class diofant.functions.special.bessel.besselk(nu, z)[source]
Modified Bessel function of the second kind.
The Bessel K function of order \(\nu\) is defined as
\[K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2} \frac{I_{-\mu}(z) -I_\mu(z)}{\sin(\pi \mu)},\]where \(I_\mu(z)\) is the modified Bessel function of the first kind.
It is a solution of the modified Bessel equation, and linearly independent from \(Y_\nu\).
Examples
>>> besselk(n, z).diff(z) -besselk(n - 1, z)/2 - besselk(n + 1, z)/2
References
- class diofant.functions.special.bessel.hankel1(nu, z)[source]
Hankel function of the first kind.
This function is defined as
\[H_\nu^{(1)} = J_\nu(z) + iY_\nu(z),\]where \(J_\nu(z)\) is the Bessel function of the first kind, and \(Y_\nu(z)\) is the Bessel function of the second kind.
It is a solution to Bessel’s equation.
Examples
>>> hankel1(n, z).diff(z) hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2
References
- class diofant.functions.special.bessel.hankel2(nu, z)[source]
Hankel function of the second kind.
This function is defined as
\[H_\nu^{(2)} = J_\nu(z) - iY_\nu(z),\]where \(J_\nu(z)\) is the Bessel function of the first kind, and \(Y_\nu(z)\) is the Bessel function of the second kind.
It is a solution to Bessel’s equation, and linearly independent from \(H_\nu^{(1)}\).
Examples
>>> hankel2(n, z).diff(z) hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2
References
- class diofant.functions.special.bessel.jn(nu, z)[source]
Spherical Bessel function of the first kind.
This function is a solution to the spherical Bessel equation
\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + 2z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu(\nu + 1)) w = 0.\]It can be defined as
\[j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu + \frac{1}{2}}(z),\]where \(J_\nu(z)\) is the Bessel function of the first kind.
Examples
>>> print(jn(0, z).expand(func=True)) sin(z)/z >>> jn(1, z).expand(func=True) == sin(z)/z**2 - cos(z)/z True >>> expand_func(jn(3, z)) (-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z)
The spherical Bessel functions of integral order are calculated using the formula:
\[j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z},\]where the coefficients \(f_n(z)\) are available as
diofant.polys.orthopolys.spherical_bessel_fn().
- class diofant.functions.special.bessel.yn(nu, z)[source]
Spherical Bessel function of the second kind.
This function is another solution to the spherical Bessel equation, and linearly independent from \(j_n\). It can be defined as
\[j_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z),\]where \(Y_\nu(z)\) is the Bessel function of the second kind.
Examples
>>> expand_func(yn(0, z)) -cos(z)/z >>> expand_func(yn(1, z)) == -cos(z)/z**2-sin(z)/z True
For integral orders \(n\), \(y_n\) is calculated using the formula:
\[y_n(z) = (-1)^{n+1} j_{-n-1}(z)\]
Airy Functions
- class diofant.functions.special.bessel.AiryBase(*args)[source]
Abstract base class for Airy functions.
This class is meant to reduce code duplication.
- class diofant.functions.special.bessel.airyai(arg)[source]
The Airy function \(\operatorname{Ai}\) of the first kind.
The Airy function \(\operatorname{Ai}(z)\) is defined to be the function satisfying Airy’s differential equation
\[\frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.\]Equivalently, for real \(z\)
\[\operatorname{Ai}(z) := \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.\]Examples
Create an Airy function object:
>>> airyai(z) airyai(z)
Several special values are known:
>>> airyai(0) root(3, 3)/(3*gamma(2/3)) >>> airyai(oo) 0 >>> airyai(-oo) 0
The Airy function obeys the mirror symmetry:
>>> conjugate(airyai(z)) airyai(conjugate(z))
Differentiation with respect to z is supported:
>>> diff(airyai(z), z) airyaiprime(z) >>> diff(airyai(z), (z, 2)) z*airyai(z)
Series expansion is also supported:
>>> airyai(z).series(z, 0, 3) root(3, 6)**5*gamma(1/3)/(6*pi) - root(3, 6)*z*gamma(2/3)/(2*pi) + O(z**3)
We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:
>>> airyai(-2).evalf(50) 0.22740742820168557599192443603787379946077222541710
Rewrite Ai(z) in terms of hypergeometric functions:
>>> airyai(z).rewrite(hyper) -root(3, 3)**2*z*hyper((), (4/3,), z**3/9)/(3*gamma(1/3)) + root(3, 3)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))
See also
airybiAiry function of the second kind.
airyaiprimeDerivative of the Airy function of the first kind.
airybiprimeDerivative of the Airy function of the second kind.
References
- class diofant.functions.special.bessel.airybi(arg)[source]
The Airy function \(\operatorname{Bi}\) of the second kind.
The Airy function \(\operatorname{Bi}(z)\) is defined to be the function satisfying Airy’s differential equation
\[\frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.\]Equivalently, for real \(z\)
\[\operatorname{Bi}(z) := \frac{1}{\pi} \int_0^\infty \exp\left(-\frac{t^3}{3} + z t\right) + \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.\]Examples
Create an Airy function object:
>>> airybi(z) airybi(z)
Several special values are known:
>>> airybi(0) root(3, 6)**5/(3*gamma(2/3)) >>> airybi(oo) oo >>> airybi(-oo) 0
The Airy function obeys the mirror symmetry:
>>> conjugate(airybi(z)) airybi(conjugate(z))
Differentiation with respect to z is supported:
>>> diff(airybi(z), z) airybiprime(z) >>> diff(airybi(z), (z, 2)) z*airybi(z)
Series expansion is also supported:
>>> airybi(z).series(z, 0, 3) root(3, 3)*gamma(1/3)/(2*pi) + root(3, 3)**2*z*gamma(2/3)/(2*pi) + O(z**3)
We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:
>>> airybi(-2).evalf(50) -0.41230258795639848808323405461146104203453483447240
Rewrite Bi(z) in terms of hypergeometric functions:
>>> airybi(z).rewrite(hyper) root(3, 6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + root(3, 6)**5*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))
See also
airyaiAiry function of the first kind.
airyaiprimeDerivative of the Airy function of the first kind.
airybiprimeDerivative of the Airy function of the second kind.
References
- class diofant.functions.special.bessel.airyaiprime(arg)[source]
The derivative \(\operatorname{Ai}^\prime\) of the Airy function of the first kind.
The Airy function \(\operatorname{Ai}^\prime(z)\) is defined to be the function
\[\operatorname{Ai}^\prime(z) := \frac{\mathrm{d} \operatorname{Ai}(z)}{\mathrm{d} z}.\]Examples
Create an Airy function object:
>>> airyaiprime(z) airyaiprime(z)
Several special values are known:
>>> airyaiprime(0) -root(3, 3)**2/(3*gamma(1/3)) >>> airyaiprime(oo) 0
The Airy function obeys the mirror symmetry:
>>> conjugate(airyaiprime(z)) airyaiprime(conjugate(z))
Differentiation with respect to z is supported:
>>> diff(airyaiprime(z), z) z*airyai(z) >>> diff(airyaiprime(z), (z, 2)) z*airyaiprime(z) + airyai(z)
Series expansion is also supported:
>>> airyaiprime(z).series(z, 0, 3) -root(3, 3)**2/(3*gamma(1/3)) + root(3, 3)*z**2/(6*gamma(2/3)) + O(z**3)
We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:
>>> airyaiprime(-2).evalf(50) 0.61825902074169104140626429133247528291577794512415
Rewrite Ai’(z) in terms of hypergeometric functions:
>>> airyaiprime(z).rewrite(hyper) root(3, 3)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) - root(3, 3)**2*hyper((), (1/3,), z**3/9)/(3*gamma(1/3))
See also
airyaiAiry function of the first kind.
airybiAiry function of the second kind.
airybiprimeDerivative of the Airy function of the second kind.
References
- class diofant.functions.special.bessel.airybiprime(arg)[source]
The derivative \(\operatorname{Bi}^\prime\) of the Airy function of the first kind.
The Airy function \(\operatorname{Bi}^\prime(z)\) is defined to be the function
\[\operatorname{Bi}^\prime(z) := \frac{\mathrm{d} \operatorname{Bi}(z)}{\mathrm{d} z}.\]Examples
Create an Airy function object:
>>> airybiprime(z) airybiprime(z)
Several special values are known:
>>> airybiprime(0) root(3, 6)/gamma(1/3) >>> airybiprime(oo) oo >>> airybiprime(-oo) 0
The Airy function obeys the mirror symmetry:
>>> conjugate(airybiprime(z)) airybiprime(conjugate(z))
Differentiation with respect to z is supported:
>>> diff(airybiprime(z), z) z*airybi(z) >>> diff(airybiprime(z), (z, 2)) z*airybiprime(z) + airybi(z)
Series expansion is also supported:
>>> airybiprime(z).series(z, 0, 3) root(3, 6)/gamma(1/3) + root(3, 6)**5*z**2/(6*gamma(2/3)) + O(z**3)
We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:
>>> airybiprime(-2).evalf(50) 0.27879516692116952268509756941098324140300059345163
Rewrite Bi’(z) in terms of hypergeometric functions:
>>> airybiprime(z).rewrite(hyper) root(3, 6)**5*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + root(3, 6)*hyper((), (1/3,), z**3/9)/gamma(1/3)
See also
airyaiAiry function of the first kind.
airybiAiry function of the second kind.
airyaiprimeDerivative of the Airy function of the first kind.
References
B-Splines
- diofant.functions.special.bsplines.bspline_basis(d, knots, n, x, close=True)[source]
The \(n\)-th B-spline at \(x\) of degree \(d\) with knots.
B-Splines are piecewise polynomials of degree \(d\). They are defined on a set of knots, which is a sequence of integers or floats.
The 0th degree splines have a value of one on a single interval:
>>> d = 0 >>> knots = range(5) >>> bspline_basis(d, knots, 0, x) Piecewise((1, (x >= 0) & (x <= 1)), (0, true))
For a given
(d, knots)there arelen(knots)-d-1B-splines defined, that are indexed byn(starting at 0).Here is an example of a cubic B-spline:
>>> bspline_basis(3, range(5), 0, x) Piecewise((x**3/6, (x >= 0) & (x < 1)), (-x**3/2 + 2*x**2 - 2*x + 2/3, (x >= 1) & (x < 2)), (x**3/2 - 4*x**2 + 10*x - 22/3, (x >= 2) & (x < 3)), (-x**3/6 + 2*x**2 - 8*x + 32/3, (x >= 3) & (x <= 4)), (0, true))
By repeating knot points, you can introduce discontinuities in the B-splines and their derivatives:
>>> d = 1 >>> knots = [0, 0, 2, 3, 4] >>> bspline_basis(d, knots, 0, x) Piecewise((-x/2 + 1, (x >= 0) & (x <= 2)), (0, true))
It is quite time consuming to construct and evaluate B-splines. If you need to evaluate a B-splines many times, it is best to lambdify them first:
>>> d = 3 >>> knots = range(10) >>> b0 = bspline_basis(d, knots, 0, x) >>> f = lambdify(x, b0) >>> y = f(0.5)
References
- diofant.functions.special.bsplines.bspline_basis_set(d, knots, x)[source]
Return the
len(knots)-d-1B-splines atxof degreedwithknots.This function returns a list of Piecewise polynomials that are the
len(knots)-d-1B-splines of degreedfor the given knots. This function callsbspline_basis(d, knots, n, x)for different values ofn.Examples
>>> d = 2 >>> knots = range(5) >>> splines = bspline_basis_set(d, knots, x) >>> splines [Piecewise((x**2/2, (x >= 0) & (x < 1)), (-x**2 + 3*x - 3/2, (x >= 1) & (x < 2)), (x**2/2 - 3*x + 9/2, (x >= 2) & (x <= 3)), (0, true)), Piecewise((x**2/2 - x + 1/2, (x >= 1) & (x < 2)), (-x**2 + 5*x - 11/2, (x >= 2) & (x < 3)), (x**2/2 - 4*x + 8, (x >= 3) & (x <= 4)), (0, true))]
Riemann Zeta and Related Functions
- class diofant.functions.special.zeta_functions.zeta(z, a=None)[source]
Hurwitz zeta function (or Riemann zeta function).
For \(\operatorname{Re}(a) > 0\) and \(\operatorname{Re}(s) > 1\), this function is defined as
\[\zeta(s, a) = \sum_{n=0}^\infty \frac{1}{(n + a)^s},\]where the standard choice of argument for \(n + a\) is used. For fixed \(a\) with \(\operatorname{Re}(a) > 0\) the Hurwitz zeta function admits a meromorphic continuation to all of \(\mathbb{C}\), it is an unbranched function with a simple pole at \(s = 1\).
Analytic continuation to other \(a\) is possible under some circumstances, but this is not typically done.
The Hurwitz zeta function is a special case of the Lerch transcendent:
\[\zeta(s, a) = \Phi(1, s, a).\]This formula defines an analytic continuation for all possible values of \(s\) and \(a\) (also \(\operatorname{Re}(a) < 0\)), see the documentation of
lerchphifor a description of the branching behavior.If no value is passed for \(a\), by this function assumes a default value of \(a = 1\), yielding the Riemann zeta function.
See also
References
Examples
For \(a = 1\) the Hurwitz zeta function reduces to the famous Riemann zeta function:
\[\zeta(s, 1) = \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.\]>>> from diofant.abc import s >>> zeta(s, 1) zeta(s) >>> zeta(s) zeta(s)
The Riemann zeta function can also be expressed using the Dirichlet eta function:
>>> zeta(s).rewrite(dirichlet_eta) dirichlet_eta(s)/(-2**(-s + 1) + 1)
The Riemann zeta function at positive even integer and negative odd integer values is related to the Bernoulli numbers:
>>> zeta(2) pi**2/6 >>> zeta(4) pi**4/90 >>> zeta(-1) -1/12
The specific formulae are:
\[\zeta(2n) = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!}\]\[\zeta(-n) = -\frac{B_{n+1}}{n+1}\]At negative even integers the Riemann zeta function is zero:
>>> zeta(-4) 0
No closed-form expressions are known at positive odd integers, but numerical evaluation is possible:
>>> zeta(3).evalf() 1.20205690315959
The derivative of \(\zeta(s, a)\) with respect to \(a\) is easily computed:
>>> zeta(s, a).diff(a) -s*zeta(s + 1, a)
However the derivative with respect to \(s\) has no useful closed form expression:
>>> zeta(s, a).diff(s) Derivative(zeta(s, a), s)
The Hurwitz zeta function can be expressed in terms of the Lerch transcendent,
diofant.functions.special.zeta_functions.lerchphi:>>> zeta(s, a).rewrite(lerchphi) lerchphi(1, s, a)
- class diofant.functions.special.zeta_functions.dirichlet_eta(s)[source]
Dirichlet eta function.
For \(\operatorname{Re}(s) > 0\), this function is defined as
\[\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}.\]It admits a unique analytic continuation to all of \(\mathbb{C}\). It is an entire, unbranched function.
See also
References
Examples
The Dirichlet eta function is closely related to the Riemann zeta function:
>>> from diofant.abc import s >>> dirichlet_eta(s).rewrite(zeta) (-2**(-s + 1) + 1)*zeta(s)
- class diofant.functions.special.zeta_functions.polylog(s, z)[source]
Polylogarithm function.
For \(|z| < 1\) and \(s \in \mathbb{C}\), the polylogarithm is defined by
\[\operatorname{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s},\]where the standard branch of the argument is used for \(n\). It admits an analytic continuation which is branched at \(z=1\) (notably not on the sheet of initial definition), \(z=0\) and \(z=\infty\).
The name polylogarithm comes from the fact that for \(s=1\), the polylogarithm is related to the ordinary logarithm (see examples), and that
\[\operatorname{Li}_{s+1}(z) = \int_0^z \frac{\operatorname{Li}_s(t)}{t} \mathrm{d}t.\]The polylogarithm is a special case of the Lerch transcendent:
\[\operatorname{Li}_{s}(z) = z \Phi(z, s, 1)\]Examples
For \(z \in \{0, 1, -1\}\), the polylogarithm is automatically expressed using other functions:
>>> from diofant.abc import s >>> polylog(s, 0) 0 >>> polylog(s, 1) zeta(s) >>> polylog(s, -1) -dirichlet_eta(s)
If \(s\) is a negative integer, \(0\) or \(1\), the polylogarithm can be expressed using elementary functions. This can be done using expand_func():
>>> expand_func(polylog(1, z)) -log(-z + 1) >>> expand_func(polylog(0, z)) z/(-z + 1)
The derivative with respect to \(z\) can be computed in closed form:
>>> polylog(s, z).diff(z) polylog(s - 1, z)/z
The polylogarithm can be expressed in terms of the lerch transcendent:
>>> polylog(s, z).rewrite(lerchphi) z*lerchphi(z, s, 1)
References
- class diofant.functions.special.zeta_functions.lerchphi(*args)[source]
Lerch transcendent (Lerch phi function).
For \(\operatorname{Re}(a) > 0\), \(|z| < 1\) and \(s \in \mathbb{C}\), the Lerch transcendent is defined as
\[\Phi(z, s, a) = \sum_{n=0}^\infty \frac{z^n}{(n + a)^s},\]where the standard branch of the argument is used for \(n + a\), and by analytic continuation for other values of the parameters.
A commonly used related function is the Lerch zeta function, defined by
\[L(q, s, a) = \Phi(e^{2\pi i q}, s, a).\]Analytic Continuation and Branching Behavior
It can be shown that
\[\Phi(z, s, a) = z\Phi(z, s, a+1) + a^{-s}.\]This provides the analytic continuation to \(\operatorname{Re}(a) \le 0\).
Assume now \(\operatorname{Re}(a) > 0\). The integral representation
\[\Phi_0(z, s, a) = \int_0^\infty \frac{t^{s-1} e^{-at}}{1 - ze^{-t}} \frac{\mathrm{d}t}{\Gamma(s)}\]provides an analytic continuation to \(\mathbb{C} - [1, \infty)\). Finally, for \(x \in (1, \infty)\) we find
\[\lim_{\epsilon \to 0^+} \Phi_0(x + i\epsilon, s, a) -\lim_{\epsilon \to 0^+} \Phi_0(x - i\epsilon, s, a) = \frac{2\pi i \log^{s-1}{x}}{x^a \Gamma(s)},\]using the standard branch for both \(\log{x}\) and \(\log{\log{x}}\) (a branch of \(\log{\log{x}}\) is needed to evaluate \(\log{x}^{s-1}\)). This concludes the analytic continuation. The Lerch transcendent is thus branched at \(z \in \{0, 1, \infty\}\) and \(a \in \mathbb{Z}_{\le 0}\). For fixed \(z, a\) outside these branch points, it is an entire function of \(s\).
References
Bateman, H.; Erdélyi, A. (1953), Higher Transcendental Functions, Vol. I, New York: McGraw-Hill. Section 1.11.
Examples
The Lerch transcendent is a fairly general function, for this reason it does not automatically evaluate to simpler functions. Use expand_func() to achieve this.
If \(z=1\), the Lerch transcendent reduces to the Hurwitz zeta function:
>>> from diofant.abc import s >>> expand_func(lerchphi(1, s, a)) zeta(s, a)
More generally, if \(z\) is a root of unity, the Lerch transcendent reduces to a sum of Hurwitz zeta functions:
>>> expand_func(lerchphi(-1, s, a)) 2**(-s)*zeta(s, a/2) - 2**(-s)*zeta(s, a/2 + 1/2)
If \(a=1\), the Lerch transcendent reduces to the polylogarithm:
>>> expand_func(lerchphi(z, s, 1)) polylog(s, z)/z
More generally, if \(a\) is rational, the Lerch transcendent reduces to a sum of polylogarithms:
>>> expand_func(lerchphi(z, s, Rational(1, 2))) 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) - polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z)) >>> expand_func(lerchphi(z, s, Rational(3, 2))) -2**s/z + 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) - polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))/z
The derivatives with respect to \(z\) and \(a\) can be computed in closed form:
>>> lerchphi(z, s, a).diff(z) (-a*lerchphi(z, s, a) + lerchphi(z, s - 1, a))/z >>> lerchphi(z, s, a).diff(a) -s*lerchphi(z, s + 1, a)
Hypergeometric Functions
- class diofant.functions.special.hyper.hyper(ap, bq, z)[source]
The (generalized) hypergeometric function is defined by a series where the ratios of successive terms are a rational function of the summation index. When convergent, it is continued analytically to the largest possible domain.
The hypergeometric function depends on two vectors of parameters, called the numerator parameters \(a_p\), and the denominator parameters \(b_q\). It also has an argument \(z\). The series definition is
\[\begin{split}{}_pF_q\left(\begin{matrix} a_1, \ldots, a_p \\ b_1, \ldots, b_q \end{matrix} \middle| z \right) = \sum_{n=0}^\infty \frac{(a_1)_n \ldots (a_p)_n}{(b_1)_n \ldots (b_q)_n} \frac{z^n}{n!},\end{split}\]where \((a)_n = (a)(a+1)\ldots(a+n-1)\) denotes the rising factorial.
If one of the \(b_q\) is a non-positive integer then the series is undefined unless one of the \(a_p\) is a larger (i.e. smaller in magnitude) non-positive integer. If none of the \(b_q\) is a non-positive integer and one of the \(a_p\) is a non-positive integer, then the series reduces to a polynomial. To simplify the following discussion, we assume that none of the \(a_p\) or \(b_q\) is a non-positive integer. For more details, see the references.
The series converges for all \(z\) if \(p \le q\), and thus defines an entire single-valued function in this case. If \(p = q+1\) the series converges for \(|z| < 1\), and can be continued analytically into a half-plane. If \(p > q+1\) the series is divergent for all \(z\).
Note: The hypergeometric function constructor currently does not check if the parameters actually yield a well-defined function.
Examples
The parameters \(a_p\) and \(b_q\) can be passed as arbitrary iterables, for example:
>>> hyper((1, 2, 3), [3, 4], x) hyper((1, 2, 3), (3, 4), x)
There is also pretty printing:
>>> pprint(hyper((1, 2, 3), [3, 4], x)) ┌─ ⎛1, 2, 3 │ ⎞ ├─ ⎜ │ x⎟ 3╵ 2 ⎝ 3, 4 │ ⎠
The parameters must always be iterables, even if they are vectors of length one or zero:
>>> hyper([1], [], x) hyper((1,), (), x)
But of course they may be variables (but if they depend on x then you should not expect much implemented functionality):
>>> hyper([n, a], [n**2], x) hyper((n, a), (n**2,), x)
The hypergeometric function generalizes many named special functions. The function hyperexpand() tries to express a hypergeometric function using named special functions. For example:
>>> hyperexpand(hyper([], [], x)) E**x
You can also use expand_func:
>>> expand_func(x*hyper([1, 1], [2], -x)) log(x + 1)
More examples:
>>> hyperexpand(hyper([], [Rational(1, 2)], -x**2/4)) cos(x) >>> hyperexpand(x*hyper([Rational(1, 2), Rational(1, 2)], [Rational(3, 2)], x**2)) asin(x)
We can also sometimes hyperexpand parametric functions:
>>> hyperexpand(hyper([-a], [], x)) (-x + 1)**a
See also
diofant.simplify.hyperexpand,diofant.functions.special.gamma_functions.gamma,diofant.functions.special.hyper.meijergReferences
Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1
https://en.wikipedia.org/wiki/Generalized_hypergeometric_function
- property ap
Numerator parameters of the hypergeometric function.
- property argument
Argument of the hypergeometric function.
- property bq
Denominator parameters of the hypergeometric function.
- property convergence_statement
Return a condition on z under which the series converges.
- property eta
A quantity related to the convergence of the series.
- property radius_of_convergence
Compute the radius of convergence of the defining series.
Note that even if this is not oo, the function may still be evaluated outside of the radius of convergence by analytic continuation. But if this is zero, then the function is not actually defined anywhere else.
>>> hyper((1, 2), [3], z).radius_of_convergence 1 >>> hyper((1, 2, 3), [4], z).radius_of_convergence 0 >>> hyper((1, 2), (3, 4), z).radius_of_convergence oo
- class diofant.functions.special.hyper.meijerg(*args)[source]
The Meijer G-function is defined by a Mellin-Barnes type integral that resembles an inverse Mellin transform. It generalizes the hypergeometric functions.
The Meijer G-function depends on four sets of parameters. There are “numerator parameters” \(a_1, \ldots, a_n\) and \(a_{n+1}, \ldots, a_p\), and there are “denominator parameters” \(b_1, \ldots, b_m\) and \(b_{m+1}, \ldots, b_q\). Confusingly, it is traditionally denoted as follows (note the position of \(m\), \(n\), \(p\), \(q\), and how they relate to the lengths of the four parameter vectors):
\[\begin{split}G_{p,q}^{m,n} \left(\begin{matrix}a_1, \ldots, a_n & a_{n+1}, \ldots, a_p \\ b_1, \ldots, b_m & b_{m+1}, \ldots, b_q \end{matrix} \middle| z \right).\end{split}\]However, in diofant the four parameter vectors are always available separately (see examples), so that there is no need to keep track of the decorating sub- and super-scripts on the G symbol.
The G function is defined as the following integral:
\[\frac{1}{2 \pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j - s) \prod_{j=1}^n \Gamma(1 - a_j + s)}{\prod_{j=m+1}^q \Gamma(1- b_j +s) \prod_{j=n+1}^p \Gamma(a_j - s)} z^s \mathrm{d}s,\]where \(\Gamma(z)\) is the gamma function. There are three possible contours which we will not describe in detail here (see the references). If the integral converges along more than one of them the definitions agree. The contours all separate the poles of \(\Gamma(1-a_j+s)\) from the poles of \(\Gamma(b_k-s)\), so in particular the G function is undefined if \(a_j - b_k \in \mathbb{Z}_{>0}\) for some \(j \le n\) and \(k \le m\).
The conditions under which one of the contours yields a convergent integral are complicated and we do not state them here, see the references.
Note: Currently the Meijer G-function constructor does not check any convergence conditions.
Examples
You can pass the parameters either as four separate vectors:
>>> pprint(meijerg([1, 2], [a, 4], [5], [], x)) ╭─╮1, 2 ⎛1, 2 a, 4 │ ⎞ │╶┐ ⎜ │ x⎟ ╰─╯4, 1 ⎝ 5 │ ⎠
or as two nested vectors:
>>> pprint(meijerg(([1, 2], [3, 4]), ([5], []), x)) ╭─╮1, 2 ⎛1, 2 3, 4 │ ⎞ │╶┐ ⎜ │ x⎟ ╰─╯4, 1 ⎝ 5 │ ⎠
As with the hypergeometric function, the parameters may be passed as arbitrary iterables. Vectors of length zero and one also have to be passed as iterables. The parameters need not be constants, but if they depend on the argument then not much implemented functionality should be expected.
All the subvectors of parameters are available:
>>> g = meijerg([1], [2], [3], [4], x) >>> pprint(g) ╭─╮1, 1 ⎛1 2 │ ⎞ │╶┐ ⎜ │ x⎟ ╰─╯2, 2 ⎝3 4 │ ⎠ >>> g.an (1,) >>> g.ap (1, 2) >>> g.aother (2,) >>> g.bm (3,) >>> g.bq (3, 4) >>> g.bother (4,)
The Meijer G-function generalizes the hypergeometric functions. In some cases it can be expressed in terms of hypergeometric functions, using Slater’s theorem. For example:
>>> hyperexpand(meijerg([a], [], [c], [b], x), allow_hyper=True) x**c*gamma(-a + c + 1)*hyper((-a + c + 1,), (-b + c + 1,), -x)/gamma(-b + c + 1)
Thus the Meijer G-function also subsumes many named functions as special cases. You can use expand_func or hyperexpand to (try to) rewrite a Meijer G-function in terms of named special functions. For example:
>>> expand_func(meijerg([[], []], [[0], []], -x)) E**x >>> hyperexpand(meijerg([[], []], [[Rational(1, 2)], [0]], (x/2)**2)) sin(x)/sqrt(pi)
References
Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1
- property an
First set of numerator parameters.
- property aother
Second set of numerator parameters.
- property ap
Combined numerator parameters.
- property argument
Argument of the Meijer G-function.
- property bm
First set of denominator parameters.
- property bother
Second set of denominator parameters.
- property bq
Combined denominator parameters.
- property delta
A quantity related to the convergence region of the integral, c.f. references.
- get_period()[source]
Return a number P such that G(x*exp(I*P)) == G(x).
>>> meijerg([1], [], [], [], z).get_period() 2*pi >>> meijerg([pi], [], [], [], z).get_period() oo >>> meijerg([1, 2], [], [], [], z).get_period() oo >>> meijerg([1, 1], [2], [1, Rational(1, 2), Rational(1, 3)], [1], z).get_period() 12*pi
- property nu
A quantity related to the convergence region of the integral, c.f. references.
Elliptic integrals
- class diofant.functions.special.elliptic_integrals.elliptic_k(m)[source]
The complete elliptic integral of the first kind, defined by
\[K(m) = F\left(\tfrac{\pi}{2}\middle| m\right)\]where \(F\left(z\middle| m\right)\) is the Legendre incomplete elliptic integral of the first kind.
The function \(K(m)\) is a single-valued function on the complex plane with branch cut along the interval \((1, \infty)\).
Examples
>>> elliptic_k(0) pi/2 >>> elliptic_k(1.0 + I) 1.50923695405127 + 0.625146415202697*I >>> elliptic_k(m).series(m, n=3) pi/2 + pi*m/8 + 9*pi*m**2/128 + O(m**3)
References
See also
- class diofant.functions.special.elliptic_integrals.elliptic_f(z, m)[source]
The Legendre incomplete elliptic integral of the first kind, defined by
\[F\left(z\middle| m\right) = \int_0^z \frac{dt}{\sqrt{1 - m \sin^2 t}}\]This function reduces to a complete elliptic integral of the first kind, \(K(m)\), when \(z = \pi/2\).
Examples
>>> elliptic_f(z, m).series(z) z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6) >>> elliptic_f(3.0 + I/2, 1.0 + I) 2.909449841483 + 1.74720545502474*I
References
See also
- class diofant.functions.special.elliptic_integrals.elliptic_e(z, m=None)[source]
Called with two arguments \(z\) and \(m\), evaluates the incomplete elliptic integral of the second kind, defined by
\[E\left(z\middle| m\right) = \int_0^z \sqrt{1 - m \sin^2 t} dt\]Called with a single argument \(m\), evaluates the Legendre complete elliptic integral of the second kind
\[E(m) = E\left(\tfrac{\pi}{2}\middle| m\right)\]The function \(E(m)\) is a single-valued function on the complex plane with branch cut along the interval \((1, \infty)\).
Examples
>>> elliptic_e(z, m).series(z) z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6) >>> elliptic_e(m).series(m, n=4) pi/2 - pi*m/8 - 3*pi*m**2/128 - 5*pi*m**3/512 + O(m**4) >>> elliptic_e(1 + I, 2 - I/2).evalf() 1.55203744279187 + 0.290764986058437*I >>> elliptic_e(0) pi/2 >>> elliptic_e(2.0 - I) 0.991052601328069 + 0.81879421395609*I
References
- class diofant.functions.special.elliptic_integrals.elliptic_pi(n, m, z=None)[source]
Called with three arguments \(n\), \(z\) and \(m\), evaluates the Legendre incomplete elliptic integral of the third kind, defined by
\[\Pi\left(n; z\middle| m\right) = \int_0^z \frac{dt} {\left(1 - n \sin^2 t\right) \sqrt{1 - m \sin^2 t}}\]Called with two arguments \(n\) and \(m\), evaluates the complete elliptic integral of the third kind:
\[\Pi\left(n\middle| m\right) = \Pi\left(n; \tfrac{\pi}{2}\middle| m\right)\]Examples
>>> elliptic_pi(n, z, m).series(z, n=4) z + z**3*(m/6 + n/3) + O(z**4) >>> elliptic_pi(0.5 + I, 1.0 - I, 1.2) 2.50232379629182 - 0.760939574180767*I >>> elliptic_pi(0, 0) pi/2 >>> elliptic_pi(1.0 - I/3, 2.0 + I) 3.29136443417283 + 0.32555634906645*I
References
Orthogonal Polynomials
This module mainly implements special orthogonal polynomials.
See also functions.combinatorial.numbers which contains some combinatorial polynomials.
Jacobi Polynomials
- class diofant.functions.special.polynomials.jacobi(n, a, b, x)[source]
Jacobi polynomial \(P_n^{\left(\alpha, \beta\right)}(x)\)
jacobi(n, alpha, beta, x) gives the nth Jacobi polynomial in x, \(P_n^{\left(\alpha, \beta\right)}(x)\).
The Jacobi polynomials are orthogonal on \([-1, 1]\) with respect to the weight \(\left(1-x\right)^\alpha \left(1+x\right)^\beta\).
Examples
>>> jacobi(0, a, b, x) 1 >>> jacobi(1, a, b, x) a/2 - b/2 + x*(a/2 + b/2 + 1)
>>> jacobi(n, a, b, x) jacobi(n, a, b, x)
>>> jacobi(n, a, a, x) RisingFactorial(a + 1, n)*gegenbauer(n, a + 1/2, x)/RisingFactorial(2*a + 1, n)
>>> jacobi(n, 0, 0, x) legendre(n, x)
>>> jacobi(n, Rational(1, 2), Rational(1, 2), x) RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1)
>>> jacobi(n, -Rational(1, 2), -Rational(1, 2), x) RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n)
>>> jacobi(n, a, b, -x) (-1)**n*jacobi(n, b, a, x)
>>> jacobi(n, a, b, 0) 2**(-n)*gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(factorial(n)*gamma(a + 1)) >>> jacobi(n, a, b, 1) RisingFactorial(a + 1, n)/factorial(n)
>>> conjugate(jacobi(n, a, b, x)) jacobi(n, conjugate(a), conjugate(b), conjugate(x))
>>> diff(jacobi(n, a, b, x), x) (a + b + n + 1)*jacobi(n - 1, a + 1, b + 1, x)/2
See also
gegenbauer,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,diofant.polys.orthopolys.jacobi_poly,diofant.polys.orthopolys.gegenbauer_poly,diofant.polys.orthopolys.chebyshevt_poly,diofant.polys.orthopolys.chebyshevu_poly,diofant.polys.orthopolys.hermite_poly,diofant.polys.orthopolys.legendre_poly,diofant.polys.orthopolys.laguerre_polyReferences
- diofant.functions.special.polynomials.jacobi_normalized(n, a, b, x)[source]
Jacobi polynomial \(P_n^{\left(\alpha, \beta\right)}(x)\)
jacobi_normalized(n, alpha, beta, x) gives the nth Jacobi polynomial in x, \(P_n^{\left(\alpha, \beta\right)}(x)\).
The Jacobi polynomials are orthogonal on \([-1, 1]\) with respect to the weight \(\left(1-x\right)^\alpha \left(1+x\right)^\beta\).
This functions returns the polynomials normilzed:
\[\int_{-1}^{1} P_m^{\left(\alpha, \beta\right)}(x) P_n^{\left(\alpha, \beta\right)}(x) (1-x)^{\alpha} (1+x)^{\beta} \mathrm{d}x = \delta_{m,n}\]Examples
>>> jacobi_normalized(n, a, b, x) jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1)))
See also
gegenbauer,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,diofant.polys.orthopolys.jacobi_poly,diofant.polys.orthopolys.gegenbauer_poly,diofant.polys.orthopolys.chebyshevt_poly,diofant.polys.orthopolys.chebyshevu_poly,diofant.polys.orthopolys.hermite_poly,diofant.polys.orthopolys.legendre_poly,diofant.polys.orthopolys.laguerre_polyReferences
Gegenbauer Polynomials
- class diofant.functions.special.polynomials.gegenbauer(n, a, x)[source]
Gegenbauer polynomial \(C_n^{\left(\alpha\right)}(x)\)
gegenbauer(n, alpha, x) gives the nth Gegenbauer polynomial in x, \(C_n^{\left(\alpha\right)}(x)\).
The Gegenbauer polynomials are orthogonal on \([-1, 1]\) with respect to the weight \(\left(1-x^2\right)^{\alpha-\frac{1}{2}}\).
Examples
>>> gegenbauer(0, a, x) 1 >>> gegenbauer(1, a, x) 2*a*x >>> gegenbauer(2, a, x) -a + x**2*(2*a**2 + 2*a) >>> gegenbauer(3, a, x) x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)
>>> gegenbauer(n, a, x) gegenbauer(n, a, x) >>> gegenbauer(n, a, -x) (-1)**n*gegenbauer(n, a, x)
>>> gegenbauer(n, a, 0) 2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(-n/2 + 1/2)*gamma(n + 1)) >>> gegenbauer(n, a, 1) gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))
>>> conjugate(gegenbauer(n, a, x)) gegenbauer(n, conjugate(a), conjugate(x))
>>> diff(gegenbauer(n, a, x), x) 2*a*gegenbauer(n - 1, a + 1, x)
See also
jacobi,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,diofant.polys.orthopolys.jacobi_poly,diofant.polys.orthopolys.gegenbauer_poly,diofant.polys.orthopolys.chebyshevt_poly,diofant.polys.orthopolys.chebyshevu_poly,diofant.polys.orthopolys.hermite_poly,diofant.polys.orthopolys.legendre_poly,diofant.polys.orthopolys.laguerre_polyReferences
Chebyshev Polynomials
- class diofant.functions.special.polynomials.chebyshevt(n, x)[source]
Chebyshev polynomial of the first kind, \(T_n(x)\)
chebyshevt(n, x) gives the nth Chebyshev polynomial (of the first kind) in x, \(T_n(x)\).
The Chebyshev polynomials of the first kind are orthogonal on \([-1, 1]\) with respect to the weight \(\frac{1}{\sqrt{1-x^2}}\).
Examples
>>> chebyshevt(0, x) 1 >>> chebyshevt(1, x) x >>> chebyshevt(2, x) 2*x**2 - 1
>>> chebyshevt(n, x) chebyshevt(n, x) >>> chebyshevt(n, -x) (-1)**n*chebyshevt(n, x) >>> chebyshevt(-n, x) chebyshevt(n, x)
>>> chebyshevt(n, 0) cos(pi*n/2) >>> chebyshevt(n, -1) (-1)**n
>>> diff(chebyshevt(n, x), x) n*chebyshevu(n - 1, x)
See also
jacobi,gegenbauer,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,diofant.polys.orthopolys.jacobi_poly,diofant.polys.orthopolys.gegenbauer_poly,diofant.polys.orthopolys.chebyshevt_poly,diofant.polys.orthopolys.chebyshevu_poly,diofant.polys.orthopolys.hermite_poly,diofant.polys.orthopolys.legendre_poly,diofant.polys.orthopolys.laguerre_polyReferences
- class diofant.functions.special.polynomials.chebyshevu(n, x)[source]
Chebyshev polynomial of the second kind, \(U_n(x)\)
chebyshevu(n, x) gives the nth Chebyshev polynomial of the second kind in x, \(U_n(x)\).
The Chebyshev polynomials of the second kind are orthogonal on \([-1, 1]\) with respect to the weight \(\sqrt{1-x^2}\).
Examples
>>> chebyshevu(0, x) 1 >>> chebyshevu(1, x) 2*x >>> chebyshevu(2, x) 4*x**2 - 1
>>> chebyshevu(n, x) chebyshevu(n, x) >>> chebyshevu(n, -x) (-1)**n*chebyshevu(n, x) >>> chebyshevu(-n, x) -chebyshevu(n - 2, x)
>>> chebyshevu(n, 0) cos(pi*n/2) >>> chebyshevu(n, 1) n + 1
>>> diff(chebyshevu(n, x), x) (-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)
See also
jacobi,gegenbauer,chebyshevt,chebyshevt_root,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,diofant.polys.orthopolys.jacobi_poly,diofant.polys.orthopolys.gegenbauer_poly,diofant.polys.orthopolys.chebyshevt_poly,diofant.polys.orthopolys.chebyshevu_poly,diofant.polys.orthopolys.hermite_poly,diofant.polys.orthopolys.legendre_poly,diofant.polys.orthopolys.laguerre_polyReferences
- class diofant.functions.special.polynomials.chebyshevt_root(n, k)[source]
chebyshev_root(n, k) returns the kth root (indexed from zero) of the nth Chebyshev polynomial of the first kind; that is, if 0 <= k < n, chebyshevt(n, chebyshevt_root(n, k)) == 0.
Examples
>>> chebyshevt_root(3, 2) -sqrt(3)/2 >>> chebyshevt(3, chebyshevt_root(3, 2)) 0
See also
jacobi,gegenbauer,chebyshevt,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,diofant.polys.orthopolys.jacobi_poly,diofant.polys.orthopolys.gegenbauer_poly,diofant.polys.orthopolys.chebyshevt_poly,diofant.polys.orthopolys.chebyshevu_poly,diofant.polys.orthopolys.hermite_poly,diofant.polys.orthopolys.legendre_poly,diofant.polys.orthopolys.laguerre_poly
- class diofant.functions.special.polynomials.chebyshevu_root(n, k)[source]
chebyshevu_root(n, k) returns the kth root (indexed from zero) of the nth Chebyshev polynomial of the second kind; that is, if 0 <= k < n, chebyshevu(n, chebyshevu_root(n, k)) == 0.
Examples
>>> chebyshevu_root(3, 2) -sqrt(2)/2 >>> chebyshevu(3, chebyshevu_root(3, 2)) 0
See also
chebyshevt,chebyshevt_root,chebyshevu,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,diofant.polys.orthopolys.jacobi_poly,diofant.polys.orthopolys.gegenbauer_poly,diofant.polys.orthopolys.chebyshevt_poly,diofant.polys.orthopolys.chebyshevu_poly,diofant.polys.orthopolys.hermite_poly,diofant.polys.orthopolys.legendre_poly,diofant.polys.orthopolys.laguerre_poly
Legendre Polynomials
- class diofant.functions.special.polynomials.legendre(n, x)[source]
legendre(n, x) gives the nth Legendre polynomial of x, \(P_n(x)\)
The Legendre polynomials are orthogonal on [-1, 1] with respect to the constant weight 1. They satisfy \(P_n(1) = 1\) for all n; further, \(P_n\) is odd for odd n and even for even n.
Examples
>>> legendre(0, x) 1 >>> legendre(1, x) x >>> legendre(2, x) 3*x**2/2 - 1/2 >>> legendre(n, x) legendre(n, x) >>> diff(legendre(n, x), x) n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)
See also
jacobi,gegenbauer,chebyshevt,chebyshevt_root,chebyshevu,chebyshevu_root,assoc_legendre,hermite,laguerre,assoc_laguerre,diofant.polys.orthopolys.jacobi_poly,diofant.polys.orthopolys.gegenbauer_poly,diofant.polys.orthopolys.chebyshevt_poly,diofant.polys.orthopolys.chebyshevu_poly,diofant.polys.orthopolys.hermite_poly,diofant.polys.orthopolys.legendre_poly,diofant.polys.orthopolys.laguerre_polyReferences
- class diofant.functions.special.polynomials.assoc_legendre(n, m, x)[source]
assoc_legendre(n,m, x) gives \(P_n^m(x)\), where n and m are the degree and order or an expression which is related to the nth order Legendre polynomial, \(P_n(x)\) in the following manner:
\[P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}} \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m}\]Associated Legendre polynomial are orthogonal on [-1, 1] with:
weight = 1 for the same m, and different n.
weight = 1/(1-x**2) for the same n, and different m.
Examples
>>> assoc_legendre(0, 0, x) 1 >>> assoc_legendre(1, 0, x) x >>> assoc_legendre(1, 1, x) -sqrt(-x**2 + 1) >>> assoc_legendre(n, m, x) assoc_legendre(n, m, x)
See also
jacobi,gegenbauer,chebyshevt,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,hermite,laguerre,assoc_laguerre,diofant.polys.orthopolys.jacobi_poly,diofant.polys.orthopolys.gegenbauer_poly,diofant.polys.orthopolys.chebyshevt_poly,diofant.polys.orthopolys.chebyshevu_poly,diofant.polys.orthopolys.hermite_poly,diofant.polys.orthopolys.legendre_poly,diofant.polys.orthopolys.laguerre_polyReferences
Hermite Polynomials
- class diofant.functions.special.polynomials.hermite(n, x)[source]
hermite(n, x) gives the nth Hermite polynomial in x, \(H_n(x)\)
The Hermite polynomials are orthogonal on \((-\infty, \infty)\) with respect to the weight \(\exp\left(-x^2\right)\).
Examples
>>> hermite(0, x) 1 >>> hermite(1, x) 2*x >>> hermite(2, x) 4*x**2 - 2 >>> hermite(n, x) hermite(n, x) >>> diff(hermite(n, x), x) 2*n*hermite(n - 1, x) >>> hermite(n, -x) (-1)**n*hermite(n, x)
See also
jacobi,gegenbauer,chebyshevt,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,laguerre,assoc_laguerre,diofant.polys.orthopolys.jacobi_poly,diofant.polys.orthopolys.gegenbauer_poly,diofant.polys.orthopolys.chebyshevt_poly,diofant.polys.orthopolys.chebyshevu_poly,diofant.polys.orthopolys.hermite_poly,diofant.polys.orthopolys.legendre_poly,diofant.polys.orthopolys.laguerre_polyReferences
Laguerre Polynomials
- class diofant.functions.special.polynomials.laguerre(n, x)[source]
Returns the nth Laguerre polynomial in x, \(L_n(x)\).
- Parameters:
n (int) – Degree of Laguerre polynomial. Must be
n >= 0.
Examples
>>> laguerre(0, x) 1 >>> laguerre(1, x) -x + 1 >>> laguerre(2, x) x**2/2 - 2*x + 1 >>> laguerre(3, x) -x**3/6 + 3*x**2/2 - 3*x + 1
>>> laguerre(n, x) laguerre(n, x)
>>> diff(laguerre(n, x), x) -assoc_laguerre(n - 1, 1, x)
See also
jacobi,gegenbauer,chebyshevt,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,assoc_laguerre,diofant.polys.orthopolys.jacobi_poly,diofant.polys.orthopolys.gegenbauer_poly,diofant.polys.orthopolys.chebyshevt_poly,diofant.polys.orthopolys.chebyshevu_poly,diofant.polys.orthopolys.hermite_poly,diofant.polys.orthopolys.legendre_poly,diofant.polys.orthopolys.laguerre_polyReferences
- class diofant.functions.special.polynomials.assoc_laguerre(n, alpha, x)[source]
Returns the nth generalized Laguerre polynomial in x, \(L_n(x)\).
- Parameters:
n (int) – Degree of Laguerre polynomial. Must be
n >= 0.alpha (Expr) – Arbitrary expression. For
alpha=0regular Laguerre polynomials will be generated.
Examples
>>> assoc_laguerre(0, a, x) 1 >>> assoc_laguerre(1, a, x) a - x + 1 >>> assoc_laguerre(2, a, x) a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1 >>> assoc_laguerre(3, a, x) a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) + x*(-a**2/2 - 5*a/2 - 3) + 1
>>> assoc_laguerre(n, a, 0) binomial(a + n, a)
>>> assoc_laguerre(n, a, x) assoc_laguerre(n, a, x)
>>> assoc_laguerre(n, 0, x) laguerre(n, x)
>>> diff(assoc_laguerre(n, a, x), x) -assoc_laguerre(n - 1, a + 1, x)
>>> diff(assoc_laguerre(n, a, x), a) Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1))
See also
jacobi,gegenbauer,chebyshevt,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,diofant.polys.orthopolys.jacobi_poly,diofant.polys.orthopolys.gegenbauer_poly,diofant.polys.orthopolys.chebyshevt_poly,diofant.polys.orthopolys.chebyshevu_poly,diofant.polys.orthopolys.hermite_poly,diofant.polys.orthopolys.legendre_poly,diofant.polys.orthopolys.laguerre_polyReferences
Spherical Harmonics
- class diofant.functions.special.spherical_harmonics.Ynm(n, m, theta, phi)[source]
Spherical harmonics defined as
\[Y_n^m(\theta, \varphi) := \sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}} \exp(i m \varphi) \mathrm{P}_n^m\left(\cos(\theta)\right)\]Ynm() gives the spherical harmonic function of order \(n\) and \(m\) in \(\theta\) and \(\varphi\), \(Y_n^m(\theta, \varphi)\). The four parameters are as follows: \(n \geq 0\) an integer and \(m\) an integer such that \(-n \leq m \leq n\) holds. The two angles are real-valued with \(\theta \in [0, \pi]\) and \(\varphi \in [0, 2\pi]\).
Examples
>>> theta = Symbol('theta') >>> phi = Symbol('phi')
>>> Ynm(n, m, theta, phi) Ynm(n, m, theta, phi)
Several symmetries are known, for the order
>>> theta = Symbol('theta') >>> phi = Symbol('phi')
>>> Ynm(n, -m, theta, phi) (-1)**m*E**(-2*I*m*phi)*Ynm(n, m, theta, phi)
as well as for the angles
>>> theta = Symbol('theta') >>> phi = Symbol('phi')
>>> Ynm(n, m, -theta, phi) Ynm(n, m, theta, phi)
>>> Ynm(n, m, theta, -phi) E**(-2*I*m*phi)*Ynm(n, m, theta, phi)
For specific integers n and m we can evaluate the harmonics to more useful expressions
>>> simplify(Ynm(0, 0, theta, phi).expand(func=True)) 1/(2*sqrt(pi))
>>> simplify(Ynm(1, -1, theta, phi).expand(func=True)) sqrt(6)*E**(-I*phi)*sin(theta)/(4*sqrt(pi))
>>> simplify(Ynm(1, 0, theta, phi).expand(func=True)) sqrt(3)*cos(theta)/(2*sqrt(pi))
>>> simplify(Ynm(1, 1, theta, phi).expand(func=True)) -sqrt(6)*E**(I*phi)*sin(theta)/(4*sqrt(pi))
>>> simplify(Ynm(2, -2, theta, phi).expand(func=True)) sqrt(30)*E**(-2*I*phi)*sin(theta)**2/(8*sqrt(pi))
>>> simplify(Ynm(2, -1, theta, phi).expand(func=True)) sqrt(30)*E**(-I*phi)*sin(2*theta)/(8*sqrt(pi))
>>> simplify(Ynm(2, 0, theta, phi).expand(func=True)) sqrt(5)*(3*cos(theta)**2 - 1)/(4*sqrt(pi))
>>> simplify(Ynm(2, 1, theta, phi).expand(func=True)) -sqrt(30)*E**(I*phi)*sin(2*theta)/(8*sqrt(pi))
>>> simplify(Ynm(2, 2, theta, phi).expand(func=True)) sqrt(30)*E**(2*I*phi)*sin(theta)**2/(8*sqrt(pi))
We can differentiate the functions with respect to both angles
>>> theta = Symbol('theta') >>> phi = Symbol('phi')
>>> diff(Ynm(n, m, theta, phi), theta) m*cot(theta)*Ynm(n, m, theta, phi) + E**(-I*phi)*sqrt((-m + n)*(m + n + 1))*Ynm(n, m + 1, theta, phi)
>>> diff(Ynm(n, m, theta, phi), phi) I*m*Ynm(n, m, theta, phi)
Further we can compute the complex conjugation
>>> theta = Symbol('theta') >>> phi = Symbol('phi') >>> m = Symbol('m')
>>> conjugate(Ynm(n, m, theta, phi)) (-1)**(2*m)*E**(-2*I*m*phi)*Ynm(n, m, theta, phi)
To get back the well known expressions in spherical coordinates we use full expansion
>>> theta = Symbol('theta') >>> phi = Symbol('phi')
>>> expand_func(Ynm(n, m, theta, phi)) E**(I*m*phi)*sqrt((2*n + 1)*factorial(-m + n)/factorial(m + n))*assoc_legendre(n, m, cos(theta))/(2*sqrt(pi))
See also
diofant.functions.special.spherical_harmonics.Ynm_c,diofant.functions.special.spherical_harmonics.ZnmReferences
- diofant.functions.special.spherical_harmonics.Ynm_c(n, m, theta, phi)[source]
Conjugate spherical harmonics defined as
\[\overline{Y_n^m(\theta, \varphi)} := (-1)^m Y_n^{-m}(\theta, \varphi)\]See also
diofant.functions.special.spherical_harmonics.Ynm,diofant.functions.special.spherical_harmonics.ZnmReferences
- class diofant.functions.special.spherical_harmonics.Znm(n, m, theta, phi)[source]
Real spherical harmonics defined as
\[\begin{split}Z_n^m(\theta, \varphi) := \begin{cases} \frac{Y_n^m(\theta, \varphi) + \overline{Y_n^m(\theta, \varphi)}}{\sqrt{2}} &\quad m > 0 \\ Y_n^m(\theta, \varphi) &\quad m = 0 \\ \frac{Y_n^m(\theta, \varphi) - \overline{Y_n^m(\theta, \varphi)}}{i \sqrt{2}} &\quad m < 0 \\ \end{cases}\end{split}\]which gives in simplified form
\[\begin{split}Z_n^m(\theta, \varphi) = \begin{cases} \frac{Y_n^m(\theta, \varphi) + (-1)^m Y_n^{-m}(\theta, \varphi)}{\sqrt{2}} &\quad m > 0 \\ Y_n^m(\theta, \varphi) &\quad m = 0 \\ \frac{Y_n^m(\theta, \varphi) - (-1)^m Y_n^{-m}(\theta, \varphi)}{i \sqrt{2}} &\quad m < 0 \\ \end{cases}\end{split}\]See also
diofant.functions.special.spherical_harmonics.Ynm,diofant.functions.special.spherical_harmonics.Ynm_cReferences
Tensor Functions
- diofant.functions.special.tensor_functions.Eijk(*args, **kwargs)[source]
Represent the Levi-Civita symbol.
This is just compatibility wrapper to
LeviCivita().
- diofant.functions.special.tensor_functions.eval_levicivita(*args)[source]
Evaluate Levi-Civita symbol.
- class diofant.functions.special.tensor_functions.LeviCivita(*args)[source]
Represent the Levi-Civita symbol.
For even permutations of indices it returns 1, for odd permutations -1, and for everything else (a repeated index) it returns 0.
Thus it represents an alternating pseudotensor.
Examples
>>> from diofant.abc import i, j >>> LeviCivita(1, 2, 3) 1 >>> LeviCivita(1, 3, 2) -1 >>> LeviCivita(1, 2, 2) 0 >>> LeviCivita(i, j, k) LeviCivita(i, j, k) >>> LeviCivita(i, j, i) 0
- class diofant.functions.special.tensor_functions.KroneckerDelta(i, j)[source]
The discrete, or Kronecker, delta function.
A function that takes in two integers \(i\) and \(j\). It returns \(0\) if \(i\) and \(j\) are not equal or it returns \(1\) if \(i\) and \(j\) are equal.
- Parameters:
i (Number, Symbol) – The first index of the delta function.
j (Number, Symbol) – The second index of the delta function.
Examples
A simple example with integer indices:
>>> KroneckerDelta(1, 2) 0 >>> KroneckerDelta(3, 3) 1
Symbolic indices:
>>> from diofant.abc import i, j >>> KroneckerDelta(i, j) KroneckerDelta(i, j) >>> KroneckerDelta(i, i) 1 >>> KroneckerDelta(i, i + 1) 0 >>> KroneckerDelta(i, i + 1 + k) KroneckerDelta(i, i + k + 1)
See also
diofant.functions.special.tensor_functions.KroneckerDelta.eval,diofant.functions.special.delta_functions.DiracDeltaReferences
- classmethod eval(i, j)[source]
Evaluates the discrete delta function.
Examples
>>> from diofant.abc import i, j
>>> KroneckerDelta(i, j) KroneckerDelta(i, j) >>> KroneckerDelta(i, i) 1 >>> KroneckerDelta(i, i + 1) 0 >>> KroneckerDelta(i, i + 1 + k) KroneckerDelta(i, i + k + 1)
- property indices_contain_equal_information
Returns True if indices are either both above or below fermi.
Examples
>>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, q).indices_contain_equal_information True >>> KroneckerDelta(p, q+1).indices_contain_equal_information True >>> KroneckerDelta(i, p).indices_contain_equal_information False
- property is_above_fermi
True if Delta can be non-zero above fermi
Examples
>>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, a).is_above_fermi True >>> KroneckerDelta(p, i).is_above_fermi False >>> KroneckerDelta(p, q).is_above_fermi True
- property is_below_fermi
True if Delta can be non-zero below fermi
Examples
>>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, a).is_below_fermi False >>> KroneckerDelta(p, i).is_below_fermi True >>> KroneckerDelta(p, q).is_below_fermi True
- property is_only_above_fermi
True if Delta is restricted to above fermi
Examples
>>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, a).is_only_above_fermi True >>> KroneckerDelta(p, q).is_only_above_fermi False >>> KroneckerDelta(p, i).is_only_above_fermi False
- property is_only_below_fermi
True if Delta is restricted to below fermi
Examples
>>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, i).is_only_below_fermi True >>> KroneckerDelta(p, q).is_only_below_fermi False >>> KroneckerDelta(p, a).is_only_below_fermi False
- property killable_index
Returns the index which is preferred to substitute in the final expression.
The index to substitute is the index with less information regarding fermi level. If indices contain same information, ‘a’ is preferred before ‘b’.
Examples
>>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> j = Symbol('j', below_fermi=True) >>> p = Symbol('p') >>> KroneckerDelta(p, i).killable_index p >>> KroneckerDelta(p, a).killable_index p >>> KroneckerDelta(i, j).killable_index j
- property preferred_index
Returns the index which is preferred to keep in the final expression.
The preferred index is the index with more information regarding fermi level. If indices contain same information, ‘a’ is preferred before ‘b’.
Examples
>>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> j = Symbol('j', below_fermi=True) >>> p = Symbol('p') >>> KroneckerDelta(p, i).preferred_index i >>> KroneckerDelta(p, a).preferred_index a >>> KroneckerDelta(i, j).preferred_index i