Elementary¶
This module implements elementary functions such as trigonometric, hyperbolic
as well as functions like Abs
, Max
, sqrt
etc.
diofant.functions.elementary.complexes¶
re¶
-
class
diofant.functions.elementary.complexes.
re
[source]¶ Returns real part of expression.
This function performs only elementary analysis and so it will fail to decompose properly more complicated expressions. If completely simplified result is needed then use Basic.as_real_imag() or perform complex expansion on instance of this function.
Examples
>>> re(2*E) 2*E >>> re(2*I + 17) 17 >>> re(2*I) 0 >>> re(im(x) + x*I + 2) 2
im¶
-
class
diofant.functions.elementary.complexes.
im
[source]¶ Returns imaginary part of expression.
This function performs only elementary analysis and so it will fail to decompose properly more complicated expressions. If completely simplified result is needed then use Basic.as_real_imag() or perform complex expansion on instance of this function.
Examples
>>> im(2*E) 0 >>> re(2*I + 17) 17 >>> im(x*I) re(x) >>> im(re(x) + y) im(y)
sign¶
-
class
diofant.functions.elementary.complexes.
sign
[source]¶ Returns the complex sign of an expression.
If the expression is real the sign will be:
- 1 if expression is positive
- 0 if expression is equal to zero
- -1 if expression is negative
If the expression is imaginary the sign will be:
- I if im(expression) is positive
- -I if im(expression) is negative
Otherwise an unevaluated expression will be returned. When evaluated, the result (in general) will be
cos(arg(expr)) + I*sin(arg(expr))
.Examples
>>> sign(-1) -1 >>> sign(0) 0 >>> sign(-3*I) -I >>> sign(1 + I) sign(1 + I) >>> _.evalf() 0.707106781186548 + 0.707106781186548*I
Abs¶
-
class
diofant.functions.elementary.complexes.
Abs
[source]¶ Return the absolute value of the argument.
This is an extension of the built-in function abs() to accept symbolic values. If you pass a Diofant expression to the built-in abs(), it will pass it automatically to Abs().
Examples
>>> Abs(-1) 1 >>> x = Symbol('x', real=True) >>> Abs(-x) Abs(x) >>> Abs(x**2) x**2 >>> abs(-x) # The Python built-in Abs(x)
Note that the Python built-in will return either an Expr or int depending on the argument:
>>> type(abs(-1)) <... 'int'> >>> type(abs(S.NegativeOne)) <class 'diofant.core.numbers.One'>
Abs will always return a diofant object.
adjoint¶
arg¶
conjugate¶
-
class
diofant.functions.elementary.complexes.
conjugate
[source]¶ Returns the complex conjugate of an argument.
In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part.
Thus, the conjugate of the complex number \(a + i b\) (where a and b are real numbers) is \(a - i b\)
Examples
>>> conjugate(2) 2 >>> conjugate(I) -I
References
polar_lift¶
-
class
diofant.functions.elementary.complexes.
polar_lift
[source]¶ Lift argument to the Riemann surface of the logarithm, using the standard branch.
>>> p = Symbol('p', polar=True) >>> polar_lift(4) 4*exp_polar(0) >>> polar_lift(-4) 4*exp_polar(I*pi) >>> polar_lift(-I) exp_polar(-I*pi/2) >>> polar_lift(I + 2) polar_lift(2 + I)
>>> polar_lift(4*x) 4*polar_lift(x) >>> polar_lift(4*p) 4*p
periodic_argument¶
-
class
diofant.functions.elementary.complexes.
periodic_argument
[source]¶ Represent the argument on a quotient of the Riemann surface of the logarithm. That is, given a period P, always return a value in (-P/2, P/2], by using exp(P*I) == 1.
>>> unbranched_argument(exp(5*I*pi)) pi >>> unbranched_argument(exp_polar(5*I*pi)) 5*pi >>> periodic_argument(exp_polar(5*I*pi), 2*pi) pi >>> periodic_argument(exp_polar(5*I*pi), 3*pi) -pi >>> periodic_argument(exp_polar(5*I*pi), pi) 0
See also
diofant.functions.elementary.exponential.exp_polar
diofant.functions.elementary.complexes.polar_lift
- Lift argument to the Riemann surface of the logarithm
principal_branch¶
-
class
diofant.functions.elementary.complexes.
principal_branch
[source]¶ Represent a polar number reduced to its principal branch on a quotient of the Riemann surface of the logarithm.
This is a function of two arguments. The first argument is a polar number \(z\), and the second one a positive real number of infinity, \(p\). The result is “z mod exp_polar(I*p)”.
>>> principal_branch(z, oo) z >>> principal_branch(exp_polar(2*pi*I)*3, 2*pi) 3*exp_polar(0) >>> principal_branch(exp_polar(2*pi*I)*3*z, 2*pi) 3*principal_branch(z, 2*pi)
See also
diofant.functions.elementary.exponential.exp_polar
diofant.functions.elementary.complexes.polar_lift
- Lift argument to the Riemann surface of the logarithm
diofant.functions.elementary.trigonometric¶
Trigonometric Functions¶
sin¶
-
class
diofant.functions.elementary.trigonometric.
sin
[source]¶ The sine function.
Returns the sine of x (measured in radians).
Notes
This function will evaluate automatically in the case x/pi is some rational number. For example, if x is a multiple of pi, pi/2, pi/3, pi/4 and pi/6.
Examples
>>> sin(x**2).diff(x) 2*x*cos(x**2) >>> sin(pi) 0 >>> sin(pi/2) 1 >>> sin(pi/6) 1/2 >>> sin(pi/12) -sqrt(2)/4 + sqrt(6)/4
See also
diofant.functions.elementary.trigonometric.csc
,diofant.functions.elementary.trigonometric.cos
,diofant.functions.elementary.trigonometric.sec
,diofant.functions.elementary.trigonometric.tan
,diofant.functions.elementary.trigonometric.cot
,diofant.functions.elementary.trigonometric.asin
,diofant.functions.elementary.trigonometric.acsc
,diofant.functions.elementary.trigonometric.acos
,diofant.functions.elementary.trigonometric.asec
,diofant.functions.elementary.trigonometric.atan
,diofant.functions.elementary.trigonometric.acot
,diofant.functions.elementary.trigonometric.atan2
References
cos¶
-
class
diofant.functions.elementary.trigonometric.
cos
[source]¶ The cosine function.
Returns the cosine of x (measured in radians).
Notes
See
sin()
for notes about automatic evaluation.Examples
>>> cos(x**2).diff(x) -2*x*sin(x**2) >>> cos(pi) -1 >>> cos(pi/2) 0 >>> cos(2*pi/3) -1/2 >>> cos(pi/12) sqrt(2)/4 + sqrt(6)/4
See also
diofant.functions.elementary.trigonometric.sin
,diofant.functions.elementary.trigonometric.csc
,diofant.functions.elementary.trigonometric.sec
,diofant.functions.elementary.trigonometric.tan
,diofant.functions.elementary.trigonometric.cot
,diofant.functions.elementary.trigonometric.asin
,diofant.functions.elementary.trigonometric.acsc
,diofant.functions.elementary.trigonometric.acos
,diofant.functions.elementary.trigonometric.asec
,diofant.functions.elementary.trigonometric.atan
,diofant.functions.elementary.trigonometric.acot
,diofant.functions.elementary.trigonometric.atan2
References
tan¶
-
class
diofant.functions.elementary.trigonometric.
tan
[source]¶ The tangent function.
Returns the tangent of x (measured in radians).
Notes
See
sin()
for notes about automatic evaluation.Examples
>>> tan(x**2).diff(x) 2*x*(tan(x**2)**2 + 1) >>> tan(pi/8).expand() -1 + sqrt(2)
See also
diofant.functions.elementary.trigonometric.sin
,diofant.functions.elementary.trigonometric.csc
,diofant.functions.elementary.trigonometric.cos
,diofant.functions.elementary.trigonometric.sec
,diofant.functions.elementary.trigonometric.cot
,diofant.functions.elementary.trigonometric.asin
,diofant.functions.elementary.trigonometric.acsc
,diofant.functions.elementary.trigonometric.acos
,diofant.functions.elementary.trigonometric.asec
,diofant.functions.elementary.trigonometric.atan
,diofant.functions.elementary.trigonometric.acot
,diofant.functions.elementary.trigonometric.atan2
References
cot¶
-
class
diofant.functions.elementary.trigonometric.
cot
[source]¶ The cotangent function.
Returns the cotangent of x (measured in radians).
Notes
See
sin()
for notes about automatic evaluation.Examples
>>> cot(x**2).diff(x) 2*x*(-cot(x**2)**2 - 1)
See also
diofant.functions.elementary.trigonometric.sin
,diofant.functions.elementary.trigonometric.csc
,diofant.functions.elementary.trigonometric.cos
,diofant.functions.elementary.trigonometric.sec
,diofant.functions.elementary.trigonometric.tan
,diofant.functions.elementary.trigonometric.asin
,diofant.functions.elementary.trigonometric.acsc
,diofant.functions.elementary.trigonometric.acos
,diofant.functions.elementary.trigonometric.asec
,diofant.functions.elementary.trigonometric.atan
,diofant.functions.elementary.trigonometric.acot
,diofant.functions.elementary.trigonometric.atan2
References
sec¶
-
class
diofant.functions.elementary.trigonometric.
sec
[source]¶ The secant function.
Returns the secant of x (measured in radians).
Notes
See
sin()
for notes about automatic evaluation.Examples
>>> sec(x**2).diff(x) 2*x*tan(x**2)*sec(x**2)
See also
diofant.functions.elementary.trigonometric.sin
,diofant.functions.elementary.trigonometric.csc
,diofant.functions.elementary.trigonometric.cos
,diofant.functions.elementary.trigonometric.tan
,diofant.functions.elementary.trigonometric.cot
,diofant.functions.elementary.trigonometric.asin
,diofant.functions.elementary.trigonometric.acsc
,diofant.functions.elementary.trigonometric.acos
,diofant.functions.elementary.trigonometric.asec
,diofant.functions.elementary.trigonometric.atan
,diofant.functions.elementary.trigonometric.acot
,diofant.functions.elementary.trigonometric.atan2
References
csc¶
-
class
diofant.functions.elementary.trigonometric.
csc
[source]¶ The cosecant function.
Returns the cosecant of x (measured in radians).
Notes
See
sin()
for notes about automatic evaluation.Examples
>>> csc(x**2).diff(x) -2*x*cot(x**2)*csc(x**2)
See also
diofant.functions.elementary.trigonometric.sin
,diofant.functions.elementary.trigonometric.cos
,diofant.functions.elementary.trigonometric.sec
,diofant.functions.elementary.trigonometric.tan
,diofant.functions.elementary.trigonometric.cot
,diofant.functions.elementary.trigonometric.asin
,diofant.functions.elementary.trigonometric.acsc
,diofant.functions.elementary.trigonometric.acos
,diofant.functions.elementary.trigonometric.asec
,diofant.functions.elementary.trigonometric.atan
,diofant.functions.elementary.trigonometric.acot
,diofant.functions.elementary.trigonometric.atan2
References
Trigonometric Inverses¶
asin¶
-
class
diofant.functions.elementary.trigonometric.
asin
[source]¶ The inverse sine function.
Returns the arcsine of x in radians.
Notes
asin(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1 and for some instances when the result is a rational multiple of pi (see the eval class method).
Examples
>>> asin(1) pi/2 >>> asin(-1) -pi/2
See also
diofant.functions.elementary.trigonometric.sin
,diofant.functions.elementary.trigonometric.csc
,diofant.functions.elementary.trigonometric.cos
,diofant.functions.elementary.trigonometric.sec
,diofant.functions.elementary.trigonometric.tan
,diofant.functions.elementary.trigonometric.cot
,diofant.functions.elementary.trigonometric.acsc
,diofant.functions.elementary.trigonometric.acos
,diofant.functions.elementary.trigonometric.asec
,diofant.functions.elementary.trigonometric.atan
,diofant.functions.elementary.trigonometric.acot
,diofant.functions.elementary.trigonometric.atan2
References
acos¶
-
class
diofant.functions.elementary.trigonometric.
acos
[source]¶ The inverse cosine function.
Returns the arc cosine of x (measured in radians).
Notes
acos(x)
will evaluate automatically in the casesoo
,-oo
,0
,1
,-1
.acos(zoo)
evaluates tozoo
(see note in :py:class`diofant.functions.elementary.trigonometric.asec`)Examples
>>> acos(1) 0 >>> acos(0) pi/2 >>> acos(oo) oo*I
See also
diofant.functions.elementary.trigonometric.sin
,diofant.functions.elementary.trigonometric.csc
,diofant.functions.elementary.trigonometric.cos
,diofant.functions.elementary.trigonometric.sec
,diofant.functions.elementary.trigonometric.tan
,diofant.functions.elementary.trigonometric.cot
,diofant.functions.elementary.trigonometric.asin
,diofant.functions.elementary.trigonometric.acsc
,diofant.functions.elementary.trigonometric.asec
,diofant.functions.elementary.trigonometric.atan
,diofant.functions.elementary.trigonometric.acot
,diofant.functions.elementary.trigonometric.atan2
References
asec¶
-
class
diofant.functions.elementary.trigonometric.
asec
[source]¶ The inverse secant function.
Returns the arc secant of x (measured in radians).
Notes
asec(x)
will evaluate automatically in the casesoo
,-oo
,0
,1
,-1
.asec(x)
has branch cut in the interval [-1, 1]. For complex arguments, it can be defined as\[sec^{-1}(z) = -i*(log(\sqrt{1 - z^2} + 1) / z)\]At
x = 0
, for positive branch cut, the limit evaluates tozoo
. For negative branch cut, the limit\[\lim_{z \to 0}-i*(log(-\sqrt{1 - z^2} + 1) / z)\]simplifies to \(-i*log(z/2 + O(z^3))\) which ultimately evaluates to
zoo
.As
asex(x)
=asec(1/x)
, a similar argument can be given foracos(x)
.Examples
>>> asec(1) 0 >>> asec(-1) pi
See also
diofant.functions.elementary.trigonometric.sin
,diofant.functions.elementary.trigonometric.csc
,diofant.functions.elementary.trigonometric.cos
,diofant.functions.elementary.trigonometric.sec
,diofant.functions.elementary.trigonometric.tan
,diofant.functions.elementary.trigonometric.cot
,diofant.functions.elementary.trigonometric.asin
,diofant.functions.elementary.trigonometric.acsc
,diofant.functions.elementary.trigonometric.acos
,diofant.functions.elementary.trigonometric.atan
,diofant.functions.elementary.trigonometric.acot
,diofant.functions.elementary.trigonometric.atan2
References
atan¶
-
class
diofant.functions.elementary.trigonometric.
atan
[source]¶ The inverse tangent function.
Returns the arc tangent of x (measured in radians).
Notes
atan(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1.
Examples
>>> atan(0) 0 >>> atan(1) pi/4 >>> atan(oo) pi/2
See also
diofant.functions.elementary.trigonometric.sin
,diofant.functions.elementary.trigonometric.csc
,diofant.functions.elementary.trigonometric.cos
,diofant.functions.elementary.trigonometric.sec
,diofant.functions.elementary.trigonometric.tan
,diofant.functions.elementary.trigonometric.cot
,diofant.functions.elementary.trigonometric.asin
,diofant.functions.elementary.trigonometric.acsc
,diofant.functions.elementary.trigonometric.acos
,diofant.functions.elementary.trigonometric.asec
,diofant.functions.elementary.trigonometric.acot
,diofant.functions.elementary.trigonometric.atan2
References
acot¶
-
class
diofant.functions.elementary.trigonometric.
acot
[source]¶ The inverse cotangent function.
Returns the arc cotangent of x (measured in radians). This function has a branch cut discontinuity in the complex plane running from \(-i\) to \(i\).
See also
diofant.functions.elementary.trigonometric.sin
,diofant.functions.elementary.trigonometric.csc
,diofant.functions.elementary.trigonometric.cos
,diofant.functions.elementary.trigonometric.sec
,diofant.functions.elementary.trigonometric.tan
,diofant.functions.elementary.trigonometric.cot
,diofant.functions.elementary.trigonometric.asin
,diofant.functions.elementary.trigonometric.acsc
,diofant.functions.elementary.trigonometric.acos
,diofant.functions.elementary.trigonometric.asec
,diofant.functions.elementary.trigonometric.atan
,diofant.functions.elementary.trigonometric.atan2
References
acsc¶
-
class
diofant.functions.elementary.trigonometric.
acsc
[source]¶ The inverse cosecant function.
Returns the arc cosecant of x (measured in radians).
Notes
acsc(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1.
Examples
>>> acsc(1) pi/2 >>> acsc(-1) -pi/2
See also
diofant.functions.elementary.trigonometric.sin
,diofant.functions.elementary.trigonometric.csc
,diofant.functions.elementary.trigonometric.cos
,diofant.functions.elementary.trigonometric.sec
,diofant.functions.elementary.trigonometric.tan
,diofant.functions.elementary.trigonometric.cot
,diofant.functions.elementary.trigonometric.asin
,diofant.functions.elementary.trigonometric.acos
,diofant.functions.elementary.trigonometric.asec
,diofant.functions.elementary.trigonometric.atan
,diofant.functions.elementary.trigonometric.acot
,diofant.functions.elementary.trigonometric.atan2
References
atan2¶
-
class
diofant.functions.elementary.trigonometric.
atan2
[source]¶ The function
atan2(y, x)
computes \(\operatorname{atan}(y/x)\) taking two arguments \(y\) and \(x\). Signs of both \(y\) and \(x\) are considered to determine the appropriate quadrant of \(\operatorname{atan}(y/x)\). The range is \((-\pi, \pi]\). The complete definition reads as follows:\[\begin{split}\operatorname{atan2}(y, x) = \begin{cases} \arctan\left(\frac y x\right) & \qquad x > 0 \\ \arctan\left(\frac y x\right) + \pi& \qquad y \ge 0 , x < 0 \\ \arctan\left(\frac y x\right) - \pi& \qquad y < 0 , x < 0 \\ +\frac{\pi}{2} & \qquad y > 0 , x = 0 \\ -\frac{\pi}{2} & \qquad y < 0 , x = 0 \\ \text{undefined} & \qquad y = 0, x = 0 \end{cases}\end{split}\]Attention: Note the role reversal of both arguments. The \(y\)-coordinate is the first argument and the \(x\)-coordinate the second.
Examples
Going counter-clock wise around the origin we find the following angles:
>>> atan2(0, 1) 0 >>> atan2(1, 1) pi/4 >>> atan2(1, 0) pi/2 >>> atan2(1, -1) 3*pi/4 >>> atan2(0, -1) pi >>> atan2(-1, -1) -3*pi/4 >>> atan2(-1, 0) -pi/2 >>> atan2(-1, 1) -pi/4
which are all correct. Compare this to the results of the ordinary \(\operatorname{atan}\) function for the point \((x, y) = (-1, 1)\)
>>> atan(Integer(1) / -1) -pi/4 >>> atan2(1, -1) 3*pi/4
where only the \(\operatorname{atan2}\) function returns what we expect. We can differentiate the function with respect to both arguments:
>>> diff(atan2(y, x), x) -y/(x**2 + y**2)
>>> diff(atan2(y, x), y) x/(x**2 + y**2)
We can express the \(\operatorname{atan2}\) function in terms of complex logarithms:
>>> atan2(y, x).rewrite(log) -I*log((x + I*y)/sqrt(x**2 + y**2))
and in terms of \(\operatorname(atan)\):
>>> atan2(y, x).rewrite(atan) 2*atan(y/(x + sqrt(x**2 + y**2)))
but note that this form is undefined on the negative real axis.
See also
diofant.functions.elementary.trigonometric.sin
,diofant.functions.elementary.trigonometric.csc
,diofant.functions.elementary.trigonometric.cos
,diofant.functions.elementary.trigonometric.sec
,diofant.functions.elementary.trigonometric.tan
,diofant.functions.elementary.trigonometric.cot
,diofant.functions.elementary.trigonometric.asin
,diofant.functions.elementary.trigonometric.acsc
,diofant.functions.elementary.trigonometric.acos
,diofant.functions.elementary.trigonometric.asec
,diofant.functions.elementary.trigonometric.atan
,diofant.functions.elementary.trigonometric.acot
References
diofant.functions.elementary.hyperbolic¶
Hyperbolic Functions¶
HyperbolicFunction¶
sinh¶
cosh¶
tanh¶
coth¶
sech¶
-
class
diofant.functions.elementary.hyperbolic.
sech
[source]¶ The hyperbolic secant function, \(\frac{2}{e^x + e^{-x}}\)
- sech(x) -> Returns the hyperbolic secant of x
See also
diofant.functions.elementary.hyperbolic.sinh
,diofant.functions.elementary.hyperbolic.cosh
,diofant.functions.elementary.hyperbolic.tanh
,diofant.functions.elementary.hyperbolic.coth
,diofant.functions.elementary.hyperbolic.csch
,diofant.functions.elementary.hyperbolic.asinh
,diofant.functions.elementary.hyperbolic.acosh
csch¶
Hyperbolic Inverses¶
asinh¶
acosh¶
atanh¶
diofant.functions.elementary.integers¶
ceiling¶
-
class
diofant.functions.elementary.integers.
ceiling
[source]¶ Ceiling is a univariate function which returns the smallest integer value not less than its argument. Ceiling function is generalized in this implementation to complex numbers.
Examples
>>> ceiling(17) 17 >>> ceiling(Rational(23, 10)) 3 >>> ceiling(2*E) 6 >>> ceiling(-Float(0.567)) 0 >>> ceiling(I/2) I
References
- “Concrete mathematics” by Graham, pp. 87
- http://mathworld.wolfram.com/CeilingFunction.html
floor¶
-
class
diofant.functions.elementary.integers.
floor
[source]¶ Floor is a univariate function which returns the largest integer value not greater than its argument. However this implementation generalizes floor to complex numbers.
Examples
>>> floor(17) 17 >>> floor(Rational(23, 10)) 2 >>> floor(2*E) 5 >>> floor(-Float(0.567)) -1 >>> floor(-I/2) -I
References
- “Concrete mathematics” by Graham, pp. 87
- http://mathworld.wolfram.com/FloorFunction.html
diofant.functions.elementary.exponential¶
exp¶
exp_polar¶
-
class
diofant.functions.elementary.exponential.
exp_polar
[source]¶ Represent a ‘polar number’ (see g-function Sphinx documentation).
exp_polar
represents the function \(Exp: \mathbb{C} \rightarrow \mathcal{S}\), sending the complex number \(z = a + bi\) to the polar number \(r = exp(a), \theta = b\). It is one of the main functions to construct polar numbers.The main difference is that polar numbers don’t “wrap around” at \(2 \pi\):
>>> exp(2*pi*I) 1 >>> exp_polar(2*pi*I) exp_polar(2*I*pi)
apart from that they behave mostly like classical complex numbers:
>>> exp_polar(2)*exp_polar(3) exp_polar(5)
See also
diofant.simplify.powsimp.powsimp
,diofant.functions.elementary.complexes.polar_lift
,diofant.functions.elementary.complexes.periodic_argument
,diofant.functions.elementary.complexes.principal_branch
-
exp
¶ Returns the exponent of the function.
-
LambertW¶
-
class
diofant.functions.elementary.exponential.
LambertW
[source]¶ The Lambert W function \(W(z)\) is defined as the inverse function of \(w \exp(w)\).
In other words, the value of \(W(z)\) is such that \(z = W(z) \exp(W(z))\) for any complex number \(z\). The Lambert W function is a multivalued function with infinitely many branches \(W_k(z)\), indexed by \(k \in \mathbb{Z}\). Each branch gives a different solution \(w\) of the equation \(z = w \exp(w)\).
The Lambert W function has two partially real branches: the principal branch (\(k = 0\)) is real for real \(z > -1/e\), and the \(k = -1\) branch is real for \(-1/e < z < 0\). All branches except \(k = 0\) have a logarithmic singularity at \(z = 0\).
Examples
>>> LambertW(1.2) 0.635564016364870 >>> LambertW(1.2, -1).evalf() -1.34747534407696 - 4.41624341514535*I >>> LambertW(-1).is_real False
References
log¶
-
class
diofant.functions.elementary.exponential.
log
[source]¶ The natural logarithm function \(\ln(x)\) or \(\log(x)\). Logarithms are taken with the natural base, \(e\). To get a logarithm of a different base
b
, uselog(x, b)
, which is essentially short-hand forlog(x)/log(b)
.
diofant.functions.elementary.piecewise¶
ExprCondPair¶
Piecewise¶
-
class
diofant.functions.elementary.piecewise.
Piecewise
[source]¶ Represents a piecewise function.
Usage:
- Piecewise( (expr,cond), (expr,cond), … )
- Each argument is a 2-tuple defining an expression and condition
- The conds are evaluated in turn returning the first that is True. If any of the evaluated conds are not determined explicitly False, e.g. x < 1, the function is returned in symbolic form.
- If the function is evaluated at a place where all conditions are False, a ValueError exception will be raised.
- Pairs where the cond is explicitly False, will be removed.
Examples
>>> f = x**2 >>> g = log(x) >>> p = Piecewise((0, x<-1), (f, x<=1), (g, True)) >>> p.subs({x: 1}) 1 >>> p.subs({x: 5}) log(5)
diofant.functions.elementary.miscellaneous¶
IdentityFunction¶
Min¶
-
class
diofant.functions.elementary.miscellaneous.
Min
[source]¶ Return, if possible, the minimum value of the list.
It is named
Min
and notmin
to avoid conflicts with the built-in functionmin
.Examples
>>> p = Symbol('p', positive=True) >>> n = Symbol('n', negative=True)
>>> Min(x, -2) Min(-2, x) >>> Min(x, -2).subs({x: 3}) -2 >>> Min(p, -3) -3 >>> Min(x, y) Min(x, y) >>> Min(n, 8, p, -7, p, oo) Min(-7, n)
See also
diofant.functions.elementary.miscellaneous.Max
- find maximum values
Max¶
-
class
diofant.functions.elementary.miscellaneous.
Max
[source]¶ Return, if possible, the maximum value of the list.
When number of arguments is equal one, then return this argument.
When number of arguments is equal two, then return, if possible, the value from (a, b) that is >= the other.
In common case, when the length of list greater than 2, the task is more complicated. Return only the arguments, which are greater than others, if it is possible to determine directional relation.
If is not possible to determine such a relation, return a partially evaluated result.
Assumptions are used to make the decision too.
Also, only comparable arguments are permitted.
It is named
Max
and notmax
to avoid conflicts with the built-in functionmax
.Examples
>>> p = Symbol('p', positive=True) >>> n = Symbol('n', negative=True)
>>> Max(x, -2) Max(-2, x) >>> Max(x, -2).subs({x: 3}) 3 >>> Max(p, -2) p >>> Max(x, y) Max(x, y) >>> Max(x, y) == Max(y, x) True >>> Max(x, Max(y, z)) Max(x, y, z) >>> Max(n, 8, p, 7, -oo) Max(8, p) >>> Max (1, x, oo) oo
Notes
The task can be considered as searching of supremums in the directed complete partial orders.
The source values are sequentially allocated by the isolated subsets in which supremums are searched and result as Max arguments.
If the resulted supremum is single, then it is returned.
The isolated subsets are the sets of values which are only the comparable with each other in the current set. E.g. natural numbers are comparable with each other, but not comparable with the \(x\) symbol. Another example: the symbol \(x\) with negative assumption is comparable with a natural number.
Also there are “least” elements, which are comparable with all others, and have a zero property (maximum or minimum for all elements). E.g. \(oo\). In case of it the allocation operation is terminated and only this value is returned.
- Assumption:
- if A > B > C then A > C
- if A == B then B can be removed
References
- https://en.wikipedia.org/wiki/Directed_complete_partial_order
- https://en.wikipedia.org/wiki/Lattice_%28order%29
See also
diofant.functions.elementary.miscellaneous.Min
- find minimum values
root¶
-
diofant.functions.elementary.miscellaneous.
root
(x, n, k) → Returns the k-th n-th root of x, defaulting to the[source]¶ principle root (k=0).
Examples
>>> root(x, 2) sqrt(x)
>>> root(x, 3) x**(1/3)
>>> root(x, n) x**(1/n)
>>> root(x, -Rational(2, 3)) x**(-3/2)
To get the k-th n-th root, specify k:
>>> root(-2, 3, 2) -(-1)**(2/3)*2**(1/3)
To get all n n-th roots you can use the RootOf function. The following examples show the roots of unity for n equal 2, 3 and 4:
>>> [RootOf(x**2 - 1, i) for i in range(2)] [-1, 1]
>>> [RootOf(x**3 - 1, i) for i in range(3)] [1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]
>>> [RootOf(x**4 - 1, i) for i in range(4)] [-1, 1, -I, I]
Diofant, like other symbolic algebra systems, returns the complex root of negative numbers. This is the principal root and differs from the text-book result that one might be expecting. For example, the cube root of -8 does not come back as -2:
>>> root(-8, 3) 2*(-1)**(1/3)
The real_root function can be used to either make the principle result real (or simply to return the real root directly):
>>> real_root(_) -2 >>> real_root(-32, 5) -2
Alternatively, the n//2-th n-th root of a negative number can be computed with root:
>>> root(-32, 5, 5//2) -2
See also
diofant.polys.rootoftools.RootOf()
,diofant.core.power.integer_nthroot()
,diofant.functions.elementary.miscellaneous.sqrt()
,diofant.functions.elementary.miscellaneous.real_root()
References
real_root¶
-
diofant.functions.elementary.miscellaneous.
real_root
(arg, n=None)[source]¶ Return the real nth-root of arg if possible. If n is omitted then all instances of (-n)**(1/odd) will be changed to -n**(1/odd); this will only create a real root of a principle root – the presence of other factors may cause the result to not be real.
Examples
>>> real_root(-8, 3) -2 >>> root(-8, 3) 2*(-1)**(1/3) >>> real_root(_) -2
If one creates a non-principle root and applies real_root, the result will not be real (so use with caution):
>>> root(-8, 3, 2) -2*(-1)**(2/3) >>> real_root(_) -2*(-1)**(2/3)
sqrt¶
-
diofant.functions.elementary.miscellaneous.
sqrt
(arg, **kwargs)[source]¶ The square root function
sqrt(x) -> Returns the principal square root of x.
Examples
>>> sqrt(x) sqrt(x)
>>> sqrt(x)**2 x
Note that sqrt(x**2) does not simplify to x.
>>> sqrt(x**2) sqrt(x**2)
This is because the two are not equal to each other in general. For example, consider x == -1:
>>> Eq(sqrt(x**2), x).subs({x: -1}) false
This is because sqrt computes the principal square root, so the square may put the argument in a different branch. This identity does hold if x is positive:
>>> y = Symbol('y', positive=True) >>> sqrt(y**2) y
You can force this simplification by using the powdenest() function with the force option set to True:
>>> sqrt(x**2) sqrt(x**2) >>> powdenest(sqrt(x**2), force=True) x
To get both branches of the square root you can use the RootOf function:
>>> [RootOf(x**2 - 3, i) for i in (0, 1)] [-sqrt(3), sqrt(3)]
See also
diofant.polys.rootoftools.RootOf()
,diofant.functions.elementary.miscellaneous.root()
,diofant.functions.elementary.miscellaneous.real_root()
References