Mathematical Functions

All functions support the methods documented below, inherited from diofant.core.function.Function.

class diofant.core.function.Function[source]

Base class for applied mathematical functions.

It also serves as a constructor for undefined function classes.

Examples

First example shows how to use Function as a constructor for undefined function classes:

>>> g = g(x)
>>> f
f
>>> f(x)
f(x)
>>> g
g(x)
>>> f(x).diff(x)
Derivative(f(x), x)
>>> g.diff(x)
Derivative(g(x), x)

In the following example Function is used as a base class for my_func that represents a mathematical function my_func. Suppose that it is well known, that my_func(0) is 1 and my_func at infinity goes to 0, so we want those two simplifications to occur automatically. Suppose also that my_func(x) is real exactly when x is real. Here is an implementation that honours those requirements:

>>> class my_func(Function):
...
...     @classmethod
...     def eval(cls, x):
...         if x.is_Number:
...             if x is S.Zero:
...                 return S.One
...             elif x is oo:
...                 return S.Zero
...
...     def _eval_is_real(self):
...         return self.args[0].is_real
...
>>> x = Symbol('x')
>>> my_func(0) + sin(0)
1
>>> my_func(oo)
0
>>> my_func(3.54).n() # Not yet implemented for my_func.
my_func(3.54)
>>> my_func(I).is_real
False

In order for my_func to become useful, several other methods would need to be implemented. See source code of some of the already implemented functions for more complete examples.

Also, if the function can take more than one argument, then nargs must be defined, e.g. if my_func can take one or two arguments then,

>>> class my_func(Function):
...     nargs = (1, 2)
...
>>>
as_base_exp()[source]

Returns the method as the 2-tuple (base, exponent).

classmethod class_key()[source]

Nice order of classes.

fdiff(argindex=1)[source]

Returns the first derivative of the function.