Representation of holonomic functions in SymPy
==============================================
.. currentmodule:: sympy.holonomic.holonomic
Class :class:`DifferentialOperator` is used to represent the annihilator
but we create differential operators easily using the function
:func:`DifferentialOperators`. Class :class:`HolonomicFunction` represents a holonomic function.
Let's explain this with an example:
Take `\sin(x)` for instance, the differential equation satisfied by it
is `y^{(2)}(x) + y(x) = 0`. By definition we conclude it is a holonomic
function. The general solution of this ODE is
`C_{1} \cdot \sin(x) + C_{2} \cdot \cos(x)` but to get `\sin(x)` we need to
provide initial conditions i.e. `y(0) = 0, y^{(1)}(0) = 1`.
To represent the same in this module one needs to provide the differential
equation in the form of annihilator. Basically a differential operator is an
operator on functions that differentiates them. So `D^{n} \cdot y(x) = y^{(n)}(x)`
where :math:`y^{(n)}(x)` denotes ``n`` times differentiation of :math:`y(x)` with
respect to ``x``.
So the differential equation can also be written as
:math:`D^{2} \cdot y(x) + y(x) = 0` or `(D^{2} + 1) \cdot y(x) = 0`.
The part left of :math:`y(x)` is the annihilator i.e. :math:`D^{2}+1`.
So this is how one will represent `\sin(x)` as a Holonomic Function:
>>> from sympy.holonomic import DifferentialOperators, HolonomicFunction
>>> from sympy.abc import x
>>> from sympy import ZZ
>>> R, D = DifferentialOperators(ZZ.old_poly_ring(x), 'D')
>>> HolonomicFunction(D**2 + 1, x, 0, [0, 1])
HolonomicFunction((1) + (1)*D**2, x, 0, [0, 1])
The polynomial coefficients will be members of the ring ``ZZ[x]`` in the example.
The ``D`` operator returned by the function :py:func:`DifferentialOperators` can
be used to create annihilators just like SymPy expressions.
We currently use the older implementations of rings in SymPy for priority
mechanism.
.. autoclass:: HolonomicFunction
.. autoclass:: DifferentialOperator
:members:
.. autofunction:: DifferentialOperators
.. autoclass:: DifferentialOperatorAlgebra
:members: