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1589 lines
50 KiB
1589 lines
50 KiB
""" Integral Transforms """
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from functools import reduce, wraps
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from itertools import repeat
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from sympy.core import S, pi
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from sympy.core.add import Add
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from sympy.core.function import (
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AppliedUndef, count_ops, expand, expand_mul, Function)
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from sympy.core.mul import Mul
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from sympy.core.numbers import igcd, ilcm
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from sympy.core.sorting import default_sort_key
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from sympy.core.symbol import Dummy
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from sympy.core.traversal import postorder_traversal
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from sympy.functions.combinatorial.factorials import factorial, rf
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from sympy.functions.elementary.complexes import re, arg, Abs
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from sympy.functions.elementary.exponential import exp, exp_polar
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from sympy.functions.elementary.hyperbolic import cosh, coth, sinh, tanh
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from sympy.functions.elementary.integers import ceiling
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from sympy.functions.elementary.miscellaneous import Max, Min, sqrt
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from sympy.functions.elementary.piecewise import piecewise_fold
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from sympy.functions.elementary.trigonometric import cos, cot, sin, tan
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from sympy.functions.special.bessel import besselj
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from sympy.functions.special.delta_functions import Heaviside
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from sympy.functions.special.gamma_functions import gamma
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from sympy.functions.special.hyper import meijerg
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from sympy.integrals import integrate, Integral
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from sympy.integrals.meijerint import _dummy
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from sympy.logic.boolalg import to_cnf, conjuncts, disjuncts, Or, And
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from sympy.polys.polyroots import roots
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from sympy.polys.polytools import factor, Poly
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from sympy.polys.rootoftools import CRootOf
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from sympy.utilities.iterables import iterable
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from sympy.utilities.misc import debug
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##########################################################################
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# Helpers / Utilities
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##########################################################################
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class IntegralTransformError(NotImplementedError):
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"""
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Exception raised in relation to problems computing transforms.
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Explanation
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===========
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This class is mostly used internally; if integrals cannot be computed
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objects representing unevaluated transforms are usually returned.
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The hint ``needeval=True`` can be used to disable returning transform
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objects, and instead raise this exception if an integral cannot be
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computed.
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"""
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def __init__(self, transform, function, msg):
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super().__init__(
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"%s Transform could not be computed: %s." % (transform, msg))
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self.function = function
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class IntegralTransform(Function):
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"""
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Base class for integral transforms.
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Explanation
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===========
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This class represents unevaluated transforms.
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To implement a concrete transform, derive from this class and implement
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the ``_compute_transform(f, x, s, **hints)`` and ``_as_integral(f, x, s)``
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functions. If the transform cannot be computed, raise :obj:`IntegralTransformError`.
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Also set ``cls._name``. For instance,
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>>> from sympy import LaplaceTransform
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>>> LaplaceTransform._name
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'Laplace'
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Implement ``self._collapse_extra`` if your function returns more than just a
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number and possibly a convergence condition.
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"""
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@property
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def function(self):
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""" The function to be transformed. """
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return self.args[0]
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@property
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def function_variable(self):
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""" The dependent variable of the function to be transformed. """
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return self.args[1]
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@property
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def transform_variable(self):
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""" The independent transform variable. """
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return self.args[2]
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@property
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def free_symbols(self):
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"""
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This method returns the symbols that will exist when the transform
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is evaluated.
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"""
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return self.function.free_symbols.union({self.transform_variable}) \
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- {self.function_variable}
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def _compute_transform(self, f, x, s, **hints):
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raise NotImplementedError
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def _as_integral(self, f, x, s):
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raise NotImplementedError
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def _collapse_extra(self, extra):
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cond = And(*extra)
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if cond == False:
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raise IntegralTransformError(self.__class__.name, None, '')
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return cond
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def _try_directly(self, **hints):
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T = None
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try_directly = not any(func.has(self.function_variable)
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for func in self.function.atoms(AppliedUndef))
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if try_directly:
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try:
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T = self._compute_transform(self.function,
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self.function_variable, self.transform_variable, **hints)
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except IntegralTransformError:
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debug('[IT _try ] Caught IntegralTransformError, returns None')
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T = None
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fn = self.function
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if not fn.is_Add:
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fn = expand_mul(fn)
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return fn, T
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def doit(self, **hints):
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"""
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Try to evaluate the transform in closed form.
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Explanation
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===========
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This general function handles linearity, but apart from that leaves
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pretty much everything to _compute_transform.
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Standard hints are the following:
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- ``simplify``: whether or not to simplify the result
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- ``noconds``: if True, do not return convergence conditions
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- ``needeval``: if True, raise IntegralTransformError instead of
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returning IntegralTransform objects
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The default values of these hints depend on the concrete transform,
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usually the default is
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``(simplify, noconds, needeval) = (True, False, False)``.
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"""
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needeval = hints.pop('needeval', False)
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simplify = hints.pop('simplify', True)
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hints['simplify'] = simplify
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fn, T = self._try_directly(**hints)
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if T is not None:
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return T
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if fn.is_Add:
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hints['needeval'] = needeval
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res = [self.__class__(*([x] + list(self.args[1:]))).doit(**hints)
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for x in fn.args]
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extra = []
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ress = []
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for x in res:
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if not isinstance(x, tuple):
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x = [x]
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ress.append(x[0])
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if len(x) == 2:
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# only a condition
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extra.append(x[1])
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elif len(x) > 2:
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# some region parameters and a condition (Mellin, Laplace)
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extra += [x[1:]]
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if simplify==True:
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res = Add(*ress).simplify()
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else:
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res = Add(*ress)
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if not extra:
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return res
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try:
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extra = self._collapse_extra(extra)
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if iterable(extra):
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return (res,) + tuple(extra)
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else:
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return (res, extra)
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except IntegralTransformError:
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pass
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if needeval:
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raise IntegralTransformError(
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self.__class__._name, self.function, 'needeval')
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# TODO handle derivatives etc
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# pull out constant coefficients
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coeff, rest = fn.as_coeff_mul(self.function_variable)
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return coeff*self.__class__(*([Mul(*rest)] + list(self.args[1:])))
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@property
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def as_integral(self):
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return self._as_integral(self.function, self.function_variable,
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self.transform_variable)
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def _eval_rewrite_as_Integral(self, *args, **kwargs):
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return self.as_integral
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def _simplify(expr, doit):
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if doit:
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from sympy.simplify import simplify
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from sympy.simplify.powsimp import powdenest
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return simplify(powdenest(piecewise_fold(expr), polar=True))
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return expr
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def _noconds_(default):
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"""
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This is a decorator generator for dropping convergence conditions.
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Explanation
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===========
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Suppose you define a function ``transform(*args)`` which returns a tuple of
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the form ``(result, cond1, cond2, ...)``.
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Decorating it ``@_noconds_(default)`` will add a new keyword argument
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``noconds`` to it. If ``noconds=True``, the return value will be altered to
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be only ``result``, whereas if ``noconds=False`` the return value will not
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be altered.
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The default value of the ``noconds`` keyword will be ``default`` (i.e. the
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argument of this function).
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"""
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def make_wrapper(func):
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@wraps(func)
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def wrapper(*args, noconds=default, **kwargs):
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res = func(*args, **kwargs)
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if noconds:
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return res[0]
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return res
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return wrapper
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return make_wrapper
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_noconds = _noconds_(False)
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##########################################################################
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# Mellin Transform
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##########################################################################
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def _default_integrator(f, x):
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return integrate(f, (x, S.Zero, S.Infinity))
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@_noconds
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def _mellin_transform(f, x, s_, integrator=_default_integrator, simplify=True):
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""" Backend function to compute Mellin transforms. """
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# We use a fresh dummy, because assumptions on s might drop conditions on
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# convergence of the integral.
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s = _dummy('s', 'mellin-transform', f)
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F = integrator(x**(s - 1) * f, x)
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if not F.has(Integral):
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return _simplify(F.subs(s, s_), simplify), (S.NegativeInfinity, S.Infinity), S.true
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if not F.is_Piecewise: # XXX can this work if integration gives continuous result now?
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raise IntegralTransformError('Mellin', f, 'could not compute integral')
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F, cond = F.args[0]
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if F.has(Integral):
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raise IntegralTransformError(
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'Mellin', f, 'integral in unexpected form')
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def process_conds(cond):
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"""
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Turn ``cond`` into a strip (a, b), and auxiliary conditions.
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"""
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from sympy.solvers.inequalities import _solve_inequality
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a = S.NegativeInfinity
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b = S.Infinity
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aux = S.true
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conds = conjuncts(to_cnf(cond))
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t = Dummy('t', real=True)
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for c in conds:
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a_ = S.Infinity
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b_ = S.NegativeInfinity
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aux_ = []
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for d in disjuncts(c):
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d_ = d.replace(
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re, lambda x: x.as_real_imag()[0]).subs(re(s), t)
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if not d.is_Relational or \
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d.rel_op in ('==', '!=') \
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or d_.has(s) or not d_.has(t):
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aux_ += [d]
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continue
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soln = _solve_inequality(d_, t)
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if not soln.is_Relational or \
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soln.rel_op in ('==', '!='):
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aux_ += [d]
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continue
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if soln.lts == t:
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b_ = Max(soln.gts, b_)
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else:
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a_ = Min(soln.lts, a_)
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if a_ is not S.Infinity and a_ != b:
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a = Max(a_, a)
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elif b_ is not S.NegativeInfinity and b_ != a:
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b = Min(b_, b)
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else:
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aux = And(aux, Or(*aux_))
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return a, b, aux
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conds = [process_conds(c) for c in disjuncts(cond)]
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conds = [x for x in conds if x[2] != False]
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conds.sort(key=lambda x: (x[0] - x[1], count_ops(x[2])))
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if not conds:
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raise IntegralTransformError('Mellin', f, 'no convergence found')
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a, b, aux = conds[0]
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return _simplify(F.subs(s, s_), simplify), (a, b), aux
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class MellinTransform(IntegralTransform):
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"""
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Class representing unevaluated Mellin transforms.
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For usage of this class, see the :class:`IntegralTransform` docstring.
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For how to compute Mellin transforms, see the :func:`mellin_transform`
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docstring.
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"""
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_name = 'Mellin'
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def _compute_transform(self, f, x, s, **hints):
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return _mellin_transform(f, x, s, **hints)
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def _as_integral(self, f, x, s):
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return Integral(f*x**(s - 1), (x, S.Zero, S.Infinity))
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def _collapse_extra(self, extra):
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a = []
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b = []
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cond = []
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for (sa, sb), c in extra:
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a += [sa]
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b += [sb]
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cond += [c]
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res = (Max(*a), Min(*b)), And(*cond)
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if (res[0][0] >= res[0][1]) == True or res[1] == False:
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raise IntegralTransformError(
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'Mellin', None, 'no combined convergence.')
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return res
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def mellin_transform(f, x, s, **hints):
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r"""
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Compute the Mellin transform `F(s)` of `f(x)`,
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.. math :: F(s) = \int_0^\infty x^{s-1} f(x) \mathrm{d}x.
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For all "sensible" functions, this converges absolutely in a strip
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`a < \operatorname{Re}(s) < b`.
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Explanation
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===========
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The Mellin transform is related via change of variables to the Fourier
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transform, and also to the (bilateral) Laplace transform.
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This function returns ``(F, (a, b), cond)``
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where ``F`` is the Mellin transform of ``f``, ``(a, b)`` is the fundamental strip
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(as above), and ``cond`` are auxiliary convergence conditions.
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If the integral cannot be computed in closed form, this function returns
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an unevaluated :class:`MellinTransform` object.
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For a description of possible hints, refer to the docstring of
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:func:`sympy.integrals.transforms.IntegralTransform.doit`. If ``noconds=False``,
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then only `F` will be returned (i.e. not ``cond``, and also not the strip
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``(a, b)``).
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Examples
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========
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>>> from sympy import mellin_transform, exp
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>>> from sympy.abc import x, s
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>>> mellin_transform(exp(-x), x, s)
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(gamma(s), (0, oo), True)
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See Also
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========
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inverse_mellin_transform, laplace_transform, fourier_transform
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hankel_transform, inverse_hankel_transform
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"""
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return MellinTransform(f, x, s).doit(**hints)
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def _rewrite_sin(m_n, s, a, b):
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"""
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Re-write the sine function ``sin(m*s + n)`` as gamma functions, compatible
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with the strip (a, b).
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Return ``(gamma1, gamma2, fac)`` so that ``f == fac/(gamma1 * gamma2)``.
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Examples
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========
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>>> from sympy.integrals.transforms import _rewrite_sin
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>>> from sympy import pi, S
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>>> from sympy.abc import s
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>>> _rewrite_sin((pi, 0), s, 0, 1)
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(gamma(s), gamma(1 - s), pi)
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>>> _rewrite_sin((pi, 0), s, 1, 0)
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(gamma(s - 1), gamma(2 - s), -pi)
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>>> _rewrite_sin((pi, 0), s, -1, 0)
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(gamma(s + 1), gamma(-s), -pi)
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>>> _rewrite_sin((pi, pi/2), s, S(1)/2, S(3)/2)
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(gamma(s - 1/2), gamma(3/2 - s), -pi)
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>>> _rewrite_sin((pi, pi), s, 0, 1)
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(gamma(s), gamma(1 - s), -pi)
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>>> _rewrite_sin((2*pi, 0), s, 0, S(1)/2)
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(gamma(2*s), gamma(1 - 2*s), pi)
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>>> _rewrite_sin((2*pi, 0), s, S(1)/2, 1)
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(gamma(2*s - 1), gamma(2 - 2*s), -pi)
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"""
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# (This is a separate function because it is moderately complicated,
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# and I want to doctest it.)
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# We want to use pi/sin(pi*x) = gamma(x)*gamma(1-x).
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# But there is one comlication: the gamma functions determine the
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# inegration contour in the definition of the G-function. Usually
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# it would not matter if this is slightly shifted, unless this way
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# we create an undefined function!
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# So we try to write this in such a way that the gammas are
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# eminently on the right side of the strip.
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m, n = m_n
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m = expand_mul(m/pi)
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n = expand_mul(n/pi)
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r = ceiling(-m*a - n.as_real_imag()[0]) # Don't use re(n), does not expand
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return gamma(m*s + n + r), gamma(1 - n - r - m*s), (-1)**r*pi
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class MellinTransformStripError(ValueError):
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"""
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Exception raised by _rewrite_gamma. Mainly for internal use.
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"""
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pass
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def _rewrite_gamma(f, s, a, b):
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"""
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Try to rewrite the product f(s) as a product of gamma functions,
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so that the inverse Mellin transform of f can be expressed as a meijer
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G function.
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Explanation
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===========
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Return (an, ap), (bm, bq), arg, exp, fac such that
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G((an, ap), (bm, bq), arg/z**exp)*fac is the inverse Mellin transform of f(s).
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Raises IntegralTransformError or MellinTransformStripError on failure.
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It is asserted that f has no poles in the fundamental strip designated by
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(a, b). One of a and b is allowed to be None. The fundamental strip is
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important, because it determines the inversion contour.
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This function can handle exponentials, linear factors, trigonometric
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functions.
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This is a helper function for inverse_mellin_transform that will not
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attempt any transformations on f.
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Examples
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========
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>>> from sympy.integrals.transforms import _rewrite_gamma
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>>> from sympy.abc import s
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>>> from sympy import oo
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>>> _rewrite_gamma(s*(s+3)*(s-1), s, -oo, oo)
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(([], [-3, 0, 1]), ([-2, 1, 2], []), 1, 1, -1)
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>>> _rewrite_gamma((s-1)**2, s, -oo, oo)
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(([], [1, 1]), ([2, 2], []), 1, 1, 1)
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Importance of the fundamental strip:
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>>> _rewrite_gamma(1/s, s, 0, oo)
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(([1], []), ([], [0]), 1, 1, 1)
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>>> _rewrite_gamma(1/s, s, None, oo)
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(([1], []), ([], [0]), 1, 1, 1)
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>>> _rewrite_gamma(1/s, s, 0, None)
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(([1], []), ([], [0]), 1, 1, 1)
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>>> _rewrite_gamma(1/s, s, -oo, 0)
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(([], [1]), ([0], []), 1, 1, -1)
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>>> _rewrite_gamma(1/s, s, None, 0)
|
|
(([], [1]), ([0], []), 1, 1, -1)
|
|
>>> _rewrite_gamma(1/s, s, -oo, None)
|
|
(([], [1]), ([0], []), 1, 1, -1)
|
|
|
|
>>> _rewrite_gamma(2**(-s+3), s, -oo, oo)
|
|
(([], []), ([], []), 1/2, 1, 8)
|
|
"""
|
|
# Our strategy will be as follows:
|
|
# 1) Guess a constant c such that the inversion integral should be
|
|
# performed wrt s'=c*s (instead of plain s). Write s for s'.
|
|
# 2) Process all factors, rewrite them independently as gamma functions in
|
|
# argument s, or exponentials of s.
|
|
# 3) Try to transform all gamma functions s.t. they have argument
|
|
# a+s or a-s.
|
|
# 4) Check that the resulting G function parameters are valid.
|
|
# 5) Combine all the exponentials.
|
|
|
|
a_, b_ = S([a, b])
|
|
|
|
def left(c, is_numer):
|
|
"""
|
|
Decide whether pole at c lies to the left of the fundamental strip.
|
|
"""
|
|
# heuristically, this is the best chance for us to solve the inequalities
|
|
c = expand(re(c))
|
|
if a_ is None and b_ is S.Infinity:
|
|
return True
|
|
if a_ is None:
|
|
return c < b_
|
|
if b_ is None:
|
|
return c <= a_
|
|
if (c >= b_) == True:
|
|
return False
|
|
if (c <= a_) == True:
|
|
return True
|
|
if is_numer:
|
|
return None
|
|
if a_.free_symbols or b_.free_symbols or c.free_symbols:
|
|
return None # XXX
|
|
#raise IntegralTransformError('Inverse Mellin', f,
|
|
# 'Could not determine position of singularity %s'
|
|
# ' relative to fundamental strip' % c)
|
|
raise MellinTransformStripError('Pole inside critical strip?')
|
|
|
|
# 1)
|
|
s_multipliers = []
|
|
for g in f.atoms(gamma):
|
|
if not g.has(s):
|
|
continue
|
|
arg = g.args[0]
|
|
if arg.is_Add:
|
|
arg = arg.as_independent(s)[1]
|
|
coeff, _ = arg.as_coeff_mul(s)
|
|
s_multipliers += [coeff]
|
|
for g in f.atoms(sin, cos, tan, cot):
|
|
if not g.has(s):
|
|
continue
|
|
arg = g.args[0]
|
|
if arg.is_Add:
|
|
arg = arg.as_independent(s)[1]
|
|
coeff, _ = arg.as_coeff_mul(s)
|
|
s_multipliers += [coeff/pi]
|
|
s_multipliers = [Abs(x) if x.is_extended_real else x for x in s_multipliers]
|
|
common_coefficient = S.One
|
|
for x in s_multipliers:
|
|
if not x.is_Rational:
|
|
common_coefficient = x
|
|
break
|
|
s_multipliers = [x/common_coefficient for x in s_multipliers]
|
|
if not (all(x.is_Rational for x in s_multipliers) and
|
|
common_coefficient.is_extended_real):
|
|
raise IntegralTransformError("Gamma", None, "Nonrational multiplier")
|
|
s_multiplier = common_coefficient/reduce(ilcm, [S(x.q)
|
|
for x in s_multipliers], S.One)
|
|
if s_multiplier == common_coefficient:
|
|
if len(s_multipliers) == 0:
|
|
s_multiplier = common_coefficient
|
|
else:
|
|
s_multiplier = common_coefficient \
|
|
*reduce(igcd, [S(x.p) for x in s_multipliers])
|
|
|
|
f = f.subs(s, s/s_multiplier)
|
|
fac = S.One/s_multiplier
|
|
exponent = S.One/s_multiplier
|
|
if a_ is not None:
|
|
a_ *= s_multiplier
|
|
if b_ is not None:
|
|
b_ *= s_multiplier
|
|
|
|
# 2)
|
|
numer, denom = f.as_numer_denom()
|
|
numer = Mul.make_args(numer)
|
|
denom = Mul.make_args(denom)
|
|
args = list(zip(numer, repeat(True))) + list(zip(denom, repeat(False)))
|
|
|
|
facs = []
|
|
dfacs = []
|
|
# *_gammas will contain pairs (a, c) representing Gamma(a*s + c)
|
|
numer_gammas = []
|
|
denom_gammas = []
|
|
# exponentials will contain bases for exponentials of s
|
|
exponentials = []
|
|
|
|
def exception(fact):
|
|
return IntegralTransformError("Inverse Mellin", f, "Unrecognised form '%s'." % fact)
|
|
while args:
|
|
fact, is_numer = args.pop()
|
|
if is_numer:
|
|
ugammas, lgammas = numer_gammas, denom_gammas
|
|
ufacs = facs
|
|
else:
|
|
ugammas, lgammas = denom_gammas, numer_gammas
|
|
ufacs = dfacs
|
|
|
|
def linear_arg(arg):
|
|
""" Test if arg is of form a*s+b, raise exception if not. """
|
|
if not arg.is_polynomial(s):
|
|
raise exception(fact)
|
|
p = Poly(arg, s)
|
|
if p.degree() != 1:
|
|
raise exception(fact)
|
|
return p.all_coeffs()
|
|
|
|
# constants
|
|
if not fact.has(s):
|
|
ufacs += [fact]
|
|
# exponentials
|
|
elif fact.is_Pow or isinstance(fact, exp):
|
|
if fact.is_Pow:
|
|
base = fact.base
|
|
exp_ = fact.exp
|
|
else:
|
|
base = exp_polar(1)
|
|
exp_ = fact.exp
|
|
if exp_.is_Integer:
|
|
cond = is_numer
|
|
if exp_ < 0:
|
|
cond = not cond
|
|
args += [(base, cond)]*Abs(exp_)
|
|
continue
|
|
elif not base.has(s):
|
|
a, b = linear_arg(exp_)
|
|
if not is_numer:
|
|
base = 1/base
|
|
exponentials += [base**a]
|
|
facs += [base**b]
|
|
else:
|
|
raise exception(fact)
|
|
# linear factors
|
|
elif fact.is_polynomial(s):
|
|
p = Poly(fact, s)
|
|
if p.degree() != 1:
|
|
# We completely factor the poly. For this we need the roots.
|
|
# Now roots() only works in some cases (low degree), and CRootOf
|
|
# only works without parameters. So try both...
|
|
coeff = p.LT()[1]
|
|
rs = roots(p, s)
|
|
if len(rs) != p.degree():
|
|
rs = CRootOf.all_roots(p)
|
|
ufacs += [coeff]
|
|
args += [(s - c, is_numer) for c in rs]
|
|
continue
|
|
a, c = p.all_coeffs()
|
|
ufacs += [a]
|
|
c /= -a
|
|
# Now need to convert s - c
|
|
if left(c, is_numer):
|
|
ugammas += [(S.One, -c + 1)]
|
|
lgammas += [(S.One, -c)]
|
|
else:
|
|
ufacs += [-1]
|
|
ugammas += [(S.NegativeOne, c + 1)]
|
|
lgammas += [(S.NegativeOne, c)]
|
|
elif isinstance(fact, gamma):
|
|
a, b = linear_arg(fact.args[0])
|
|
if is_numer:
|
|
if (a > 0 and (left(-b/a, is_numer) == False)) or \
|
|
(a < 0 and (left(-b/a, is_numer) == True)):
|
|
raise NotImplementedError(
|
|
'Gammas partially over the strip.')
|
|
ugammas += [(a, b)]
|
|
elif isinstance(fact, sin):
|
|
# We try to re-write all trigs as gammas. This is not in
|
|
# general the best strategy, since sometimes this is impossible,
|
|
# but rewriting as exponentials would work. However trig functions
|
|
# in inverse mellin transforms usually all come from simplifying
|
|
# gamma terms, so this should work.
|
|
a = fact.args[0]
|
|
if is_numer:
|
|
# No problem with the poles.
|
|
gamma1, gamma2, fac_ = gamma(a/pi), gamma(1 - a/pi), pi
|
|
else:
|
|
gamma1, gamma2, fac_ = _rewrite_sin(linear_arg(a), s, a_, b_)
|
|
args += [(gamma1, not is_numer), (gamma2, not is_numer)]
|
|
ufacs += [fac_]
|
|
elif isinstance(fact, tan):
|
|
a = fact.args[0]
|
|
args += [(sin(a, evaluate=False), is_numer),
|
|
(sin(pi/2 - a, evaluate=False), not is_numer)]
|
|
elif isinstance(fact, cos):
|
|
a = fact.args[0]
|
|
args += [(sin(pi/2 - a, evaluate=False), is_numer)]
|
|
elif isinstance(fact, cot):
|
|
a = fact.args[0]
|
|
args += [(sin(pi/2 - a, evaluate=False), is_numer),
|
|
(sin(a, evaluate=False), not is_numer)]
|
|
else:
|
|
raise exception(fact)
|
|
|
|
fac *= Mul(*facs)/Mul(*dfacs)
|
|
|
|
# 3)
|
|
an, ap, bm, bq = [], [], [], []
|
|
for gammas, plus, minus, is_numer in [(numer_gammas, an, bm, True),
|
|
(denom_gammas, bq, ap, False)]:
|
|
while gammas:
|
|
a, c = gammas.pop()
|
|
if a != -1 and a != +1:
|
|
# We use the gamma function multiplication theorem.
|
|
p = Abs(S(a))
|
|
newa = a/p
|
|
newc = c/p
|
|
if not a.is_Integer:
|
|
raise TypeError("a is not an integer")
|
|
for k in range(p):
|
|
gammas += [(newa, newc + k/p)]
|
|
if is_numer:
|
|
fac *= (2*pi)**((1 - p)/2) * p**(c - S.Half)
|
|
exponentials += [p**a]
|
|
else:
|
|
fac /= (2*pi)**((1 - p)/2) * p**(c - S.Half)
|
|
exponentials += [p**(-a)]
|
|
continue
|
|
if a == +1:
|
|
plus.append(1 - c)
|
|
else:
|
|
minus.append(c)
|
|
|
|
# 4)
|
|
# TODO
|
|
|
|
# 5)
|
|
arg = Mul(*exponentials)
|
|
|
|
# for testability, sort the arguments
|
|
an.sort(key=default_sort_key)
|
|
ap.sort(key=default_sort_key)
|
|
bm.sort(key=default_sort_key)
|
|
bq.sort(key=default_sort_key)
|
|
|
|
return (an, ap), (bm, bq), arg, exponent, fac
|
|
|
|
|
|
@_noconds_(True)
|
|
def _inverse_mellin_transform(F, s, x_, strip, as_meijerg=False):
|
|
""" A helper for the real inverse_mellin_transform function, this one here
|
|
assumes x to be real and positive. """
|
|
x = _dummy('t', 'inverse-mellin-transform', F, positive=True)
|
|
# Actually, we won't try integration at all. Instead we use the definition
|
|
# of the Meijer G function as a fairly general inverse mellin transform.
|
|
F = F.rewrite(gamma)
|
|
for g in [factor(F), expand_mul(F), expand(F)]:
|
|
if g.is_Add:
|
|
# do all terms separately
|
|
ress = [_inverse_mellin_transform(G, s, x, strip, as_meijerg,
|
|
noconds=False)
|
|
for G in g.args]
|
|
conds = [p[1] for p in ress]
|
|
ress = [p[0] for p in ress]
|
|
res = Add(*ress)
|
|
if not as_meijerg:
|
|
res = factor(res, gens=res.atoms(Heaviside))
|
|
return res.subs(x, x_), And(*conds)
|
|
|
|
try:
|
|
a, b, C, e, fac = _rewrite_gamma(g, s, strip[0], strip[1])
|
|
except IntegralTransformError:
|
|
continue
|
|
try:
|
|
G = meijerg(a, b, C/x**e)
|
|
except ValueError:
|
|
continue
|
|
if as_meijerg:
|
|
h = G
|
|
else:
|
|
try:
|
|
from sympy.simplify import hyperexpand
|
|
h = hyperexpand(G)
|
|
except NotImplementedError:
|
|
raise IntegralTransformError(
|
|
'Inverse Mellin', F, 'Could not calculate integral')
|
|
|
|
if h.is_Piecewise and len(h.args) == 3:
|
|
# XXX we break modularity here!
|
|
h = Heaviside(x - Abs(C))*h.args[0].args[0] \
|
|
+ Heaviside(Abs(C) - x)*h.args[1].args[0]
|
|
# We must ensure that the integral along the line we want converges,
|
|
# and return that value.
|
|
# See [L], 5.2
|
|
cond = [Abs(arg(G.argument)) < G.delta*pi]
|
|
# Note: we allow ">=" here, this corresponds to convergence if we let
|
|
# limits go to oo symmetrically. ">" corresponds to absolute convergence.
|
|
cond += [And(Or(len(G.ap) != len(G.bq), 0 >= re(G.nu) + 1),
|
|
Abs(arg(G.argument)) == G.delta*pi)]
|
|
cond = Or(*cond)
|
|
if cond == False:
|
|
raise IntegralTransformError(
|
|
'Inverse Mellin', F, 'does not converge')
|
|
return (h*fac).subs(x, x_), cond
|
|
|
|
raise IntegralTransformError('Inverse Mellin', F, '')
|
|
|
|
_allowed = None
|
|
|
|
|
|
class InverseMellinTransform(IntegralTransform):
|
|
"""
|
|
Class representing unevaluated inverse Mellin transforms.
|
|
|
|
For usage of this class, see the :class:`IntegralTransform` docstring.
|
|
|
|
For how to compute inverse Mellin transforms, see the
|
|
:func:`inverse_mellin_transform` docstring.
|
|
"""
|
|
|
|
_name = 'Inverse Mellin'
|
|
_none_sentinel = Dummy('None')
|
|
_c = Dummy('c')
|
|
|
|
def __new__(cls, F, s, x, a, b, **opts):
|
|
if a is None:
|
|
a = InverseMellinTransform._none_sentinel
|
|
if b is None:
|
|
b = InverseMellinTransform._none_sentinel
|
|
return IntegralTransform.__new__(cls, F, s, x, a, b, **opts)
|
|
|
|
@property
|
|
def fundamental_strip(self):
|
|
a, b = self.args[3], self.args[4]
|
|
if a is InverseMellinTransform._none_sentinel:
|
|
a = None
|
|
if b is InverseMellinTransform._none_sentinel:
|
|
b = None
|
|
return a, b
|
|
|
|
def _compute_transform(self, F, s, x, **hints):
|
|
# IntegralTransform's doit will cause this hint to exist, but
|
|
# InverseMellinTransform should ignore it
|
|
hints.pop('simplify', True)
|
|
global _allowed
|
|
if _allowed is None:
|
|
_allowed = {
|
|
exp, gamma, sin, cos, tan, cot, cosh, sinh, tanh, coth,
|
|
factorial, rf}
|
|
for f in postorder_traversal(F):
|
|
if f.is_Function and f.has(s) and f.func not in _allowed:
|
|
raise IntegralTransformError('Inverse Mellin', F,
|
|
'Component %s not recognised.' % f)
|
|
strip = self.fundamental_strip
|
|
return _inverse_mellin_transform(F, s, x, strip, **hints)
|
|
|
|
def _as_integral(self, F, s, x):
|
|
c = self.__class__._c
|
|
return Integral(F*x**(-s), (s, c - S.ImaginaryUnit*S.Infinity, c +
|
|
S.ImaginaryUnit*S.Infinity))/(2*S.Pi*S.ImaginaryUnit)
|
|
|
|
|
|
def inverse_mellin_transform(F, s, x, strip, **hints):
|
|
r"""
|
|
Compute the inverse Mellin transform of `F(s)` over the fundamental
|
|
strip given by ``strip=(a, b)``.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
This can be defined as
|
|
|
|
.. math:: f(x) = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} x^{-s} F(s) \mathrm{d}s,
|
|
|
|
for any `c` in the fundamental strip. Under certain regularity
|
|
conditions on `F` and/or `f`,
|
|
this recovers `f` from its Mellin transform `F`
|
|
(and vice versa), for positive real `x`.
|
|
|
|
One of `a` or `b` may be passed as ``None``; a suitable `c` will be
|
|
inferred.
|
|
|
|
If the integral cannot be computed in closed form, this function returns
|
|
an unevaluated :class:`InverseMellinTransform` object.
|
|
|
|
Note that this function will assume x to be positive and real, regardless
|
|
of the SymPy assumptions!
|
|
|
|
For a description of possible hints, refer to the docstring of
|
|
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import inverse_mellin_transform, oo, gamma
|
|
>>> from sympy.abc import x, s
|
|
>>> inverse_mellin_transform(gamma(s), s, x, (0, oo))
|
|
exp(-x)
|
|
|
|
The fundamental strip matters:
|
|
|
|
>>> f = 1/(s**2 - 1)
|
|
>>> inverse_mellin_transform(f, s, x, (-oo, -1))
|
|
x*(1 - 1/x**2)*Heaviside(x - 1)/2
|
|
>>> inverse_mellin_transform(f, s, x, (-1, 1))
|
|
-x*Heaviside(1 - x)/2 - Heaviside(x - 1)/(2*x)
|
|
>>> inverse_mellin_transform(f, s, x, (1, oo))
|
|
(1/2 - x**2/2)*Heaviside(1 - x)/x
|
|
|
|
See Also
|
|
========
|
|
|
|
mellin_transform
|
|
hankel_transform, inverse_hankel_transform
|
|
"""
|
|
return InverseMellinTransform(F, s, x, strip[0], strip[1]).doit(**hints)
|
|
|
|
|
|
##########################################################################
|
|
# Fourier Transform
|
|
##########################################################################
|
|
|
|
@_noconds_(True)
|
|
def _fourier_transform(f, x, k, a, b, name, simplify=True):
|
|
r"""
|
|
Compute a general Fourier-type transform
|
|
|
|
.. math::
|
|
|
|
F(k) = a \int_{-\infty}^{\infty} e^{bixk} f(x)\, dx.
|
|
|
|
For suitable choice of *a* and *b*, this reduces to the standard Fourier
|
|
and inverse Fourier transforms.
|
|
"""
|
|
F = integrate(a*f*exp(b*S.ImaginaryUnit*x*k), (x, S.NegativeInfinity, S.Infinity))
|
|
|
|
if not F.has(Integral):
|
|
return _simplify(F, simplify), S.true
|
|
|
|
integral_f = integrate(f, (x, S.NegativeInfinity, S.Infinity))
|
|
if integral_f in (S.NegativeInfinity, S.Infinity, S.NaN) or integral_f.has(Integral):
|
|
raise IntegralTransformError(name, f, 'function not integrable on real axis')
|
|
|
|
if not F.is_Piecewise:
|
|
raise IntegralTransformError(name, f, 'could not compute integral')
|
|
|
|
F, cond = F.args[0]
|
|
if F.has(Integral):
|
|
raise IntegralTransformError(name, f, 'integral in unexpected form')
|
|
|
|
return _simplify(F, simplify), cond
|
|
|
|
|
|
class FourierTypeTransform(IntegralTransform):
|
|
""" Base class for Fourier transforms."""
|
|
|
|
def a(self):
|
|
raise NotImplementedError(
|
|
"Class %s must implement a(self) but does not" % self.__class__)
|
|
|
|
def b(self):
|
|
raise NotImplementedError(
|
|
"Class %s must implement b(self) but does not" % self.__class__)
|
|
|
|
def _compute_transform(self, f, x, k, **hints):
|
|
return _fourier_transform(f, x, k,
|
|
self.a(), self.b(),
|
|
self.__class__._name, **hints)
|
|
|
|
def _as_integral(self, f, x, k):
|
|
a = self.a()
|
|
b = self.b()
|
|
return Integral(a*f*exp(b*S.ImaginaryUnit*x*k), (x, S.NegativeInfinity, S.Infinity))
|
|
|
|
|
|
class FourierTransform(FourierTypeTransform):
|
|
"""
|
|
Class representing unevaluated Fourier transforms.
|
|
|
|
For usage of this class, see the :class:`IntegralTransform` docstring.
|
|
|
|
For how to compute Fourier transforms, see the :func:`fourier_transform`
|
|
docstring.
|
|
"""
|
|
|
|
_name = 'Fourier'
|
|
|
|
def a(self):
|
|
return 1
|
|
|
|
def b(self):
|
|
return -2*S.Pi
|
|
|
|
|
|
def fourier_transform(f, x, k, **hints):
|
|
r"""
|
|
Compute the unitary, ordinary-frequency Fourier transform of ``f``, defined
|
|
as
|
|
|
|
.. math:: F(k) = \int_{-\infty}^\infty f(x) e^{-2\pi i x k} \mathrm{d} x.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
If the transform cannot be computed in closed form, this
|
|
function returns an unevaluated :class:`FourierTransform` object.
|
|
|
|
For other Fourier transform conventions, see the function
|
|
:func:`sympy.integrals.transforms._fourier_transform`.
|
|
|
|
For a description of possible hints, refer to the docstring of
|
|
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
|
|
Note that for this transform, by default ``noconds=True``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import fourier_transform, exp
|
|
>>> from sympy.abc import x, k
|
|
>>> fourier_transform(exp(-x**2), x, k)
|
|
sqrt(pi)*exp(-pi**2*k**2)
|
|
>>> fourier_transform(exp(-x**2), x, k, noconds=False)
|
|
(sqrt(pi)*exp(-pi**2*k**2), True)
|
|
|
|
See Also
|
|
========
|
|
|
|
inverse_fourier_transform
|
|
sine_transform, inverse_sine_transform
|
|
cosine_transform, inverse_cosine_transform
|
|
hankel_transform, inverse_hankel_transform
|
|
mellin_transform, laplace_transform
|
|
"""
|
|
return FourierTransform(f, x, k).doit(**hints)
|
|
|
|
|
|
class InverseFourierTransform(FourierTypeTransform):
|
|
"""
|
|
Class representing unevaluated inverse Fourier transforms.
|
|
|
|
For usage of this class, see the :class:`IntegralTransform` docstring.
|
|
|
|
For how to compute inverse Fourier transforms, see the
|
|
:func:`inverse_fourier_transform` docstring.
|
|
"""
|
|
|
|
_name = 'Inverse Fourier'
|
|
|
|
def a(self):
|
|
return 1
|
|
|
|
def b(self):
|
|
return 2*S.Pi
|
|
|
|
|
|
def inverse_fourier_transform(F, k, x, **hints):
|
|
r"""
|
|
Compute the unitary, ordinary-frequency inverse Fourier transform of `F`,
|
|
defined as
|
|
|
|
.. math:: f(x) = \int_{-\infty}^\infty F(k) e^{2\pi i x k} \mathrm{d} k.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
If the transform cannot be computed in closed form, this
|
|
function returns an unevaluated :class:`InverseFourierTransform` object.
|
|
|
|
For other Fourier transform conventions, see the function
|
|
:func:`sympy.integrals.transforms._fourier_transform`.
|
|
|
|
For a description of possible hints, refer to the docstring of
|
|
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
|
|
Note that for this transform, by default ``noconds=True``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import inverse_fourier_transform, exp, sqrt, pi
|
|
>>> from sympy.abc import x, k
|
|
>>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x)
|
|
exp(-x**2)
|
|
>>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x, noconds=False)
|
|
(exp(-x**2), True)
|
|
|
|
See Also
|
|
========
|
|
|
|
fourier_transform
|
|
sine_transform, inverse_sine_transform
|
|
cosine_transform, inverse_cosine_transform
|
|
hankel_transform, inverse_hankel_transform
|
|
mellin_transform, laplace_transform
|
|
"""
|
|
return InverseFourierTransform(F, k, x).doit(**hints)
|
|
|
|
|
|
##########################################################################
|
|
# Fourier Sine and Cosine Transform
|
|
##########################################################################
|
|
|
|
@_noconds_(True)
|
|
def _sine_cosine_transform(f, x, k, a, b, K, name, simplify=True):
|
|
"""
|
|
Compute a general sine or cosine-type transform
|
|
F(k) = a int_0^oo b*sin(x*k) f(x) dx.
|
|
F(k) = a int_0^oo b*cos(x*k) f(x) dx.
|
|
|
|
For suitable choice of a and b, this reduces to the standard sine/cosine
|
|
and inverse sine/cosine transforms.
|
|
"""
|
|
F = integrate(a*f*K(b*x*k), (x, S.Zero, S.Infinity))
|
|
|
|
if not F.has(Integral):
|
|
return _simplify(F, simplify), S.true
|
|
|
|
if not F.is_Piecewise:
|
|
raise IntegralTransformError(name, f, 'could not compute integral')
|
|
|
|
F, cond = F.args[0]
|
|
if F.has(Integral):
|
|
raise IntegralTransformError(name, f, 'integral in unexpected form')
|
|
|
|
return _simplify(F, simplify), cond
|
|
|
|
|
|
class SineCosineTypeTransform(IntegralTransform):
|
|
"""
|
|
Base class for sine and cosine transforms.
|
|
Specify cls._kern.
|
|
"""
|
|
|
|
def a(self):
|
|
raise NotImplementedError(
|
|
"Class %s must implement a(self) but does not" % self.__class__)
|
|
|
|
def b(self):
|
|
raise NotImplementedError(
|
|
"Class %s must implement b(self) but does not" % self.__class__)
|
|
|
|
|
|
def _compute_transform(self, f, x, k, **hints):
|
|
return _sine_cosine_transform(f, x, k,
|
|
self.a(), self.b(),
|
|
self.__class__._kern,
|
|
self.__class__._name, **hints)
|
|
|
|
def _as_integral(self, f, x, k):
|
|
a = self.a()
|
|
b = self.b()
|
|
K = self.__class__._kern
|
|
return Integral(a*f*K(b*x*k), (x, S.Zero, S.Infinity))
|
|
|
|
|
|
class SineTransform(SineCosineTypeTransform):
|
|
"""
|
|
Class representing unevaluated sine transforms.
|
|
|
|
For usage of this class, see the :class:`IntegralTransform` docstring.
|
|
|
|
For how to compute sine transforms, see the :func:`sine_transform`
|
|
docstring.
|
|
"""
|
|
|
|
_name = 'Sine'
|
|
_kern = sin
|
|
|
|
def a(self):
|
|
return sqrt(2)/sqrt(pi)
|
|
|
|
def b(self):
|
|
return S.One
|
|
|
|
|
|
def sine_transform(f, x, k, **hints):
|
|
r"""
|
|
Compute the unitary, ordinary-frequency sine transform of `f`, defined
|
|
as
|
|
|
|
.. math:: F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \sin(2\pi x k) \mathrm{d} x.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
If the transform cannot be computed in closed form, this
|
|
function returns an unevaluated :class:`SineTransform` object.
|
|
|
|
For a description of possible hints, refer to the docstring of
|
|
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
|
|
Note that for this transform, by default ``noconds=True``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import sine_transform, exp
|
|
>>> from sympy.abc import x, k, a
|
|
>>> sine_transform(x*exp(-a*x**2), x, k)
|
|
sqrt(2)*k*exp(-k**2/(4*a))/(4*a**(3/2))
|
|
>>> sine_transform(x**(-a), x, k)
|
|
2**(1/2 - a)*k**(a - 1)*gamma(1 - a/2)/gamma(a/2 + 1/2)
|
|
|
|
See Also
|
|
========
|
|
|
|
fourier_transform, inverse_fourier_transform
|
|
inverse_sine_transform
|
|
cosine_transform, inverse_cosine_transform
|
|
hankel_transform, inverse_hankel_transform
|
|
mellin_transform, laplace_transform
|
|
"""
|
|
return SineTransform(f, x, k).doit(**hints)
|
|
|
|
|
|
class InverseSineTransform(SineCosineTypeTransform):
|
|
"""
|
|
Class representing unevaluated inverse sine transforms.
|
|
|
|
For usage of this class, see the :class:`IntegralTransform` docstring.
|
|
|
|
For how to compute inverse sine transforms, see the
|
|
:func:`inverse_sine_transform` docstring.
|
|
"""
|
|
|
|
_name = 'Inverse Sine'
|
|
_kern = sin
|
|
|
|
def a(self):
|
|
return sqrt(2)/sqrt(pi)
|
|
|
|
def b(self):
|
|
return S.One
|
|
|
|
|
|
def inverse_sine_transform(F, k, x, **hints):
|
|
r"""
|
|
Compute the unitary, ordinary-frequency inverse sine transform of `F`,
|
|
defined as
|
|
|
|
.. math:: f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \sin(2\pi x k) \mathrm{d} k.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
If the transform cannot be computed in closed form, this
|
|
function returns an unevaluated :class:`InverseSineTransform` object.
|
|
|
|
For a description of possible hints, refer to the docstring of
|
|
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
|
|
Note that for this transform, by default ``noconds=True``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import inverse_sine_transform, exp, sqrt, gamma
|
|
>>> from sympy.abc import x, k, a
|
|
>>> inverse_sine_transform(2**((1-2*a)/2)*k**(a - 1)*
|
|
... gamma(-a/2 + 1)/gamma((a+1)/2), k, x)
|
|
x**(-a)
|
|
>>> inverse_sine_transform(sqrt(2)*k*exp(-k**2/(4*a))/(4*sqrt(a)**3), k, x)
|
|
x*exp(-a*x**2)
|
|
|
|
See Also
|
|
========
|
|
|
|
fourier_transform, inverse_fourier_transform
|
|
sine_transform
|
|
cosine_transform, inverse_cosine_transform
|
|
hankel_transform, inverse_hankel_transform
|
|
mellin_transform, laplace_transform
|
|
"""
|
|
return InverseSineTransform(F, k, x).doit(**hints)
|
|
|
|
|
|
class CosineTransform(SineCosineTypeTransform):
|
|
"""
|
|
Class representing unevaluated cosine transforms.
|
|
|
|
For usage of this class, see the :class:`IntegralTransform` docstring.
|
|
|
|
For how to compute cosine transforms, see the :func:`cosine_transform`
|
|
docstring.
|
|
"""
|
|
|
|
_name = 'Cosine'
|
|
_kern = cos
|
|
|
|
def a(self):
|
|
return sqrt(2)/sqrt(pi)
|
|
|
|
def b(self):
|
|
return S.One
|
|
|
|
|
|
def cosine_transform(f, x, k, **hints):
|
|
r"""
|
|
Compute the unitary, ordinary-frequency cosine transform of `f`, defined
|
|
as
|
|
|
|
.. math:: F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \cos(2\pi x k) \mathrm{d} x.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
If the transform cannot be computed in closed form, this
|
|
function returns an unevaluated :class:`CosineTransform` object.
|
|
|
|
For a description of possible hints, refer to the docstring of
|
|
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
|
|
Note that for this transform, by default ``noconds=True``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import cosine_transform, exp, sqrt, cos
|
|
>>> from sympy.abc import x, k, a
|
|
>>> cosine_transform(exp(-a*x), x, k)
|
|
sqrt(2)*a/(sqrt(pi)*(a**2 + k**2))
|
|
>>> cosine_transform(exp(-a*sqrt(x))*cos(a*sqrt(x)), x, k)
|
|
a*exp(-a**2/(2*k))/(2*k**(3/2))
|
|
|
|
See Also
|
|
========
|
|
|
|
fourier_transform, inverse_fourier_transform,
|
|
sine_transform, inverse_sine_transform
|
|
inverse_cosine_transform
|
|
hankel_transform, inverse_hankel_transform
|
|
mellin_transform, laplace_transform
|
|
"""
|
|
return CosineTransform(f, x, k).doit(**hints)
|
|
|
|
|
|
class InverseCosineTransform(SineCosineTypeTransform):
|
|
"""
|
|
Class representing unevaluated inverse cosine transforms.
|
|
|
|
For usage of this class, see the :class:`IntegralTransform` docstring.
|
|
|
|
For how to compute inverse cosine transforms, see the
|
|
:func:`inverse_cosine_transform` docstring.
|
|
"""
|
|
|
|
_name = 'Inverse Cosine'
|
|
_kern = cos
|
|
|
|
def a(self):
|
|
return sqrt(2)/sqrt(pi)
|
|
|
|
def b(self):
|
|
return S.One
|
|
|
|
|
|
def inverse_cosine_transform(F, k, x, **hints):
|
|
r"""
|
|
Compute the unitary, ordinary-frequency inverse cosine transform of `F`,
|
|
defined as
|
|
|
|
.. math:: f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \cos(2\pi x k) \mathrm{d} k.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
If the transform cannot be computed in closed form, this
|
|
function returns an unevaluated :class:`InverseCosineTransform` object.
|
|
|
|
For a description of possible hints, refer to the docstring of
|
|
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
|
|
Note that for this transform, by default ``noconds=True``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import inverse_cosine_transform, sqrt, pi
|
|
>>> from sympy.abc import x, k, a
|
|
>>> inverse_cosine_transform(sqrt(2)*a/(sqrt(pi)*(a**2 + k**2)), k, x)
|
|
exp(-a*x)
|
|
>>> inverse_cosine_transform(1/sqrt(k), k, x)
|
|
1/sqrt(x)
|
|
|
|
See Also
|
|
========
|
|
|
|
fourier_transform, inverse_fourier_transform,
|
|
sine_transform, inverse_sine_transform
|
|
cosine_transform
|
|
hankel_transform, inverse_hankel_transform
|
|
mellin_transform, laplace_transform
|
|
"""
|
|
return InverseCosineTransform(F, k, x).doit(**hints)
|
|
|
|
|
|
##########################################################################
|
|
# Hankel Transform
|
|
##########################################################################
|
|
|
|
@_noconds_(True)
|
|
def _hankel_transform(f, r, k, nu, name, simplify=True):
|
|
r"""
|
|
Compute a general Hankel transform
|
|
|
|
.. math:: F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r.
|
|
"""
|
|
F = integrate(f*besselj(nu, k*r)*r, (r, S.Zero, S.Infinity))
|
|
|
|
if not F.has(Integral):
|
|
return _simplify(F, simplify), S.true
|
|
|
|
if not F.is_Piecewise:
|
|
raise IntegralTransformError(name, f, 'could not compute integral')
|
|
|
|
F, cond = F.args[0]
|
|
if F.has(Integral):
|
|
raise IntegralTransformError(name, f, 'integral in unexpected form')
|
|
|
|
return _simplify(F, simplify), cond
|
|
|
|
|
|
class HankelTypeTransform(IntegralTransform):
|
|
"""
|
|
Base class for Hankel transforms.
|
|
"""
|
|
|
|
def doit(self, **hints):
|
|
return self._compute_transform(self.function,
|
|
self.function_variable,
|
|
self.transform_variable,
|
|
self.args[3],
|
|
**hints)
|
|
|
|
def _compute_transform(self, f, r, k, nu, **hints):
|
|
return _hankel_transform(f, r, k, nu, self._name, **hints)
|
|
|
|
def _as_integral(self, f, r, k, nu):
|
|
return Integral(f*besselj(nu, k*r)*r, (r, S.Zero, S.Infinity))
|
|
|
|
@property
|
|
def as_integral(self):
|
|
return self._as_integral(self.function,
|
|
self.function_variable,
|
|
self.transform_variable,
|
|
self.args[3])
|
|
|
|
|
|
class HankelTransform(HankelTypeTransform):
|
|
"""
|
|
Class representing unevaluated Hankel transforms.
|
|
|
|
For usage of this class, see the :class:`IntegralTransform` docstring.
|
|
|
|
For how to compute Hankel transforms, see the :func:`hankel_transform`
|
|
docstring.
|
|
"""
|
|
|
|
_name = 'Hankel'
|
|
|
|
|
|
def hankel_transform(f, r, k, nu, **hints):
|
|
r"""
|
|
Compute the Hankel transform of `f`, defined as
|
|
|
|
.. math:: F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
If the transform cannot be computed in closed form, this
|
|
function returns an unevaluated :class:`HankelTransform` object.
|
|
|
|
For a description of possible hints, refer to the docstring of
|
|
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
|
|
Note that for this transform, by default ``noconds=True``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import hankel_transform, inverse_hankel_transform
|
|
>>> from sympy import exp
|
|
>>> from sympy.abc import r, k, m, nu, a
|
|
|
|
>>> ht = hankel_transform(1/r**m, r, k, nu)
|
|
>>> ht
|
|
2*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/(2**m*gamma(m/2 + nu/2))
|
|
|
|
>>> inverse_hankel_transform(ht, k, r, nu)
|
|
r**(-m)
|
|
|
|
>>> ht = hankel_transform(exp(-a*r), r, k, 0)
|
|
>>> ht
|
|
a/(k**3*(a**2/k**2 + 1)**(3/2))
|
|
|
|
>>> inverse_hankel_transform(ht, k, r, 0)
|
|
exp(-a*r)
|
|
|
|
See Also
|
|
========
|
|
|
|
fourier_transform, inverse_fourier_transform
|
|
sine_transform, inverse_sine_transform
|
|
cosine_transform, inverse_cosine_transform
|
|
inverse_hankel_transform
|
|
mellin_transform, laplace_transform
|
|
"""
|
|
return HankelTransform(f, r, k, nu).doit(**hints)
|
|
|
|
|
|
class InverseHankelTransform(HankelTypeTransform):
|
|
"""
|
|
Class representing unevaluated inverse Hankel transforms.
|
|
|
|
For usage of this class, see the :class:`IntegralTransform` docstring.
|
|
|
|
For how to compute inverse Hankel transforms, see the
|
|
:func:`inverse_hankel_transform` docstring.
|
|
"""
|
|
|
|
_name = 'Inverse Hankel'
|
|
|
|
|
|
def inverse_hankel_transform(F, k, r, nu, **hints):
|
|
r"""
|
|
Compute the inverse Hankel transform of `F` defined as
|
|
|
|
.. math:: f(r) = \int_{0}^\infty F_\nu(k) J_\nu(k r) k \mathrm{d} k.
|
|
|
|
Explanation
|
|
===========
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If the transform cannot be computed in closed form, this
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function returns an unevaluated :class:`InverseHankelTransform` object.
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For a description of possible hints, refer to the docstring of
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:func:`sympy.integrals.transforms.IntegralTransform.doit`.
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Note that for this transform, by default ``noconds=True``.
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Examples
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========
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>>> from sympy import hankel_transform, inverse_hankel_transform
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>>> from sympy import exp
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>>> from sympy.abc import r, k, m, nu, a
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>>> ht = hankel_transform(1/r**m, r, k, nu)
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>>> ht
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2*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/(2**m*gamma(m/2 + nu/2))
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>>> inverse_hankel_transform(ht, k, r, nu)
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r**(-m)
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>>> ht = hankel_transform(exp(-a*r), r, k, 0)
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>>> ht
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a/(k**3*(a**2/k**2 + 1)**(3/2))
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>>> inverse_hankel_transform(ht, k, r, 0)
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exp(-a*r)
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See Also
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========
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fourier_transform, inverse_fourier_transform
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sine_transform, inverse_sine_transform
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cosine_transform, inverse_cosine_transform
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hankel_transform
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mellin_transform, laplace_transform
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"""
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return InverseHankelTransform(F, k, r, nu).doit(**hints)
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##########################################################################
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# Laplace Transform
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##########################################################################
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# Stub classes and functions that used to be here
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import sympy.integrals.laplace as _laplace
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LaplaceTransform = _laplace.LaplaceTransform
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laplace_transform = _laplace.laplace_transform
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InverseLaplaceTransform = _laplace.InverseLaplaceTransform
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inverse_laplace_transform = _laplace.inverse_laplace_transform
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