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498 lines
18 KiB
498 lines
18 KiB
from sympy.core import Function, S, Mul, Pow, Add
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from sympy.core.sorting import ordered, default_sort_key
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from sympy.core.function import expand_func
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from sympy.core.symbol import Dummy
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from sympy.functions import gamma, sqrt, sin
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from sympy.polys import factor, cancel
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from sympy.utilities.iterables import sift, uniq
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def gammasimp(expr):
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r"""
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Simplify expressions with gamma functions.
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Explanation
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===========
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This function takes as input an expression containing gamma
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functions or functions that can be rewritten in terms of gamma
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functions and tries to minimize the number of those functions and
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reduce the size of their arguments.
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The algorithm works by rewriting all gamma functions as expressions
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involving rising factorials (Pochhammer symbols) and applies
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recurrence relations and other transformations applicable to rising
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factorials, to reduce their arguments, possibly letting the resulting
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rising factorial to cancel. Rising factorials with the second argument
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being an integer are expanded into polynomial forms and finally all
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other rising factorial are rewritten in terms of gamma functions.
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Then the following two steps are performed.
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1. Reduce the number of gammas by applying the reflection theorem
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gamma(x)*gamma(1-x) == pi/sin(pi*x).
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2. Reduce the number of gammas by applying the multiplication theorem
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gamma(x)*gamma(x+1/n)*...*gamma(x+(n-1)/n) == C*gamma(n*x).
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It then reduces the number of prefactors by absorbing them into gammas
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where possible and expands gammas with rational argument.
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All transformation rules can be found (or were derived from) here:
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.. [1] https://functions.wolfram.com/GammaBetaErf/Pochhammer/17/01/02/
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.. [2] https://functions.wolfram.com/GammaBetaErf/Pochhammer/27/01/0005/
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Examples
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========
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>>> from sympy.simplify import gammasimp
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>>> from sympy import gamma, Symbol
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>>> from sympy.abc import x
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>>> n = Symbol('n', integer = True)
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>>> gammasimp(gamma(x)/gamma(x - 3))
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(x - 3)*(x - 2)*(x - 1)
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>>> gammasimp(gamma(n + 3))
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gamma(n + 3)
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"""
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expr = expr.rewrite(gamma)
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# compute_ST will be looking for Functions and we don't want
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# it looking for non-gamma functions: issue 22606
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# so we mask free, non-gamma functions
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f = expr.atoms(Function)
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# take out gammas
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gammas = {i for i in f if isinstance(i, gamma)}
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if not gammas:
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return expr # avoid side effects like factoring
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f -= gammas
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# keep only those without bound symbols
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f = f & expr.as_dummy().atoms(Function)
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if f:
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dum, fun, simp = zip(*[
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(Dummy(), fi, fi.func(*[
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_gammasimp(a, as_comb=False) for a in fi.args]))
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for fi in ordered(f)])
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d = expr.xreplace(dict(zip(fun, dum)))
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return _gammasimp(d, as_comb=False).xreplace(dict(zip(dum, simp)))
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return _gammasimp(expr, as_comb=False)
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def _gammasimp(expr, as_comb):
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"""
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Helper function for gammasimp and combsimp.
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Explanation
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===========
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Simplifies expressions written in terms of gamma function. If
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as_comb is True, it tries to preserve integer arguments. See
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docstring of gammasimp for more information. This was part of
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combsimp() in combsimp.py.
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"""
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expr = expr.replace(gamma,
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lambda n: _rf(1, (n - 1).expand()))
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if as_comb:
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expr = expr.replace(_rf,
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lambda a, b: gamma(b + 1))
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else:
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expr = expr.replace(_rf,
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lambda a, b: gamma(a + b)/gamma(a))
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def rule_gamma(expr, level=0):
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""" Simplify products of gamma functions further. """
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if expr.is_Atom:
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return expr
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def gamma_rat(x):
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# helper to simplify ratios of gammas
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was = x.count(gamma)
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xx = x.replace(gamma, lambda n: _rf(1, (n - 1).expand()
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).replace(_rf, lambda a, b: gamma(a + b)/gamma(a)))
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if xx.count(gamma) < was:
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x = xx
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return x
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def gamma_factor(x):
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# return True if there is a gamma factor in shallow args
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if isinstance(x, gamma):
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return True
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if x.is_Add or x.is_Mul:
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return any(gamma_factor(xi) for xi in x.args)
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if x.is_Pow and (x.exp.is_integer or x.base.is_positive):
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return gamma_factor(x.base)
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return False
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# recursion step
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if level == 0:
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expr = expr.func(*[rule_gamma(x, level + 1) for x in expr.args])
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level += 1
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if not expr.is_Mul:
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return expr
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# non-commutative step
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if level == 1:
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args, nc = expr.args_cnc()
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if not args:
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return expr
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if nc:
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return rule_gamma(Mul._from_args(args), level + 1)*Mul._from_args(nc)
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level += 1
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# pure gamma handling, not factor absorption
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if level == 2:
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T, F = sift(expr.args, gamma_factor, binary=True)
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gamma_ind = Mul(*F)
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d = Mul(*T)
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nd, dd = d.as_numer_denom()
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for ipass in range(2):
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args = list(ordered(Mul.make_args(nd)))
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for i, ni in enumerate(args):
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if ni.is_Add:
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ni, dd = Add(*[
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rule_gamma(gamma_rat(a/dd), level + 1) for a in ni.args]
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).as_numer_denom()
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args[i] = ni
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if not dd.has(gamma):
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break
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nd = Mul(*args)
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if ipass == 0 and not gamma_factor(nd):
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break
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nd, dd = dd, nd # now process in reversed order
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expr = gamma_ind*nd/dd
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if not (expr.is_Mul and (gamma_factor(dd) or gamma_factor(nd))):
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return expr
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level += 1
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# iteration until constant
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if level == 3:
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while True:
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was = expr
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expr = rule_gamma(expr, 4)
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if expr == was:
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return expr
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numer_gammas = []
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denom_gammas = []
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numer_others = []
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denom_others = []
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def explicate(p):
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if p is S.One:
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return None, []
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b, e = p.as_base_exp()
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if e.is_Integer:
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if isinstance(b, gamma):
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return True, [b.args[0]]*e
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else:
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return False, [b]*e
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else:
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return False, [p]
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newargs = list(ordered(expr.args))
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while newargs:
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n, d = newargs.pop().as_numer_denom()
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isg, l = explicate(n)
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if isg:
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numer_gammas.extend(l)
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elif isg is False:
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numer_others.extend(l)
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isg, l = explicate(d)
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if isg:
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denom_gammas.extend(l)
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elif isg is False:
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denom_others.extend(l)
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# =========== level 2 work: pure gamma manipulation =========
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if not as_comb:
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# Try to reduce the number of gamma factors by applying the
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# reflection formula gamma(x)*gamma(1-x) = pi/sin(pi*x)
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for gammas, numer, denom in [(
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numer_gammas, numer_others, denom_others),
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(denom_gammas, denom_others, numer_others)]:
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new = []
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while gammas:
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g1 = gammas.pop()
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if g1.is_integer:
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new.append(g1)
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continue
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for i, g2 in enumerate(gammas):
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n = g1 + g2 - 1
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if not n.is_Integer:
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continue
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numer.append(S.Pi)
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denom.append(sin(S.Pi*g1))
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gammas.pop(i)
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if n > 0:
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for k in range(n):
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numer.append(1 - g1 + k)
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elif n < 0:
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for k in range(-n):
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denom.append(-g1 - k)
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break
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else:
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new.append(g1)
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# /!\ updating IN PLACE
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gammas[:] = new
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# Try to reduce the number of gammas by using the duplication
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# theorem to cancel an upper and lower: gamma(2*s)/gamma(s) =
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# 2**(2*s + 1)/(4*sqrt(pi))*gamma(s + 1/2). Although this could
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# be done with higher argument ratios like gamma(3*x)/gamma(x),
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# this would not reduce the number of gammas as in this case.
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for ng, dg, no, do in [(numer_gammas, denom_gammas, numer_others,
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denom_others),
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(denom_gammas, numer_gammas, denom_others,
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numer_others)]:
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while True:
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for x in ng:
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for y in dg:
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n = x - 2*y
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if n.is_Integer:
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break
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else:
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continue
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break
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else:
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break
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ng.remove(x)
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dg.remove(y)
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if n > 0:
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for k in range(n):
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no.append(2*y + k)
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elif n < 0:
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for k in range(-n):
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do.append(2*y - 1 - k)
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ng.append(y + S.Half)
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no.append(2**(2*y - 1))
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do.append(sqrt(S.Pi))
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# Try to reduce the number of gamma factors by applying the
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# multiplication theorem (used when n gammas with args differing
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# by 1/n mod 1 are encountered).
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#
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# run of 2 with args differing by 1/2
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#
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# >>> gammasimp(gamma(x)*gamma(x+S.Half))
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# 2*sqrt(2)*2**(-2*x - 1/2)*sqrt(pi)*gamma(2*x)
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#
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# run of 3 args differing by 1/3 (mod 1)
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#
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# >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(2)/3))
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# 6*3**(-3*x - 1/2)*pi*gamma(3*x)
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# >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(5)/3))
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# 2*3**(-3*x - 1/2)*pi*(3*x + 2)*gamma(3*x)
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#
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def _run(coeffs):
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# find runs in coeffs such that the difference in terms (mod 1)
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# of t1, t2, ..., tn is 1/n
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u = list(uniq(coeffs))
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for i in range(len(u)):
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dj = ([((u[j] - u[i]) % 1, j) for j in range(i + 1, len(u))])
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for one, j in dj:
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if one.p == 1 and one.q != 1:
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n = one.q
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got = [i]
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get = list(range(1, n))
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for d, j in dj:
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m = n*d
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if m.is_Integer and m in get:
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get.remove(m)
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got.append(j)
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if not get:
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break
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else:
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continue
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for i, j in enumerate(got):
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c = u[j]
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coeffs.remove(c)
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got[i] = c
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return one.q, got[0], got[1:]
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def _mult_thm(gammas, numer, denom):
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# pull off and analyze the leading coefficient from each gamma arg
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# looking for runs in those Rationals
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# expr -> coeff + resid -> rats[resid] = coeff
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rats = {}
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for g in gammas:
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c, resid = g.as_coeff_Add()
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rats.setdefault(resid, []).append(c)
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# look for runs in Rationals for each resid
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keys = sorted(rats, key=default_sort_key)
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for resid in keys:
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coeffs = sorted(rats[resid])
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new = []
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while True:
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run = _run(coeffs)
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if run is None:
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break
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# process the sequence that was found:
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# 1) convert all the gamma functions to have the right
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# argument (could be off by an integer)
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# 2) append the factors corresponding to the theorem
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# 3) append the new gamma function
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n, ui, other = run
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# (1)
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for u in other:
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con = resid + u - 1
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for k in range(int(u - ui)):
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numer.append(con - k)
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con = n*(resid + ui) # for (2) and (3)
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# (2)
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numer.append((2*S.Pi)**(S(n - 1)/2)*
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n**(S.Half - con))
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# (3)
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new.append(con)
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# restore resid to coeffs
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rats[resid] = [resid + c for c in coeffs] + new
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# rebuild the gamma arguments
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g = []
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for resid in keys:
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g += rats[resid]
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# /!\ updating IN PLACE
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gammas[:] = g
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for l, numer, denom in [(numer_gammas, numer_others, denom_others),
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(denom_gammas, denom_others, numer_others)]:
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_mult_thm(l, numer, denom)
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# =========== level >= 2 work: factor absorption =========
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if level >= 2:
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# Try to absorb factors into the gammas: x*gamma(x) -> gamma(x + 1)
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# and gamma(x)/(x - 1) -> gamma(x - 1)
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# This code (in particular repeated calls to find_fuzzy) can be very
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# slow.
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def find_fuzzy(l, x):
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if not l:
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return
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S1, T1 = compute_ST(x)
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for y in l:
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S2, T2 = inv[y]
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if T1 != T2 or (not S1.intersection(S2) and
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(S1 != set() or S2 != set())):
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continue
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# XXX we want some simplification (e.g. cancel or
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# simplify) but no matter what it's slow.
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a = len(cancel(x/y).free_symbols)
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b = len(x.free_symbols)
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c = len(y.free_symbols)
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# TODO is there a better heuristic?
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if a == 0 and (b > 0 or c > 0):
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return y
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# We thus try to avoid expensive calls by building the following
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# "invariants": For every factor or gamma function argument
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# - the set of free symbols S
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# - the set of functional components T
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# We will only try to absorb if T1==T2 and (S1 intersect S2 != emptyset
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# or S1 == S2 == emptyset)
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inv = {}
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def compute_ST(expr):
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if expr in inv:
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return inv[expr]
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return (expr.free_symbols, expr.atoms(Function).union(
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{e.exp for e in expr.atoms(Pow)}))
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def update_ST(expr):
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inv[expr] = compute_ST(expr)
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for expr in numer_gammas + denom_gammas + numer_others + denom_others:
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update_ST(expr)
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for gammas, numer, denom in [(
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numer_gammas, numer_others, denom_others),
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(denom_gammas, denom_others, numer_others)]:
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new = []
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while gammas:
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g = gammas.pop()
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cont = True
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while cont:
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cont = False
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y = find_fuzzy(numer, g)
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if y is not None:
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numer.remove(y)
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if y != g:
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numer.append(y/g)
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update_ST(y/g)
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g += 1
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cont = True
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y = find_fuzzy(denom, g - 1)
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if y is not None:
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denom.remove(y)
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if y != g - 1:
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numer.append((g - 1)/y)
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update_ST((g - 1)/y)
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g -= 1
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cont = True
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new.append(g)
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# /!\ updating IN PLACE
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gammas[:] = new
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# =========== rebuild expr ==================================
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return Mul(*[gamma(g) for g in numer_gammas]) \
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/ Mul(*[gamma(g) for g in denom_gammas]) \
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* Mul(*numer_others) / Mul(*denom_others)
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was = factor(expr)
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# (for some reason we cannot use Basic.replace in this case)
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expr = rule_gamma(was)
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if expr != was:
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expr = factor(expr)
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expr = expr.replace(gamma,
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lambda n: expand_func(gamma(n)) if n.is_Rational else gamma(n))
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return expr
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class _rf(Function):
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@classmethod
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def eval(cls, a, b):
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if b.is_Integer:
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if not b:
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return S.One
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n = int(b)
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if n > 0:
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return Mul(*[a + i for i in range(n)])
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elif n < 0:
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return 1/Mul(*[a - i for i in range(1, -n + 1)])
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else:
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if b.is_Add:
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c, _b = b.as_coeff_Add()
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if c.is_Integer:
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if c > 0:
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return _rf(a, _b)*_rf(a + _b, c)
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elif c < 0:
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return _rf(a, _b)/_rf(a + _b + c, -c)
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if a.is_Add:
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c, _a = a.as_coeff_Add()
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if c.is_Integer:
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if c > 0:
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return _rf(_a, b)*_rf(_a + b, c)/_rf(_a, c)
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elif c < 0:
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return _rf(_a, b)*_rf(_a + c, -c)/_rf(_a + b + c, -c)
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