def get_sol_2F1_hypergeometric(eq, func, match_object):
x = func.args[0]
from sympy.simplify.hyperexpand import hyperexpand
from sympy.polys.polytools import factor
C0, C1 = get_numbered_constants(eq, num=2)
a = match_object['a']
b = match_object['b']
c = match_object['c']
A = match_object['A']
sol = None
if c.is_integer == False:
sol = C0 * hyper([a, b], [c], x) + C1 * hyper([a - c + 1, b - c + 1],
[2 - c], x) * x**(1 - c)
elif c == 1:
y2 = Integral(
exp(Integral((-(a + b + 1) * x + c) / (x**2 - x), x)) /
(hyperexpand(hyper([a, b], [c], x))**2), x) * hyper([a, b], [c], x)
sol = C0 * hyper([a, b], [c], x) + C1 * y2
elif (c - a - b).is_integer == False:
sol = C0 * hyper([a, b], [1 + a + b - c], 1 - x) + C1 * hyper(
[c - a, c - b], [1 + c - a - b], 1 - x) * (1 - x)**(c - a - b)
if sol:
# applying transformation in the solution
subs = match_object['mobius']
dtdx = simplify(1 / (subs.diff(x)))
_B = ((a + b + 1) * x - c).subs(x, subs) * dtdx
_B = factor(_B + ((x**2 - x).subs(x, subs)) * (dtdx.diff(x) * dtdx))
_A = factor((x**2 - x).subs(x, subs) * (dtdx**2))
e = exp(logcombine(Integral(cancel(_B / (2 * _A)), x), force=True))
sol = sol.subs(x, match_object['mobius'])
sol = sol.subs(x, x**match_object['k'])
e = e.subs(x, x**match_object['k'])
if not A.is_zero:
e1 = Integral(A / 2, x)
e1 = exp(logcombine(e1, force=True))
sol = cancel((e / e1) * x**((-match_object['k'] + 1) / 2)) * sol
sol = Eq(func, sol)
return sol
sol = cancel((e) * x**((-match_object['k'] + 1) / 2)) * sol
sol = Eq(func, sol)
return sol
def sign(e, x):
"""
Returns a sign of an expression e(x) for x->oo.
::
e > 0 for x sufficiently large ... 1
e == 0 for x sufficiently large ... 0
e < 0 for x sufficiently large ... -1
The result of this function is currently undefined if e changes sign
arbitrarily often for arbitrarily large x (e.g. sin(x)).
Note that this returns zero only if e is *constantly* zero
for x sufficiently large. [If e is constant, of course, this is just
the same thing as the sign of e.]
"""
if not isinstance(e, Basic):
raise TypeError("e should be an instance of Basic")
if e.is_positive:
return 1
elif e.is_negative:
return -1
elif e.is_zero:
return 0
elif not e.has(x):
from sympy.simplify import logcombine
e = logcombine(e)
return _sign(e)
elif e == x:
return 1
elif e.is_Mul:
a, b = e.as_two_terms()
sa = sign(a, x)
if not sa:
return 0
return sa * sign(b, x)
elif isinstance(e, exp):
return 1
elif e.is_Pow:
if e.base == S.Exp1:
return 1
s = sign(e.base, x)
if s == 1:
return 1
if e.exp.is_Integer:
return s**e.exp
elif isinstance(e, log):
return sign(e.args[0] - 1, x)
# if all else fails, do it the hard way
c0, e0 = mrv_leadterm(e, x)
return sign(c0, x)