added aseries for gamma, and other series methods for incomplete gamma

fucntions Signed-off-by: arnabnandikgp <arnabnandi2002@gmail.com>
sympy · Aug 26, 2024 · 8eb7f61 · 8eb7f61
1 parent b354658
commit 8eb7f61
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diff --git a/sympy/functions/special/error_functions.py b/sympy/functions/special/error_functions.py
@@ -1434,6 +1434,15 @@ def _eval_rewrite_as_Si(self, nu, z, **kwargs):
     _eval_rewrite_as_Chi = _eval_rewrite_as_Si
     _eval_rewrite_as_Shi = _eval_rewrite_as_Si
 
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        nu, z = self.args
+        if nu == 1:
+            arg = -log(z) - EulerGamma
+        if nu.is_real and nu > 1:
+            arg = 1/(nu - 1)
+            return arg.as_leading_term(x, logx=logx)
+        return super()._eval_as_leading_term(x, logx, cdir)
+
     def _eval_nseries(self, x, n, logx, cdir=0):
         if not self.args[0].has(x):
             nu = self.args[0]

diff --git a/sympy/functions/special/gamma_functions.py b/sympy/functions/special/gamma_functions.py
@@ -214,6 +214,15 @@ def _eval_as_leading_term(self, x, logx=None, cdir=0):
             return self.func(x0)
         raise PoleError()
 
+    def _eval_aseries(self, n, args0, x, logx):
+        point = args0[0]
+        if point is S.Infinity:
+            z = self.args[0]
+            r = exp((log(z)*(z - S.Half) - z + log(2*pi)/2).expand())
+            l = [bernoulli(2*k) / (2*k*(2*k - 1)*z**(2*k - 1)) for k in range(1, n)]
+            return (exp(Add(*l))._eval_nseries(x, n, logx))*r
+        return super()._eval_aseries(n, args0, x, logx)
+
 
 ###############################################################################
 ################## LOWER and UPPER INCOMPLETE GAMMA FUNCTIONS #################
@@ -380,15 +389,30 @@ def _eval_aseries(self, n, args0, x, logx):
             return coeff*sum_expr + o
         return super()._eval_aseries(n, args0, x, logx)
 
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        from sympy.series.order import O
+        s, z = self.args
+        if z.has(x):
+            coeff = z**s*exp(-z)*gamma(s)
+            sum_expr = sum(z**k/gamma(s + k + 1) for k in range(n - 1))
+            o = O(z**s*s**(-n))
+            return coeff*sum_expr + o
+        return super()._eval_nseries(x, n, logx)
+
     def _eval_rewrite_as_uppergamma(self, s, x, **kwargs):
         return gamma(s) - uppergamma(s, x)
 
+    _eval_rewrite_as_tractable = _eval_rewrite_as_uppergamma
+
     def _eval_rewrite_as_expint(self, s, x, **kwargs):
         from sympy.functions.special.error_functions import expint
         if s.is_integer and s.is_nonpositive:
             return self
         return self.rewrite(uppergamma).rewrite(expint)
 
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        return self.rewrite(uppergamma)._eval_as_leading_term(x, logx=logx, cdir=cdir)
+
     def _eval_is_zero(self):
         x = self.args[1]
         if x.is_zero:
@@ -556,6 +580,10 @@ def _eval_rewrite_as_expint(self, s, x, **kwargs):
         from sympy.functions.special.error_functions import expint
         return expint(1 - s, x)*x**s
 
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        from sympy.functions.special.error_functions import expint
+        return self.rewrite(expint)._eval_as_leading_term(x, logx=logx, cdir=cdir)
+
 
 ###############################################################################
 ###################### POLYGAMMA and LOGGAMMA FUNCTIONS #######################

diff --git a/sympy/functions/special/tests/test_gamma_functions.py b/sympy/functions/special/tests/test_gamma_functions.py
@@ -106,6 +106,10 @@ def test_gamma_series():
         -1/(x + 1) + EulerGamma - 1 + (x + 1)*(-1 - pi**2/12 - EulerGamma**2/2 + \
        EulerGamma) + (x + 1)**2*(-1 - pi**2/12 - EulerGamma**2/2 + EulerGamma**3/6 - \
        polygamma(2, 1)/6 + EulerGamma*pi**2/12 + EulerGamma) + O((x + 1)**3, (x, -1))
+    assert gamma(x + 1).series(x, oo, 3) == sqrt(2)*sqrt(pi)*sqrt(x + 1)*(-23/(288*x**2) + \
+            1/(12*x) + 1 + O(x**(-3), (x, oo)))*exp(x*log(x + 1) - x - 1)
+    assert gamma(x).series(x, oo, 3) == sqrt(2)*sqrt(pi)*(1/(288*x**2) + 1/(12*x) + 1 + \
+            O(x**(-3), (x, oo)))*sqrt(1/x)*exp(-x*log(1/x) - x)
 
 
 def tn_branch(s, func):

diff --git a/sympy/series/tests/test_limits.py b/sympy/series/tests/test_limits.py
@@ -19,7 +19,7 @@
                                                       atan, cos, cot, csc, sec, sin, tan)
 from sympy.functions.special.bessel import (besseli, bessely, besselj, besselk)
 from sympy.functions.special.error_functions import (Ei, erf, erfc, erfi, fresnelc, fresnels)
-from sympy.functions.special.gamma_functions import (digamma, gamma, uppergamma)
+from sympy.functions.special.gamma_functions import (digamma, gamma, lowergamma, uppergamma)
 from sympy.functions.special.hyper import meijerg
 from sympy.integrals.integrals import (Integral, integrate)
 from sympy.series.limits import (Limit, limit)
@@ -392,6 +392,16 @@ def test_bessel_functions_at_infinity():
     assert limit(besselk(0, x), x, oo) == 0
 
 
+def test_lowergamma_at_origin():
+    a, x = symbols('a x')
+    assert limit(lowergamma(a, x), a, 0) == oo
+    assert limit(lowergamma(a, x), a, 0, '-') == -oo
+    assert limit(lowergamma(I*a, x), a, 0) == -oo*I
+    assert limit(lowergamma(I*a, x), a, 0, '-') == oo*I
+    assert limit(lowergamma(a, 1), a, 0) == oo
+    assert limit(x*lowergamma(x, 1)/gamma(x + 1), x, 0) == 1
+
+
 @XFAIL
 def test_doit2():
     f = Integral(2 * x, x)