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rockyrocky
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Merge pull request #1340 from mathics/gamma-fns
Split out gamma functions
2 parents b162c07 + cb56d13 commit 8f45adc

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Lines changed: 116 additions & 103 deletions

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‎mathics/builtin/arithmetic.py‎

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@@ -1858,109 +1858,6 @@ class Factorial(PostfixOperator, _MPMathFunction):
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precedence = 610
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mpmath_name = "factorial"
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class Gamma(_MPMathMultiFunction):
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"""
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<dl>
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<dt>'Gamma[$z$]'
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<dd>is the gamma function on the complex number $z$.
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<dt>'Gamma[$z$, $x$]'
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<dd>is the upper incomplete gamma function.
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<dt>'Gamma[$z$, $x0$, $x1$]'
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<dd>is equivalent to 'Gamma[$z$, $x0$] - Gamma[$z$, $x1$]'.
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</dl>
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'Gamma[$z$]' is equivalent to '($z$ - 1)!':
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>> Simplify[Gamma[z] - (z - 1)!]
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= 0
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Exact arguments:
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>> Gamma[8]
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= 5040
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>> Gamma[1/2]
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= Sqrt[Pi]
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>> Gamma[1, x]
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= E ^ (-x)
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>> Gamma[0, x]
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= ExpIntegralE[1, x]
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Numeric arguments:
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>> Gamma[123.78]
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= 4.21078*^204
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>> Gamma[1. + I]
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= 0.498016 - 0.15495 I
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Both 'Gamma' and 'Factorial' functions are continuous:
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>> Plot[{Gamma[x], x!}, {x, 0, 4}]
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= -Graphics-
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## Issue 203
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#> N[Gamma[24/10], 100]
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= 1.242169344504305404913070252268300492431517240992022966055507541481863694148882652446155342679460339
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#> N[N[Gamma[24/10],100]/N[Gamma[14/10],100],100]
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= 1.400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
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#> % // Precision
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= 100.
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#> Gamma[1.*^20]
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: Overflow occurred in computation.
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= Overflow[]
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## Needs mpmath support for lowergamma
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#> Gamma[1., 2.]
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= Gamma[1., 2.]
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"""
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mpmath_names = {
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1: "gamma",
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}
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sympy_names = {
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1: "gamma",
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2: "uppergamma",
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}
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rules = {
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"Gamma[z_, x0_, x1_]": "Gamma[z, x0] - Gamma[z, x1]",
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"Gamma[1 + z_]": "z!",
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"Derivative[1][Gamma]": "(Gamma[#1]*PolyGamma[0, #1])&",
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"Derivative[1, 0][Gamma]": "(Gamma[#1, #2]*Log[#2] + MeijerG[{{}, {1, 1}}, {{0, 0, #1}, {}}, #2])&",
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"Derivative[0, 1][Gamma]": "(-(#2^(-1 + #1)/E^#2))&",
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}
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def get_sympy_names(self):
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return ["gamma", "uppergamma", "lowergamma"]
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def from_sympy(self, sympy_name, leaves):
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if sympy_name == "lowergamma":
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# lowergamma(z, x) -> Gamma[z, 0, x]
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z, x = leaves
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return Expression(self.get_name(), z, Integer0, x)
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else:
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return Expression(self.get_name(), *leaves)
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class Pochhammer(SympyFunction):
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"""
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<dl>
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<dt>'Pochhammer[$a$, $n$]'
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<dd>is the Pochhammer symbol (a)_n.
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</dl>
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>> Pochhammer[4, 8]
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= 6652800
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"""
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attributes = ("Listable", "NumericFunction", "Protected")
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sympy_name = "RisingFactorial"
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rules = {
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"Pochhammer[a_, n_]": "Gamma[a + n] / Gamma[a]",
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"Derivative[1,0][Pochhammer]": "(Pochhammer[#1, #2]*(-PolyGamma[0, #1] + PolyGamma[0, #1 + #2]))&",
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"Derivative[0,1][Pochhammer]": "(Pochhammer[#1, #2]*PolyGamma[0, #1 + #2])&",
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}
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class HarmonicNumber(_MPMathFunction):
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"""
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<dl>

‎mathics/builtin/specialfns/gamma.py‎

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"""
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Gamma and Related Functions
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"""
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from mathics.version import __version__ # noqa used in loading to check consistency.
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from mathics.builtin.arithmetic import _MPMathMultiFunction
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from mathics.builtin.base import SympyFunction
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from mathics.core.expression import Expression, Integer0
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class Gamma(_MPMathMultiFunction):
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"""
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In number theory the logarithm of the gamma function often appears. For positive real numbers, this can be evaluated as 'Log[Gamma[$z$]]'.
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<dl>
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<dt>'Gamma[$z$]'
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<dd>is the gamma function on the complex number $z$.
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<dt>'Gamma[$z$, $x$]'
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<dd>is the upper incomplete gamma function.
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<dt>'Gamma[$z$, $x0$, $x1$]'
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<dd>is equivalent to 'Gamma[$z$, $x0$] - Gamma[$z$, $x1$]'.
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</dl>
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'Gamma[$z$]' is equivalent to '($z$ - 1)!':
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>> Simplify[Gamma[z] - (z - 1)!]
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= 0
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Exact arguments:
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>> Gamma[8]
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= 5040
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>> Gamma[1/2]
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= Sqrt[Pi]
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>> Gamma[1, x]
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= E ^ (-x)
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>> Gamma[0, x]
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= ExpIntegralE[1, x]
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Numeric arguments:
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>> Gamma[123.78]
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= 4.21078*^204
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>> Gamma[1. + I]
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= 0.498016 - 0.15495 I
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Both 'Gamma' and 'Factorial' functions are continuous:
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>> Plot[{Gamma[x], x!}, {x, 0, 4}]
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= -Graphics-
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## Issue 203
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#> N[Gamma[24/10], 100]
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= 1.242169344504305404913070252268300492431517240992022966055507541481863694148882652446155342679460339
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#> N[N[Gamma[24/10],100]/N[Gamma[14/10],100],100]
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= 1.400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
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#> % // Precision
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= 100.
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#> Gamma[1.*^20]
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: Overflow occurred in computation.
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= Overflow[]
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## Needs mpmath support for lowergamma
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#> Gamma[1., 2.]
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= Gamma[1., 2.]
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"""
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mpmath_names = {
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1: "gamma",
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}
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sympy_names = {
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1: "gamma",
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2: "uppergamma",
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}
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rules = {
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"Gamma[z_, x0_, x1_]": "Gamma[z, x0] - Gamma[z, x1]",
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"Gamma[1 + z_]": "z!",
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"Derivative[1][Gamma]": "(Gamma[#1]*PolyGamma[0, #1])&",
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"Derivative[1, 0][Gamma]": "(Gamma[#1, #2]*Log[#2] + MeijerG[{{}, {1, 1}}, {{0, 0, #1}, {}}, #2])&",
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"Derivative[0, 1][Gamma]": "(-(#2^(-1 + #1)/E^#2))&",
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}
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def get_sympy_names(self):
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return ["gamma", "uppergamma", "lowergamma"]
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def from_sympy(self, sympy_name, leaves):
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if sympy_name == "lowergamma":
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# lowergamma(z, x) -> Gamma[z, 0, x]
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z, x = leaves
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return Expression(self.get_name(), z, Integer0, x)
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else:
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return Expression(self.get_name(), *leaves)
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class Pochhammer(SympyFunction):
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"""
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The Pochhammer symbol or rising factorial often appears in series expansions for hypergeometric functions.
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The Pochammer symbol has a definie value even when the gamma functions which appear in its definition are infinite.
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<dl>
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<dt>'Pochhammer[$a$, $n$]'
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<dd>is the Pochhammer symbol (a)_n.
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</dl>
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>> Pochhammer[4, 8]
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= 6652800
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"""
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attributes = ("Listable", "NumericFunction", "Protected")
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sympy_name = "RisingFactorial"
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rules = {
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"Pochhammer[a_, n_]": "Gamma[a + n] / Gamma[a]",
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"Derivative[1,0][Pochhammer]": "(Pochhammer[#1, #2]*(-PolyGamma[0, #1] + PolyGamma[0, #1 + #2]))&",
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"Derivative[0,1][Pochhammer]": "(Pochhammer[#1, #2]*PolyGamma[0, #1 + #2])&",
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}

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