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80 lines
2.2 KiB
ReStructuredText
80 lines
2.2 KiB
ReStructuredText
Factorials and gamma functions
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------------------------------
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Factorials and factorial-like sums and products are basic tools of
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combinatorics and number theory. Much like the exponential function is
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fundamental to differential equations and analysis in general, the factorial
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function (and its extension to complex numbers, the gamma function) is
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fundamental to difference equations and functional equations.
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A large selection of factorial-like functions is implemented in mpmath. All
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functions support complex arguments, and arguments may be arbitrarily large.
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Results are numerical approximations, so to compute *exact* values a high
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enough precision must be set manually::
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>>> from mpmath import mp, fac
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>>> mp.dps = 15
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>>> mp.pretty = True
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>>> fac(100)
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9.33262154439442e+157
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>>> print(int(_)) # most digits are wrong
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93326215443944150965646704795953882578400970373184098831012889540582227238570431295066113089288327277825849664006524270554535976289719382852181865895959724032
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>>> mp.dps = 160
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>>> fac(100)
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93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000.0
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The gamma and polygamma functions are closely related to :doc:`zeta`. See also
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:doc:`qfunctions` for q-analogs of factorial-like functions.
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Factorials
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..........
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.. autofunction:: mpmath.factorial
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.. autofunction:: mpmath.fac2
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Binomial coefficients
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.....................
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.. autofunction:: mpmath.binomial
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Gamma function
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..............
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.. autofunction:: mpmath.gamma
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.. autofunction:: mpmath.rgamma
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.. autofunction:: mpmath.gammaprod
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.. autofunction:: mpmath.loggamma
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Rising and falling factorials
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.............................
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.. autofunction:: mpmath.rf
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.. autofunction:: mpmath.ff
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Beta function
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.............
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.. autofunction:: mpmath.beta
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.. autofunction:: mpmath.betainc
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Super- and hyperfactorials
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..........................
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.. autofunction:: mpmath.superfac
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.. autofunction:: mpmath.hyperfac
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.. autofunction:: mpmath.barnesg
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Polygamma functions and harmonic numbers
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........................................
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.. autofunction:: mpmath.psi
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.. autofunction:: mpmath.digamma
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.. autofunction:: mpmath.harmonic
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