Module « scipy.special »
Signature de la fonction gamma
Description
gamma.__doc__
gamma(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])
gamma(z)
gamma function.
The gamma function is defined as
.. math::
\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt
for :math:`\Re(z) > 0` and is extended to the rest of the complex
plane by analytic continuation. See [dlmf]_ for more details.
Parameters
----------
z : array_like
Real or complex valued argument
Returns
-------
scalar or ndarray
Values of the gamma function
Notes
-----
The gamma function is often referred to as the generalized
factorial since :math:`\Gamma(n + 1) = n!` for natural numbers
:math:`n`. More generally it satisfies the recurrence relation
:math:`\Gamma(z + 1) = z \cdot \Gamma(z)` for complex :math:`z`,
which, combined with the fact that :math:`\Gamma(1) = 1`, implies
the above identity for :math:`z = n`.
References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/5.2#E1
Examples
--------
>>> from scipy.special import gamma, factorial
>>> gamma([0, 0.5, 1, 5])
array([ inf, 1.77245385, 1. , 24. ])
>>> z = 2.5 + 1j
>>> gamma(z)
(0.77476210455108352+0.70763120437959293j)
>>> gamma(z+1), z*gamma(z) # Recurrence property
((1.2292740569981171+2.5438401155000685j),
(1.2292740569981158+2.5438401155000658j))
>>> gamma(0.5)**2 # gamma(0.5) = sqrt(pi)
3.1415926535897927
Plot gamma(x) for real x
>>> x = np.linspace(-3.5, 5.5, 2251)
>>> y = gamma(x)
>>> import matplotlib.pyplot as plt
>>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)')
>>> k = np.arange(1, 7)
>>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6,
... label='(x-1)!, x = 1, 2, ...')
>>> plt.xlim(-3.5, 5.5)
>>> plt.ylim(-10, 25)
>>> plt.grid()
>>> plt.xlabel('x')
>>> plt.legend(loc='lower right')
>>> plt.show()
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