Module « scipy.special »

Fonction gamma - 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()