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Module « scipy.special »

Fonction zeta - module scipy.special

Signature de la fonction zeta

def zeta(x, q=None, out=None) 

Description

help(scipy.special.zeta)

Riemann or Hurwitz zeta function.

Parameters
----------
x : array_like of float or complex.
    Input data
q : array_like of float, optional
    Input data, must be real.  Defaults to Riemann zeta. When `q` is
    ``None``, complex inputs `x` are supported. If `q` is not ``None``,
    then currently only real inputs `x` with ``x >= 1`` are supported,
    even when ``q = 1.0`` (corresponding to the Riemann zeta function).

out : ndarray, optional
    Output array for the computed values.

Returns
-------
out : array_like
    Values of zeta(x).

See Also
--------
zetac

Notes
-----
The two-argument version is the Hurwitz zeta function

.. math::

    \zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x};

see [dlmf]_ for details. The Riemann zeta function corresponds to
the case when ``q = 1``.

For complex inputs with ``q = None``, points with
``abs(z.imag) > 1e9`` and ``0 <= abs(z.real) < 2.5`` are currently not
supported due to slow convergence causing excessive runtime.

References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
    https://dlmf.nist.gov/25.11#i

Examples
--------
>>> import numpy as np
>>> from scipy.special import zeta, polygamma, factorial

Some specific values:

>>> zeta(2), np.pi**2/6
(1.6449340668482266, 1.6449340668482264)

>>> zeta(4), np.pi**4/90
(1.0823232337111381, 1.082323233711138)

First nontrivial zero:

>>> zeta(0.5 + 14.134725141734695j)
0 + 0j

Relation to the `polygamma` function:

>>> m = 3
>>> x = 1.25
>>> polygamma(m, x)
array(2.782144009188397)
>>> (-1)**(m+1) * factorial(m) * zeta(m+1, x)
2.7821440091883969

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