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Module « scipy.special »

Fonction spence - module scipy.special

Signature de la fonction spence

def spence(*args, **kwargs) 

Description

help(scipy.special.spence)

spence(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])

spence(z, out=None)

Spence's function, also known as the dilogarithm.

It is defined to be

.. math::
  \int_1^z \frac{\log(t)}{1 - t}dt

for complex :math:`z`, where the contour of integration is taken
to avoid the branch cut of the logarithm. Spence's function is
analytic everywhere except the negative real axis where it has a
branch cut.

Parameters
----------
z : array_like
    Points at which to evaluate Spence's function
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Computed values of Spence's function

Notes
-----
There is a different convention which defines Spence's function by
the integral

.. math::
  -\int_0^z \frac{\log(1 - t)}{t}dt;

this is our ``spence(1 - z)``.

Examples
--------
>>> import numpy as np
>>> from scipy.special import spence
>>> import matplotlib.pyplot as plt

The function is defined for complex inputs:

>>> spence([1-1j, 1.5+2j, 3j, -10-5j])
array([-0.20561676+0.91596559j, -0.86766909-1.39560134j,
       -0.59422064-2.49129918j, -1.14044398+6.80075924j])

For complex inputs on the branch cut, which is the negative real axis,
the function returns the limit for ``z`` with positive imaginary part.
For example, in the following, note the sign change of the imaginary
part of the output for ``z = -2`` and ``z = -2 - 1e-8j``:

>>> spence([-2 + 1e-8j, -2, -2 - 1e-8j])
array([2.32018041-3.45139229j, 2.32018042-3.4513923j ,
       2.32018041+3.45139229j])

The function returns ``nan`` for real inputs on the branch cut:

>>> spence(-1.5)
nan

Verify some particular values: ``spence(0) = pi**2/6``,
``spence(1) = 0`` and ``spence(2) = -pi**2/12``.

>>> spence([0, 1, 2])
array([ 1.64493407,  0.        , -0.82246703])
>>> np.pi**2/6, -np.pi**2/12
(1.6449340668482264, -0.8224670334241132)

Verify the identity::

    spence(z) + spence(1 - z) = pi**2/6 - log(z)*log(1 - z)

>>> z = 3 + 4j
>>> spence(z) + spence(1 - z)
(-2.6523186143876067+1.8853470951513935j)
>>> np.pi**2/6 - np.log(z)*np.log(1 - z)
(-2.652318614387606+1.885347095151394j)

Plot the function for positive real input.

>>> fig, ax = plt.subplots()
>>> x = np.linspace(0, 6, 400)
>>> ax.plot(x, spence(x))
>>> ax.grid()
>>> ax.set_xlabel('x')
>>> ax.set_title('spence(x)')
>>> plt.show()

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