Vous êtes un professionnel et vous avez besoin d'une formation ? RAG (Retrieval-Augmented Generation)
et Fine Tuning d'un LLM Voir le programme détaillé

Module « scipy.special »

Fonction shichi - module scipy.special

Signature de la fonction shichi

def shichi(*args, **kwargs) 

Description

help(scipy.special.shichi)

shichi(x[, out1, out2], / [, out=(None, None)], *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])

shichi(x, out=None)

Hyperbolic sine and cosine integrals.

The hyperbolic sine integral is

.. math::

  \int_0^x \frac{\sinh{t}}{t}dt

and the hyperbolic cosine integral is

.. math::

  \gamma + \log(x) + \int_0^x \frac{\cosh{t} - 1}{t} dt

where :math:`\gamma` is Euler's constant and :math:`\log` is the
principal branch of the logarithm [1]_.

Parameters
----------
x : array_like
    Real or complex points at which to compute the hyperbolic sine
    and cosine integrals.
out : tuple of ndarray, optional
    Optional output arrays for the function results

Returns
-------
si : scalar or ndarray
    Hyperbolic sine integral at ``x``
ci : scalar or ndarray
    Hyperbolic cosine integral at ``x``

See Also
--------
sici : Sine and cosine integrals.
exp1 : Exponential integral E1.
expi : Exponential integral Ei.

Notes
-----
For real arguments with ``x < 0``, ``chi`` is the real part of the
hyperbolic cosine integral. For such points ``chi(x)`` and ``chi(x
+ 0j)`` differ by a factor of ``1j*pi``.

For real arguments the function is computed by calling Cephes'
[2]_ *shichi* routine. For complex arguments the algorithm is based
on Mpmath's [3]_ *shi* and *chi* routines.

References
----------
.. [1] Milton Abramowitz and Irene A. Stegun, eds.
       Handbook of Mathematical Functions with Formulas,
       Graphs, and Mathematical Tables. New York: Dover, 1972.
       (See Section 5.2.)
.. [2] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/
.. [3] Fredrik Johansson and others.
       "mpmath: a Python library for arbitrary-precision floating-point
       arithmetic" (Version 0.19) http://mpmath.org/

Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import shichi, sici

`shichi` accepts real or complex input:

>>> shichi(0.5)
(0.5069967498196671, -0.05277684495649357)
>>> shichi(0.5 + 2.5j)
((0.11772029666668238+1.831091777729851j),
 (0.29912435887648825+1.7395351121166562j))

The hyperbolic sine and cosine integrals Shi(z) and Chi(z) are
related to the sine and cosine integrals Si(z) and Ci(z) by

* Shi(z) = -i*Si(i*z)
* Chi(z) = Ci(-i*z) + i*pi/2

>>> z = 0.25 + 5j
>>> shi, chi = shichi(z)
>>> shi, -1j*sici(1j*z)[0]            # Should be the same.
((-0.04834719325101729+1.5469354086921228j),
 (-0.04834719325101729+1.5469354086921228j))
>>> chi, sici(-1j*z)[1] + 1j*np.pi/2  # Should be the same.
((-0.19568708973868087+1.556276312103824j),
 (-0.19568708973868087+1.556276312103824j))

Plot the functions evaluated on the real axis:

>>> xp = np.geomspace(1e-8, 4.0, 250)
>>> x = np.concatenate((-xp[::-1], xp))
>>> shi, chi = shichi(x)

>>> fig, ax = plt.subplots()
>>> ax.plot(x, shi, label='Shi(x)')
>>> ax.plot(x, chi, '--', label='Chi(x)')
>>> ax.set_xlabel('x')
>>> ax.set_title('Hyperbolic Sine and Cosine Integrals')
>>> ax.legend(shadow=True, framealpha=1, loc='lower right')
>>> ax.grid(True)
>>> plt.show()

Vous êtes un professionnel et vous avez besoin d'une formation ? Coder avec une
Intelligence Artificielle Voir le programme détaillé