scipy.special.shichi#
- scipy.special.shichi(x, out=None) = <ufunc 'shichi'>#
- Hyperbolic sine and cosine integrals. - The hyperbolic sine integral is \[\int_0^x \frac{\sinh{t}}{t}dt\]- and the hyperbolic cosine integral is \[\gamma + \log(x) + \int_0^x \frac{\cosh{t} - 1}{t} dt\]- where \(\gamma\) is Euler’s constant and \(\log\) is the principal branch of the logarithm [1]. - Parameters:
- xarray_like
- Real or complex points at which to compute the hyperbolic sine and cosine integrals. 
- outtuple of ndarray, optional
- Optional output arrays for the function results 
 
- Returns:
- siscalar or ndarray
- Hyperbolic sine integral at - x
- ciscalar or ndarray
- Hyperbolic cosine integral at - x
 
 - Notes - For real arguments with - x < 0,- chiis 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 - shichiaccepts 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() 