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# 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] (see also [2]).

  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’ [3] shichi routine. For complex arguments the algorithm is based on Mpmath’s [4] 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]

  NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/6.2.E15 and https://dlmf.nist.gov/6.2.E16

  [3]

  Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

  [4]

  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()