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# scipy.special.sici

scipy.special.sici(x, out=None) = <ufunc 'sici'>

  Sine and cosine integrals.

  The sine integral is

  \[\int_0^x \frac{\sin{t}}{t}dt\]

  and the cosine integral is

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

  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 sine and cosine integrals.

    out tuple of ndarray, optional

      Optional output arrays for the function results

  Returns:
    si scalar or ndarray

      Sine integral at `x`

    ci scalar or ndarray

      Cosine integral at `x`

  See also

  `shichi`

    Hyperbolic sine and cosine integrals.

  `exp1`

    Exponential integral E1.

  `expi`

    Exponential integral Ei.

  Notes

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

  For real arguments the function is computed by calling Cephes’ [3] sici routine. For complex arguments the algorithm is based on Mpmath’s [4] si and ci routines.

  References

  [1] (1,2)

  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.E9, https://dlmf.nist.gov/6.2.E12, and https://dlmf.nist.gov/6.2.E13

  [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 sici, exp1

  `sici` accepts real or complex input:

      >>> sici(2.5)
      (1.7785201734438267, 0.2858711963653835)
      >>> sici(2.5 + 3j)
      ((4.505735874563953+0.06863305018999577j),
      (0.0793644206906966-2.935510262937543j))

  For z in the right half plane, the sine and cosine integrals are related to the exponential integral E1 (implemented in SciPy as `scipy.special.exp1`) by

  - Si(z) = (E1(i*z) - E1(-i*z))/2i + pi/2

  - Ci(z) = -(E1(i*z) + E1(-i*z))/2

  See [1] (equations 5.2.21 and 5.2.23).

  We can verify these relations:

      >>> z = 2 - 3j
      >>> sici(z)
      ((4.54751388956229-1.3991965806460565j),
      (1.408292501520851+2.9836177420296055j))

      >>> (exp1(1j*z) - exp1(-1j*z))/2j + np.pi/2  # Same as sine integral
      (4.54751388956229-1.3991965806460565j)

      >>> -(exp1(1j*z) + exp1(-1j*z))/2            # Same as cosine integral
      (1.408292501520851+2.9836177420296055j)

  Plot the functions evaluated on the real axis; the dotted horizontal lines are at pi/2 and -pi/2:

      >>> x = np.linspace(-16, 16, 150)
      >>> si, ci = sici(x)

      >>> fig, ax = plt.subplots()
      >>> ax.plot(x, si, label='Si(x)')
      >>> ax.plot(x, ci, '--', label='Ci(x)')
      >>> ax.legend(shadow=True, framealpha=1, loc='upper left')
      >>> ax.set_xlabel('x')
      >>> ax.set_title('Sine and Cosine Integrals')
      >>> ax.axhline(np.pi/2, linestyle=':', alpha=0.5, color='k')
      >>> ax.axhline(-np.pi/2, linestyle=':', alpha=0.5, color='k')
      >>> ax.grid(True)
      >>> plt.show()