# 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 .

Parameters:
xarray_like

Real or complex points at which to compute the sine and cosine integrals.

outtuple of ndarray, optional

Optional output arrays for the function results

Returns:
siscalar or ndarray

Sine integral at x

ciscalar or ndarray

Cosine integral at x

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’  sici routine. For complex arguments the algorithm is based on Mpmath’s  si and ci routines.

References

 (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.)



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



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  (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)')