Integral cosine

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The special function defined, for real , by

where is the Euler constant. Its graph is:

Figure: i051370a

The graphs of the functions and .

Some integrals related to the integral cosine are:

where is the integral sine minus .

For small:

The asymptotic representation, for large, is:

The integral cosine has the series representation:


As a function of the complex variable , , defined by (*), is a single-valued analytic function in the -plane with slit along the relative negative real axis . The value of here is taken to be . The behaviour of near the slit is determined by the limits

The integral cosine is related to the integral exponential function by

One sometimes uses the notation .

See also Si-ci-spiral.


[1] H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[2] E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)
[3] A. Kratzer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960)
[4] N.N. Lebedev, "Special functions and their applications" , Prentice-Hall (1965) (Translated from Russian)


The function is better known as the cosine integral. It can, of course, be defined by the integral (as above) in .

How to Cite This Entry:
Integral cosine. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article