The special function defined, for real , by
where is the Euler constant. Its graph is:
The graphs of the functions and .
Some integrals related to the integral cosine are:
where is the integral sine minus .
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.
|||H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)|
|||E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)|
|||A. Kratzer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960)|
|||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 .
Integral cosine. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_cosine&oldid=16619