# 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 .