Integral sine

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The special function defined for real by

For one has

One sometimes uses the notation

Some particular values are:

Some special relations:

where is the integral cosine. For small,

The asymptotic representation for large is


The integral sine has the series representation


As a function of the complex variable , , defined by (*), is an entire function of in the -plane.

The integral sine is related to the integral exponential function by

See also Si-ci-spiral.

For references, and the graph of the integral sine, see Integral cosine.


This function is better known as the sine integral.

How to Cite This Entry:
Integral sine. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098