Computing the sine integral: Difference between revisions

Revision as of 20:32, 28 April 2008

This article studies the computation of the following improper real integral:

$\int_{- \infty}^{\infty} \frac{\sin x}{x} = π$

Here, the value at $0$ is assigned to be 1. (The function being integrated is termed the sinc function and its indefinite integral is termed the sine integral.

Computation

We first consider the function:

$z \mapsto \frac{e^{i z}}{z}$

This is a holomorphic function and its imaginary part is $(\sin x) / (x)$ .

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 This article studies the computation of the following improper real integral:
-<math>\int_{\-infty}^\infty \frac{\sin x}{x} = \pi</math>
+<math>\int_{-\infty}^\infty \frac{\sin x}{x} = \pi</math>
 Here, the value at <math>0</math> is assigned to be 1. (The function being integrated is termed the [[sinc function]] and its indefinite integral is termed the [[sine integral]].