# Changes

## Sinc function

, 18:36, 19 December 2011
Taylor series and power series
$\operatorname{sinc}''(x) = \left\lbrace \begin{array}{rl} \frac{-1}{3}, & x = 0 \\ \frac{(2 - x^2)\sin x - 2x\cos x}{x^3}, & x \ne 0 \\\end{array}\right.$

==Integration==

===First antiderivative===

These is no antiderivative for this among elementary functions. However, we can create a new function that serves as the antiderivative:

$\operatorname{Si}(x) := \int_0^x \operatorname{sinc} t \, dt$

The function $\operatorname{Si}$ is termed the [[sine integral]].

===Integration of transformed versions of function===

We have:

$\int_0^x \operatorname{sinc}(mt) \, dt = \frac{1}{m} \operatorname{Si}(mx)$

In indefinite integral form:

$\int \operatorname{sinc}(mx + \varphi) \, dx = \frac{1}{m} \operatorname{Si}(mx + \varphi)$

===Improper definite integrals===

The following fact is true, but not easy to prove:

$\int_0^\infty \operatorname{sinc} x \, dx = \frac{\pi}{2}$

Because the function is an even function, this is equivalent to the following:

$\int_{-\infty}^0 \operatorname{sinc} x \, dx = \frac{\pi}{2}$

and

$\int_{-\infty}^\infty \operatorname{sinc} x \, dx = \pi$
==Taylor series and power series==
3,033
edits