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Sinc function

1,014 bytes added, 18:36, 19 December 2011
Taylor series and power series
<math>\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.</math>
===First antiderivative===
These is no antiderivative for this among elementary functions. However, we can create a new function that serves as the antiderivative:
<math>\operatorname{Si}(x) := \int_0^x \operatorname{sinc} t \, dt</math>
The function <math>\operatorname{Si}</math> is termed the [[sine integral]].
===Integration of transformed versions of function===
We have:
<math>\int_0^x \operatorname{sinc}(mt) \, dt = \frac{1}{m} \operatorname{Si}(mx)</math>
In indefinite integral form:
<math>\int \operatorname{sinc}(mx + \varphi) \, dx = \frac{1}{m} \operatorname{Si}(mx + \varphi)</math>
===Improper definite integrals===
The following fact is true, but not easy to prove:
<math>\int_0^\infty \operatorname{sinc} x \, dx = \frac{\pi}{2}</math>
Because the function is an even function, this is equivalent to the following:
<math>\int_{-\infty}^0 \operatorname{sinc} x \, dx = \frac{\pi}{2}</math>
<math>\int_{-\infty}^\infty \operatorname{sinc} x \, dx = \pi</math>
==Taylor series and power series==

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