funzione sinc

Key facts

General definition

sinc ⁡ x = { sin ⁡ x x , x ≠ 0 1 , x = 0 {\displaystyle \operatorname {sinc} x={\begin{cases}{\dfrac {\sin x}{x}},&x\neq 0\\1,&x=0\end{cases}}}

Fields of application

Signal processing, spectroscopy

Domain

R {\displaystyle \mathbb {R} }

Image

[ − 0.217234 … , 1 ] {\displaystyle [-0.217234\ldots ,1]}

Parity

Even

Maxima

1 at x = 0 {\displaystyle x=0}

Minima

− 0.21723 … {\displaystyle -0.21723\ldots } at x = ± 4.49341 … {\displaystyle x=\pm 4.49341\ldots }

Root

π k , k ∈ Z ≠ 0 {\displaystyle \pi k,k\in \mathbb {Z} _{\neq 0}}

Reciprocal

{ x csc ⁡ x , x ≠ 0 1 , x = 0 {\displaystyle {\begin{cases}x\csc x,&x\neq 0\\1,&x=0\end{cases}}}

Derivative

sinc ′ ⁡ x = { cos ⁡ x − sinc ⁡ x x , x ≠ 0 0 , x = 0 {\displaystyle \operatorname {sinc} 'x={\begin{cases}{\dfrac {\cos x-\operatorname {sinc} x}{x}},&x\neq 0\\0,&x=0\end{cases}}}

Antiderivative

∫ sinc ⁡ x d x = Si ⁡ ( x ) + C {\displaystyle \int \operatorname {sinc} x\,dx=\operatorname {Si} (x)+C}

Taylor series

sinc ⁡ x = ∑ k = 0 ∞ ( − 1 ) k x 2 k ( 2 k + 1 ) ! {\displaystyle \operatorname {sinc} x=\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k+1)!}}}

via Wikipedia infobox