Exploring sinc function

Mathematical Software. Mathematical Research. Mathematical Education. Tvalx Products.


Sinc function is a simple special function given by formula sin(x)/x on R\{0} and sinc(0)=1. As we see the only difference sinc(x) from sin(x)/x is that sinc(x) is defined at zero whereas sin(x)/x is not. The graphs of sinc can be easy obtained in Math Center Level 1, for example:

grapf of sinc(x) function


As we see the graph is smooth everywhere. The sinc function is also smooth in mathematical sense, that is has derivatives of all orders. Let's show that sinc(x) is continuous at zero. By l'Hôpital's rule limit of sin(x)/x at zero is equal to limit of derivative(sin(x))/derivative(x) at zero, that is cos(0)/1=1.

The normalized sinc function is similar to sinc(x) and is given by formula sin(πx)/(πx) on R\{0} and sinc(0)=1. Compare graphs of sinc function and normalized sinc function:

Graphs of Sa and NSa


Graphs of two first derivatives can be obtained in Math Center Level 2 :

Derivatives of normalized sinc function


Apply AntiAlias and HighQuality:

Graph of derivatives of normalized sinc function


Sinc function can be calculated with help of Scientific Calculator Precision 81 .


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