sinc function centered at $x=c$ that goes to zero at $x=0$?
The sinc function is ordinarily defined as
$$ \operatorname{sinc}(x) = \frac{\sin x} x \text{ if } x \neq 0 \text{ else } 1. $$
I want a sinc function that is shifted away from the origin such that it's centered at some value $c$, and also equals zero at $x=0$. How can I define this function?
$\endgroup$ 33 Answers
$\begingroup$Hint:
You can use $$ \frac{\sin\left(\frac{\pi}{c}x-\pi \right)}{\frac{\pi}{c}x-\pi} $$
But, given the periodicity of $\sin x$, there are other possible functions.
$\endgroup$ $\begingroup$$$ \operatorname{sinc} (x - c) = \begin{cases} \dfrac{\sin (x-c)}{x-c} & x\ne c\\[8pt] \qquad 1 & x = c \end{cases} $$
Below sinc $(x-\pi)$:
Shifting any function at all is done as follows: Let $g(x) = f(x-c).$ Then the graph of $g$ is the graph of $f$ shifted so that whatever $f$ does at $x=0$, $g$ does at $x=c.$
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