Definition of Sinc function

Question

I just want to make clear of the definition of sinc(x). I know there is a normalized and unnormalized definition for the sinc function. If we have unnormalized sinc then we have: $$\sin(x)/x=\text{sinc}(x) \hspace{0.2in}\textbf{unnormalized sinc function}$$

And for the normalized sinc we have: $$\sin(\pi x)/\pi x = \text{sinc}(x) \hspace{0.2in}\textbf{normalized sinc function}$$

My question is: If we have something like: $\dfrac{\sin\left(\frac{200\pi x}{500}\right)}{200\pi x }$, if we divide and multiply the equation by $500$, will this convert to be: $500\,\text{sinc}(x)$? It just a tad bit confusing about what constants stay if inside the argument of the sine (if any).

Answer 1

In the engineering literature, those who define the Fourier transform as $$X(\omega) = \int_{-\infty}^{\infty} x(t)\exp(-i\omega t)\ \mathrm dt$$ tend to use the unnormalized version, while those who define the Fourier transform as $$X(f) = \int_{-\infty}^{\infty} x(t)\exp(-i2\pi f t)\ \mathrm dt$$ tend to use the normalized version. My personal preference is for the normalized version because the zeroes of the function are the nonzero integers, but, like most people, I have learned to live with both definitions and figure out which one an author is using even if it is not explicitly stated. There are, of course, zealots who say that people who use the convention they do not happen to prefer are in a state of sin.

Incidentally, I would like to say that my preference for the definition of the sinc (or sine cardinal) function is $$\mathrm{sinc}(x) = \begin{cases}\frac{\sin(\pi x)}{\pi x}, & x \neq 0,\\ 1, & x = 0,\end{cases}$$ and not simply $\mathrm{sinc}(x) = \sin(\pi x)/(\pi x)$ the way the OP and Wikipedia states it.

Answer 2

$$\frac{\sin\left(\frac{200}{500}\pi x\right)}{200\pi x} = \frac{1}{500} \frac{\sin\left(\frac{200}{500}\pi x\right)}{\frac{200}{500}\pi x}=\begin{cases} \frac{1}{500}\operatorname{sinc}\left(\frac{200}{500}\pi x\right) & \rm nonnormalized~sinc~convention \\ \color{White}X \\ \frac{1}{500}\operatorname{sinc}\left(\frac{200}{500}x\right) & \rm ~~~normalized~sinc~convention. \end{cases}$$

If you're doing mathematical writing and utilize a standard function that has multiple different versions (this happens a lot with normalizations, see also the Fourier transform), what you need to do is pick one of the conventions, and begin by making explicit to the reader which convention you have chose to use. Which version you pick is of course up to you, but a lot of the time there might be pros or cons to the different versions, depending on context and motivations.