<s>[INST] How to derive FWHM of sinc function So this is probably a simple question, but I am unable to get my head around it. If we have $\operatorname{sinc}(2 \pi v L)$, what is the width of that $\operatorname{sinc}$ in terms of $v$ at half the maximum point.
Normally I'd just say that the maximum is $1$, thus half point is.... $1/2$
$$
\operatorname{sinc}(2\pi v L) = \frac{1}{2}
\\  
\frac{ \sin(2 \pi vL) } { 2 \pi v L} =  \frac{1}{2}
\\
\sin(2\pi vL) = \pi v L
$$
EDIT: How can you analytically find this?
Thanks.
 [/INST] I cannot resist the pleasure of providing an approximate solution for $Lv \leq 1$. This approximation was made by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ So $$\frac{\sin(x)}x\simeq \frac{16 (\pi -x) }{5 \pi ^2-4 (\pi -x) x}$$ and for a value equal to $\frac 12$, the solution is given by $$x=\frac{\pi }{2}+\sqrt{16-(\pi -4) \pi }-4\approx 1.89477$$ while the exact solution would be $\approx 1.89549426$.
Equations which mix polynomial and trigonometric functions do not show explicit solutions and numerical methods (or sophisticated approximations) should be used.
The solution being close to $\frac{3\pi}5$, we can build the simplest Pade approximant of the left hand side and using exact values for the trigonometric function of the angle get $$x=\frac{250 \left(5+\sqrt{5}\right)+3 \pi  \left(100 \sqrt{10-2 \sqrt{5}}-3 \pi 
   \left(-55+5 \sqrt{5}+3 \sqrt{2 \left(5+\sqrt{5}\right)} \pi \right)\right)}{500
   \sqrt{5+2 \sqrt{5}}+75 \left(9+\sqrt{5}\right) \pi -45 \sqrt{2
   \left(5+\sqrt{5}\right)} \pi ^2}$$ which is $\approx 1.89549416$
Edit
In a more general manner, if we expand $\frac{\sin(x)}x$ as its simplest approximant at $x=\theta$, the solution of $\frac{\sin(x)}x=\frac 12$ is given by $$x=\frac{-3 \theta ^2-\left(\theta ^2-2\right) \cos (2 \theta )+\theta  \sin (\theta )
   \left(\theta ^2+8 \cos (\theta )\right)-2}{\left(\theta ^2-2\right) \sin (\theta
   )-3 \theta +2 \sin (2 \theta )+2 \theta  \cos (\theta )-\theta  \cos (2 \theta )}$$ $\theta=\frac{29 \pi}{48}$ would have been a very good choice since the values of its trigonometric  functions are known (with many radicals). 
Using it, the result would have been $\approx 1.89549427$
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