# [Physics] Why is there no exact formula for position of maxima in single slit interference

diffractioninterferencewaves

On this website (http://www.ualberta.ca/~pogosyan/teaching/PHYS_130/FALL_2010/lectures/lect35/lecture35.html) it says that the formula $\sin(\theta_{max}) = (m+\frac{1}{2})\lambda$ derived for the maxima of a single slit interference is only approximate, while the formula for the minima $\sin(\theta_{max}) = m\lambda$ is exact (if we neglect other approximations like interference of parallel light waves). Why are the maxima not just in the middle of two minima?

We can clearly divide a light ray in three parts from which two cancel and thus get the mentioned formula for the maxima. Why is this formula not exact?

The function describing the interference pattern is $$f(x)=(\sin (x)/x)^2$$, where $$x=\frac{\pi a}{\lambda} \sin(\theta).$$ To find the extremes you need to differentiate and equate to zero. For the minima it is easy, you find the condition $$\sin x=0$$, but for the maxima, you get $$\tan x=x$$ which has to be solved numerically.