I am struggling to understand how the highlighted two integrals are equal to each other. I changed $\sin(2bx)$ into exponential form to help, but it doesn't seem to be helping at all. Also $b$ is just a number greater than $>0$.
2026-03-28 16:12:43.1774714363
How are these integrals equal?
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It comes from equating/comparing the real and imaginary parts (the imaginary part being 0) of the previous line. Not sure where that line comes from though...
The imaginary parts must cancel, giving $$ e^{b^2} \int_0^\infty e^{-x^2} \sin 2bx \ dx = \int_0^b e^{y^2} \ dy. $$ Now shuffle the $e^{b^2}$ over to the other side by dividing it, giving the negated exponent.