1

I am working through John Todd's Introduction to the Constructive Theory of Functions (1963). On page 47, he computes the following integral (due to Stieltjes): \begin{align*} \mu_n&=\int_0^\infty x^ne^{-x^{1/4}}\sin(x^{1/4})\;dx\\ &=4\int_0^\infty e^{-y}y^{4n+3}\sin(y)\;dy\\ &=4\;\text{Im}\bigg[\int_0^\infty e^{-(1-i)y}y^{4n+3}\;dy \bigg]\\ &\overset{\mathrm{?}}{=}4\;\text{Im}\bigg[ (1-i)^{-4n-4}\int_0^\infty e^{-y}y^{4n+3}\;dy \bigg]\\ &=4\;\text{Im}\bigg[ (-4)^{-n-1}\int_0^\infty e^{-y}y^{4n+3}\;dy \bigg]\\ &=0. \end{align*}

I understand every equality except the one with the ? over it. Does anyone see how it works?

InterestingFool
  • 11
  • Set $t=(1-i)y$ and maybe argue something about deforming the contour in the complex plane since the integrand is analytic? By the way, here is another way of computing this integral. – Hans Lundmark Nov 18 '21 at 11:47

0 Answers0