Math 6720 • Applied Complex Variables and Asymptotic Methods

Use the keyhole contour below to show that on the principal branch of \(x^k\)
\[
I(a)=\int_0^\infty \frac{x^{k-1}}{x+a}\,dx
=\frac{\pi}{\sin(k\pi)}a^{k-1},\qquad 0<k<2,\quad a>0.
\]

\emph{Figure description: A keyhole contour about the positive real axis with outer arc \(C_R\), inner arc \(C_\epsilon\) about \(z=0\), a branch cut along the positive real axis ending at \(x=R\), and marked points \(z=-b\) and \(z=-a\) on the negative real axis.}

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