The contour traverses |z| = 1/2 three times counterclockwise. Only z = 0 is singular inside it. The Laurent expansion $1/(e^z-1)=1/z-1/2+z/12+\cdots$ gives $1/[z^2(e^z-1)]=1/z^3-1/(2z^2)+1/(12z)+\cdots$. Its residue is 1/12. Hence the integral is $3(2\pi i)(1/12)=\pi i/2$, giving β = 0.5.
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