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Thus we have 2 z2 a2 z2 b2 2 i lim lim 2 z ia z ia z2 b2 z ib z a2 z ib 2ab a b In the above calculation we assumed that a b. Such contours are often called keyhole contours. The integrals along C2 and C4 are f x log xdx 2 i f x dx. Even function this integral is equal to a2 x2 b2 x2 The complex version of the integrand 2 a z b2 z2 has poles in the upper half-plane at z ia and ib. X 1 We can consider this integral to be the imaginary part of eix So consider eiz dz. z2 z 1 We want to choose...
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