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Logarithmic moments of the Cauchy kernel

For

0 < a < 2

,

I (a) = \int_{0}^{\infty} \frac{x ^{a - 1}}{1 + x ^{2}} d x = \frac{π}{2} csc \frac{πa}{2} .

Differentiating at

a = 1

and expanding the secant gives

\int_{0}^{\infty} \frac{( l o g x ) ^{2}}{1 + x ^{2}} d x = \frac{π ^{3}}{8}, \int_{0}^{\infty} \frac{( l o g x ) ^{4}}{1 + x ^{2}} d x = \frac{5 π ^{5}}{32} .

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