I need to calculate the following singular integral:

NIntegrate[Log[1 + y^2]/Cos[Pi y], {y, 0, 1}]

However, it is failing to converge. I have tried to specify MaxRecursion, PrecisionGoal, and Exclusions but neither helped. How does one calculate singular integrals in Mathematica?


As it stands the integral does not converge! To see that note that

Series[Log[1 + y^2]/Cos[Pi y], {y, 1/2, 0}]

returns $$-\frac{\log \left(\frac{5}{4}\right)}{\pi \left(y-\frac{1}{2}\right)}-\frac{4}{5 \pi }+O\left(y-\frac{1}{2}\right)$$

and a simple pole is not integrable.

What you maybe want to know is Cauchy's principal value of the integral (which is finite). For that, evaluate

NIntegrate[Log[1 + y^2]/Cos[Pi y], {y, 0, 1/2, 1}, Method -> "PrincipalValue"]

which returns -0.281589.

  • 2
    $\begingroup$ Same with NIntegrate[Log[1 + y^2]/Cos[Pi y], {y, 0, 1}, Method -> "PrincipalValue", Exclusions -> {1/2}] $\endgroup$ Dec 13 '12 at 12:51

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