How to evaluate $\int_0^1 \int_0^1 \frac{1}{1-u^2+v^2}dudv$?

Asked Dec 06 '19 at 11:32

Active Aug 31 '20 at 01:52

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For (i) I did it by turning the integrand into infinite series, but what is the method by applying binomial theorem?

For (iii) I'm unsure as to a good "trigonometric substitution".

edited Aug 31 '20 at 01:52

Integrand

asked Dec 06 '19 at 11:32

helios321

Please see https://proofwiki.org/wiki/Basel_Problem/Proof_1 – Maximilian Janisch Dec 06 '19 at 11:40
For the first question, the binomial expansion of a similar function $$ \frac{1}{1-x} = 1 + x + \frac{1}{2}\cdot (1)(2)x^2+\cdots $$
or negative binomial expansion
– Chinny84 Dec 06 '19 at 11:41

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