I'm new to topology but I must understand how it works to progress in my research. First, can anybody point me to a document that introduces topology in a "gentle" way. What pre-requisites do I need to understand the topic? I need an intuitive understanding of manifolds for now.

Show that $\displaystyle{\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2} = \frac{\pi}{2}\log \left(\sqrt{2\pi} \Gamma\left(\frac{3}{4}\right) / \Gamma\left(\frac{1}{4}\right)\right)}$ This question was posted as part of this question: Solve the integral $\displaystyle{S_k = (-1)^k...

Show that $\displaystyle{\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2} = \frac{\pi}{2}\log \left(\sqrt{2\pi} \Gamma\left(\frac{3}{4}\right) / \Gamma\left(\frac{1}{4}\right)\right)}$ This question was posted as part of this question: Solve the integral $\displaystyle{S_k = (-1)^k...

I am reading a paper (sorry, no e-copy) with a number of infinite series, in which each term of the series is an integral of a complicated transcendental function like the one in the title. There are no fewer than a dozen of these infinite sums of integrals in the paper. For instance, the auth...