Book to learn about Integration

Question

Question: Does anyone know a book that might help me in Integration?

I want to learn about advanced integration, such as integrating$$\int\limits_{0}^{\infty}\dfrac {\left|\Gamma\left(a+xi\right)\right|^2}{\left|\Gamma\left(b+xi\right)\right|^2}\,dx=\sqrt\pi\dfrac {\Gamma(a)\Gamma\left(a+\frac 12\right)\Gamma\left(b-a-\frac 12\right)}{\Gamma\left(b-\frac 12\right)\Gamma(b)\Gamma(b-a)}$$$$\int\limits_0^\infty\dfrac {1+\frac {x^2}{(b+1)^2}}{1+\frac {x^2}{a^2}}\dfrac {1+\frac {x^2}{(b+1)^2}}{1+\frac {x^2}{(a+1)^2}}\cdots\,dx=\dfrac {\sqrt\pi}{2}\dfrac {\Gamma\left(a+\frac 12\right)\Gamma(b+1)\Gamma\left(b-a+\frac 12\right)}{\Gamma(a)\Gamma\left(b+\frac 12\right)\Gamma(b-a+1)}$$ But I just don't know what book/type of book to look for. I have already learned the basics of integration.

Answer 1

I don't know about those specific equations, but A Course of Modern Analysis by Whittaker and Watson is well-known for its discussion of special functions. Its first edition was published in 1902.

Some knowledge of complex analysis is needed, and this can be gleaned either from the same book or from a more modern treatment such as Ahlfors' textbook.

Answer 2

In addition to the above, I would also recommend Advanced Calculus by Buck and Mathematical Analysis by Apostol for some (somewhat forgotten) integration techniques down these lines.

Answer 3

You can get help using MIT resources. Rd-sharma could be a good book. also you can try some Arihant publication books for IIT JEE.