Can you recommend a good book (with theoretical results with proofs, and with plenty of solved problems and examples) on the topics of improper integrals, (improper) integrals with parameters, special functions (Beta, Gamma, ...)?
You may have a look at the affordable and accessible Schaum's Outline of Advanced Calculus. Not too much advanced :)
I like Hildebrand's 'Advanced Calculus for Applications'. It has many applications of special functions for differential equations and shows how to use them to solve problems.