Reference Book on Special Functions

The Bateman Manuscript Project is a good source, as is the NIST Digital Library of Mathematical Functions.

As OP stated he was a physics student in a comment, there are a lot of books called (something similar) to "Mathematical Methods for Physicists", which have relevant references, details and techniques for physicists. I'd probably start with the first two.

• Arfken, "Mathematical Methods for Physicists"

• Boas, "Mathematical Methods in the Physical Sciences "

• Courant and Hilbert, "Methods of Mathematical Physics"

• Shankar, "Basic Training in Mathematics: A Fitness Program for Science Students" (probably too basic)

• Abramowitz and Stegun, Handbook of Mathematical Functions (not proofs, but there are a lot of useful things there, which you can look up elsewhere)

• (Website) Wolfram Mathworld

And for PDE's, you can look at specific PDE books:

• Zauderer, Partial Differential Equations of Applied Mathematics

• One of the many basic linear PDE books, e.g. Haberman's Applied PDE

You may consider looking at Oldham, Myland, and Spanier's book An Atlas of Functions which can be purchased used quite cheaply on Amazon. I would dare suggest it is a beautiful book, with very nice graphs on glossy pages. It is large and it is organized by classes of functions. There aren't many proofs in the book, so I'm not sure if that would deter you from this book.

Another route to go is to get Schaum's Outline of Mathematical Formulas, though it may not be detailed enough for you.