Engineering Mathematics and Mathematical Physics Books

Honestly, there is no one book which gives you the whole mathematics used in physics and engineering with enough details. If you want to dive deeply into concepts you should read special books for special concepts. However, here are some books similar to Arfken 's book you mentioned where some of them contain more details.

$1$. Mathematical Physics, A Modern Introduction to Its Foundations By Sadri Hassani. This book provides a nice balance between the details and spectrum of the concepts. It is much better than Arfken in my opinion. It tries to convey modern mathematical thinking. The materials are introduced in a smooth integrated way. I think it is a good choice.

$2.$ Mathematical Methods of Physics, Volume 1 and Volume 2 By R. Courant and D. Hilbert. This is a classic which still worth reading as has been written by two well known mathematicians. It provides nice rigor and a vast of concepts.

$3.$ Advanced Calculus for Applications By Francis B. Hildebrand. This is also a classic which is taught in some universities.

$4.$ Mathematical Methods for Engineers and Scientists, Volume 1, Volume 2 and Volume 3 By Kwong Ting Tong. A rather new book which worth looking at.

$5.$ Advanced Engineering Mathematics By Erwin Kreyszig. It is more like an encyclopedia than a book.

$6.$ Fundamentals of Mathematical Physics By Edgar A. Kraut. This one is a popular book in mathematical physics and people are satisfied by reading it. The text is user friendly. You can take a look at reviews in Amazon.

I should emphasize again that none of such books replaces a special book written for a special mathematical subject. They have to sacrifice details for saving brevity. So do not expect these books to provide a systematic step by step rigorous approach.