I'm an undergrad, and I was looking for books to study fractional calculus. I want books that has proofs, but not very rigorous ones -pure math level-. Your recommendations is highly appreciated.
There are two Brazilians authors who published the first Fractional Calculus book in Brazil. His name is Rubens de Figueiredo Camargo and Edmundo Capelas de Oliveira, the name is: "Cálculo Fracionário". You may not findd in english, only in Portuguese. But you can find Rubens's doctoral thesis Cálculo Fracionário e Aplicações.
The text is in portuguese but you can translate it with Google. His approach is easy to understand for undergraduate level. Hope that helped you.
Despite its vast size of more than 1000 pages, I would definitely recommend the book of Samko, Kilbas and Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, 1993. In my opinion, this is a very good starting point and the presentation is accessible at undergrad level (it requires a solid background of real analysis). Although several books have been published in the last decade on the topic, I still find this to be the best reference.
Another good reference -- especially if you are interested in analytical solutions of fractional-order systems -- is the book of Kaczorek, Selected Problems of Fractional Systems Theory, Springer, 2011.
I highly recommend The Fractional Calculus by Oldham & Spanier, Dover (1974). It's brief (~200 pages) and to the point.