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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.

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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.

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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.

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I highly recommend The Fractional Calculus by Oldham & Spanier, Dover (1974). It's brief (~200 pages) and to the point.

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