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Lately I've been interested in Lev Landau's Theoretical Minimum exams and I would like to study a bit at this level (or at least try to do so). For the Physics exams, it is easy to find appropriate literature (Landau's Course of Theoretical Physics). However, I would like to know which would be good places to study for the Mathematics exams. According to this post, the subjects of the mathematical exams are:

  1. Mathematics I: Integration, ordinary differential equations, vector algebra and tensor analysis.
  2. Mathematics II: The theory of functions of a complex variable, residues, solving equations by means of contour integrals (Laplace's method), the computation of the asymptotics of integrals, special functions (Legendre, Bessel, elliptic, hypergeometric, gamma function).

I've looked around at the websites of the Landau Institute for Theoretical Physics (which applies the exams nowadays) and at the Moscow Institute of Physics And Technology (whose students have to pass some of the exams), but couldn't find any information on literature, only the contents of the exams...

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  • $\begingroup$ Any book is fine really, but if by any chance you are confortable with italian I suggest E.Lanconelli, Lezioni di Analisi Matematica (1 and 2). For the special functions part there is hardly any book that covers them in a pedagogical fashion. The internet is PACKED with informations on the topic. It would be a nice chance to learn how to do bibliographic research. $\endgroup$
    – SO_32
    Commented Jan 17, 2022 at 6:24

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One of the resources recommended to enter the Landau Institute as a graduate student is the "A Course of Higher Mathematics" series, by Vladimir I. Smirnov. I've been reading it and, so far, it seems to be quite complete, besides some of its exercises appear on Demidovitch's book of exercises on Analysis.

Edit

While this question is in the environment of Landau's Theoretical Minimum, it can be interpreted as a question asking for a couple of resources on Mathematics for physicists. Hence, I'll further complete this answer by providing a couple of links to other relevant posts. Since the question has been made Community Wiki, I hope other users will fill in some of the blanks left over here. Most of these links are in fact coming over from the overarching books question Resource recommendations.

Mathematics I

Integration and ordinary differential equations are common themes studied in Calculus. Most Calculus textbooks, if not all, will cover these themes. One of these, of course, is Smirnov's text, mentioned in the original answer.

Vector algebra is a common theme in introductory classes on vector ans analytic geometric, but also quite common when one is studying linear algebra. A quick review aimed straight at physicists is available in Chapter 1 of Griffiths' Introduction to Electrodynamics.

Tensor analysis can be a bit more subtle, but there is already a question about that: Learn about tensors for physics.

Mathematics II

The theory of functions of a complex variable, residues, and solving equations by means of contour integrals can all be summarized as Complex Analysis. This theme is already covered on Complex Variable Book Suggestion.

As of the time I'm writing this answer, I am not sure what is meant by "the computation of asymptotics of integrals" and hope someone will fill in this gap.

Special functions are a common theme in courses covering mathematical methods for physicists. At the moment I haven't found a question dealing specifically with this, but Book recommendations for Fourier Series, Dirac Delta Function and Differential Equations? is a related topic. In order to not leave the point unaswered, one of the most commonly used references is the gargantuan Mathematical Methods for Physicists: A Comprehensive Guide, by Arfken, Weber, and Harris, usually referred to simply as "Arfken". As the name suggests, it is quite comprehensive and covers some of the other topics mentioned on the question as well. On the other hand, due to its encyclopedic nature, it covers the themes with less detail.

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