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I am searching for an introductory book in the field of complex analysis for self learning, that would contain the following:

  • Analytic number theory : the connection between complex analysis and number theory and most importantly the Riemann hypothesis
  • Some proofs : I prefer to look behind the proofs of the theorem that I study before accepting it fully
  • The book must be suitable for self study, so it would be appreciated if it is clear enough and contains everything needed
  • The book must contain the necessary parts of complex analysis that are used in physics, given that I'm interested in physics too
  • I would like the algebraic approach of complex analysis but it would be good if the book contained the geometric beauty of complex numbers too like fractals (Julia set, Mandelbrot set)

I will use complex analysis later in two mathematical fields: Partial Differential Equations and Number Theory, so I am searching for a book that covers the fundamentals of complex analysis on an undergraduate level that will make me comfortable in studying PDE and Number Theory later on.

My background in mathematics: ODE - Real analysis - Multivariable analysis - Vector Analysis - Linear algebra (matrices and vector space)

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    $\begingroup$ 1) For the basics, Complex Variables and Applications, 7'th Ed. by Brown and Churchill, 2) For geometric intuition, Visual Complex Analysis by Needham, 3) The Theory of the Riemann Zeta Function by Titchmarsh $\endgroup$ – Tom Copeland Aug 15 at 20:16
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    $\begingroup$ Ahlfors' or Freitag & Busam's or Lang's Complex Analysis should do, but IIRC they don't contain any fractals. In fact I don't know of any introductory book on complex analysis that treats both Riemann's zeta function and fractals, but this might just be my ignorance. However, all three books treat Riemann's Zeta Function and connections to Analytic Number Theory to a various degree. $\endgroup$ – M.G. Aug 15 at 20:34
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    $\begingroup$ And at some point you should probably explore Ramanujan's Master Theorem/Formuls as it can be used to relate the Bernoulli polynomials to Dirichlet series and Riemann zeta--type functions through the Mellin transform. $\endgroup$ – Tom Copeland Aug 15 at 20:34
  • $\begingroup$ I think you might want to splurge and get two different sources for zeta and fractals. So far as we seem to know at this point, the ideas are not much related to each other, beyond very basic complex analysis... $\endgroup$ – paul garrett Aug 15 at 22:54
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All your requirements (except "geometric beauty") are satisfied by the book of

John Stalker, Complex Analysis, Springer 1998.

It is written by a mathematical physicist who loves formulas and number theory but thinks and explicitly says that geometry is useless in the subject:-)

(I did not understand your requirement about "some proofs". Books in Complex Analysis, like other mathematics textbooks usually contain complete proofs of all statements). For fractals and "geometric beauty" in general, I recommend another book, Milnor, Dynamics in one complex variable.

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