Are too many books on a subject a boon or a bane?

I am about to appear in a competitive exam in $3$ months for my PhD where one of the topics is complex analysis.

Background:Pure Mathematics

The questions will be of subjective type.

I am trying to learn complex analysis on my own.Though I studied it I am not so strong in it.

In the mean time whenever I am trying to solve the exercises,I am having difficulty in solving them.I am going on downloading books for the exam which will have good exercises.

The more books I am getting,instead of helping me they are making me anxious because there are so many to solve in such a short time.

What should I do now?I am unable to find out that particular book with good exercises that would help me in my exam.

I have bought Gamelin,Bak-Newmann,Shakarchi and Conway,but which should I choose between the three;Neither are helping me to solve the sample questions.Conway and Gamelin are so difficult to solve.Should I get more books?

Please help me what should I do?Where is the problem lying?

  • $\begingroup$ I can't recommend books but I will tell you this, pick the easiest (for you) of the bunch and finish reading the whole book. The if you have time, you can move on to harder ones. $\endgroup$ – Priyatham Jan 11 '17 at 13:55
  • $\begingroup$ I've heard that the best book for complex analysis is A. I. Markushevich "Theory of Functions of a Complex Variable". It has 3 volumes, you might find there the things that you need. $\endgroup$ – Mat Dyl Jan 11 '17 at 13:55

You mainly need a book of exercises with solutions. No longer read theory ; come back to theory when a difficulty springs out in an exercice.

If your test checks plainly that the basics are known, make sure that you:

  • (of course) are OK with complex numbers. For example the $n$th roots of unity, computation of modules and arguments with, as possible, a geometric view on the question (example: what are the module and argument of $e^{ia}+e^{ib}$ ?)

  • are able to work on functions $Z=f(z)$ in a basic way, for example be able to compute the image of the imaginary axis by $Z=4z^2$, or the pre-image of the unit-circle.

  • can switch between a vision $f: \mathbb{C} \to \mathbb{C}$ and a vision $f: \mathbb{R^2} \to \mathbb{R^2}$ ; in particular, make sure you understand Cauchy-Riemann equations and what they mean (i.e., complex derivation is a similitude with ratio $|f'(z)|$, thus preserves angles),

  • are apt at expanding series in an annulus (Laurent+Taylor expansions) and understand what it means,

  • understand the principle of analytic continuation (with cuts, it means understanding what complex logarithm is and is useful at),

  • master the meaning of complex integration over a contour and its application to the computation of integrals by the method of residues for simple contours,

  • are able to use some basic theorems in a thorough manner: maximums' principle, Rouché's theorem, Liouville's theorem.

    Beyond that, it's specialization. For example, don't dwell into Picard's theorem, conformal mapping, Schwarz-Christoffel, etc.

See the rich exchange here.

  • $\begingroup$ When I am searching in MSE for good books on Complex Analysis the ones which are recommend don't contain solutions ;So which one to choose ;Please help $\endgroup$ – Learnmore Jan 11 '17 at 14:27
  • $\begingroup$ An old book that has a rather simple approach "Complex variables and applications" by R. V. Churchill, Mc Graw Hill. Much more sophisticated but absolutely excellent, with many figures: "Applied and Computational Complex Analysis" by P. Henrici (2 volumes) Wiley Interscience $\endgroup$ – Jean Marie Jan 11 '17 at 14:39
  • $\begingroup$ Do they all contain solutions as you suggested $\endgroup$ – Learnmore Jan 11 '17 at 17:15
  • $\begingroup$ @JeanMarie what source do you recommend to best learn analytic continuation properly? I've take a complex analysis course and read one or two big Rudin chapters on complex analysis but I've never come across it. $\endgroup$ – juan arroyo Jan 11 '17 at 17:22
  • 1
    $\begingroup$ @Ben Stokes For many (corrected) exercices: "Complex Variables" Schaum's collection. Authors: Spiegel, Lipschutz, Schiller, Spellman $\endgroup$ – Jean Marie Jan 11 '17 at 17:38

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