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

Question

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?

Answer 1

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.