11.1 The Graduate Syllabus Behind These Problems

These problems are set against a published graduate syllabus, reproduced here so that a reader can see what they assume and what to read next. Items marked \(\checkmark \) are covered in the chapters above; the rest are where the subject continues.

1.

Undergraduate material.

(a).: Complex numbers and their geometry \(\checkmark \)
(b).: The Riemann–Stieltjes integral
(c).: Green’s formula in two dimensions \(\checkmark \)
(d).: Uniform convergence and equicontinuity of sequences of functions; integrals of functions depending on parameters \(\checkmark \) (equicontinuity excepted)

2.

Elementary analytic functions and their mapping properties.

(a).: Linear fractional transformations and the Riemann sphere \(\checkmark \)
(b).: Cross-ratio \(\checkmark \)
(c).: The exponential and the logarithm \(\checkmark \)
(d).: Trigonometric functions \(\checkmark \)

3.

The Cauchy–Riemann equations.

(a).: The operators \(\partial \) and \(\overline {\partial }\) in Cartesian and polar coordinates
(b).: The homogeneous equation \(\overline {\partial }u = 0\); properties of \(\operatorname {Re}(u)\) and \(\operatorname {Im}(u)\) \(\checkmark \)
(c).: The inhomogeneous equation \(\overline {\partial }u = f\)

4.

Cauchy’s theorem and its consequences.

(a).: Proofs and Cauchy’s formula \(\checkmark \)
(b).: Cauchy’s inequalities \(\checkmark \)
(c).: The uniqueness principle \(\checkmark \)
(d).: The maximum modulus principle and Schwarz’s lemma \(\checkmark \)
(e).: The open mapping principle \(\checkmark \)
(f).: Liouville’s theorem and the fundamental theorem of algebra \(\checkmark \)
(g).: Winding numbers, the argument principle and Rouché’s theorem \(\checkmark \)

5.

Singularities of analytic functions.

(a).: Classification of singularities \(\checkmark \)
(b).: Casorati–Weierstrass theorem \(\checkmark \)
(c).: Residue theorem \(\checkmark \)
(d).: Computation of definite integrals \(\checkmark \)

6.

Taylor and Laurent series.

(a).: Cauchy–Hadamard formula for the radius of convergence \(\checkmark \)
(b).: Abel’s theorem
(c).: Laurent series \(\checkmark \)
(d).: Infinite products
(e).: Expansions of elementary functions in infinite series and infinite products

7.

Conformal transformations.

(a).: Riemann’s mapping theorem
(b).: The reflection principle \(\checkmark \)
(c).: Elementary conformal transformations \(\checkmark \)

8.

Harmonic functions.

(a).: Maximum principle \(\checkmark \)
(b).: Mean value theorem
(c).: Poisson and Jensen formulae
(d).: Dirichlet problem
(e).: Subharmonic functions

The reading list set with the syllabus

1).: L. Ahlfors, Complex Analysis, McGraw–Hill.
2).: J. Conway, Functions of One Complex Variable, 2nd edition, Springer.
3).: K. Knopp, Theory of Functions, Parts I and II; Problem Book, Volumes I and II, Dover.
4).: R. Narasimhan, Complex Analysis in One Variable, Birkhäuser.
5).: W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw–Hill (undergraduate material, chapters 6 and 7 only).
6).: S. Saks and A. Zygmund, Analytic Functions, Warsaw.

Conway is the closest of these to the present notes and is already in the bibliography; Ahlfors is the standard reference for the material in sections 7 and 8 above.

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