Literature: S. Lang: Complex Analysis 4th ed. 1999. Corr. 2nd printing 2001. XIV, 485 pp. 139 figs. Hardcover ISBN 0-387-98592-1

Schedule: Weekly on Mondays from 11:00 till 13:00 hrs. in Room BBL 105B, starting Sept 6. The exercise sessions are run by Ettay Weiss (weiss@math.uu.nl) with the assistance of Jan Willem de Jong. They take place on Wednesdays from 9:00 till 11:00 hrs in Room BBL 105B, starting Sept 8.

Material covered:

Sept 6: elementary properties of complex numbers, complex derivative and anti-complex derivative of a differentiable map.

Sept 13: holomorphic function, Cauchy-Riemann equations, Laplace operator, harmonic function, conformal property of a holomorphic function.

Sept 27: line integrals in the complex plane, existence of a holomorphic primitive of a holomorphic function on an open rectangle, Morera's theorem, Cauchy's integral formula for a disk, integral representation of the derivatives of a holomorphic function.

Oct 4: Cauchy's estimate for the derivatives of a holomorphic function, Liouville's theorem, main theorem of algebra, various notions of convergence (absolute, uniform, normal), a normally convergent sequence of holomorphic function has a holomorphic limit, convergence issues for series, Abel's theorem, radius of convergence, representation of a holomorphic function by a Taylor series.

Oct 11: a holomorphic function at z_0 is identically zero on a neighborhood of z_0 or is nowhere zero on a punctured neighborhood of z_0, the zero set of a holomorphic function on a connected open subset U is either all of U or has no accumulation point in U, uniqueness of analytic continuation, Riemann removable singularity theorem, meromorphicity (and its characterization by growth behavior), order of a meromorphic function at a point, essential singularity, Casorati-Weierstrass theorem.

Oct 18: punctured neighborhood of infinity and related notions, a function meromorphic on the complex plane and infinity is a rational function. Convergence of Laurent series on an open annulus, Cauchy formulae for a homolomorphic function on an open annulus.

Oct 25: residues, winding number, residue formula.

Nov 1: applications of residue formula: evaluation of definite integrals

Nov 15: applications of residue formula: Fourier and Mellin transform, Rouche's theorem.

Nov 22: biholomorphic maps, $n$-th root and logarithm as a holomorphic function, geometry of a holomorphic function, Schwarz lemma.

Nov 29: biholomorphic automorphisms of the unit disk, upper half plane and of the Riemann sphere (fractional linear transformations).

Dec 6: generalized circles on the Riemann sphere, cross ratio and its projective invariance, some examples of biholomorphisms.

Problems (from Lang's book, unless stated otherwise):

Sept 6: I-1: 1h, 2c, 7, 10c; I-2: 1h, 2b, 3, 9, 10.

Sept 13: exercises given in class and Lang: I-3:1,2,3.

Sept 20: exercise given in class (prove that the exponential is holomorphic and equal to its own derivative) and Lang: III-2:1a and III-7:1.

Sept 27: exercise given in class (prove that a holomorphic function defined on the whole complex plane whose absolute value is bounded by an m-th power of |z| for |z| large, has a derivative with the same property with respect to m-1 (here m may be any real number); prove also that such a function must be a polynomial) and Lang: V-1:1, 2, 3a, 3b with 4 and 5 as extra.

Oct 4: II-2: 2, 3, 4a,b,d,g, 6, 10.

Oct 11: exercise given in class (prove that for every nonzero complex number w there exists a sequence (z_n)_n of complex numbers converging to zero such that exp(1/z_n)=w for all n); V-3: 1a,d, 3.

Oct 18: V-3: 4, 5, 6, 8, 9.

Nov 1: VI-2: 5, 7, 10.

Nov 15: VI-2: 11, 12, 15; VI-1: 31.

Nov 22: VI-2: 19, 22; VI-1: 32, 33.

Nov 29: VII-2: 1, 5; VII-3: 1, 2, 5.

Dec 6: VII-5: 9a,b,c,d,e, 10a,b,c, 12b; VII-4: 2a,b,c.

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