Literature: S. Lang:
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 (email@example.com) 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.
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,
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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