Math 213a: Complex analysis Problem Set #5 (22 October 2003):

Series of analytic functions; partial fractions and product formula

1. i) [Ahlfors IV.3.1, Ex.2 (p.153)] Express ∑_{n=−∞}^{∞} 1/(z^{3} − n^{3}) in closed form.

ii) [Seen in B. Amend’s Foxtrot cartoon, 6.ii.1996] Evaluate in closed form: ∑_{k=1}^{∞}(−1)^{k+1}k^{2}/(k^{3} + 1).

2. We showed that if f_{n} are analytic functions with f_{n} → f uniformly on compact subsets of Ω then f is analytic and f′ = lim_{n→∞} f′_{n} uniformly on compacta. We noted that as it stands this result requires that Ω ⊆ C and that f_{n}, f take values in C.

i) Give a definition for “uniform convergence on compacta” for sequences of maps between any two Riemann surfaces S, S′, and show that your definition is reasonable in that it agrees with the usual definition for subsets of C and does not depend on any choices made in the definition (such as local coordiante patches).

ii) Do the same for “f′ = lim_{n→∞} f′_{n} uniformly on compacta”. iii) Now extend “our” theorem to sequences of analytic maps from S to S′.

[This should be the easy part.]

3. [Suggested by the remark in Ahlfors p.155] For a_{n} ∈ C such that |a_{n}| < 1 for each n, let P and S be respectively the infinite product ∏_{n=1}^{∞}(1 + a_{n}) and the infinite series ∑_{n=1}^{∞} a_{n}.

i) Assume ∑_{n=1}^{∞} |a_{n}|^{2} < ∞. Prove then P converges if and only if S converges.

ii) Without the hypothesis on ∑_{n=1}^{∞} |a_{n}|^{2}, show that there exist a_{n} for which P converges but S does not, and a_{n} for which P does not converge but S does. Can all a_{n} be real?

4. [Prelude to doubly periodic functions] Let A_{0} be the ring of meromorphic functions f on C such that f(z + 1) = f(z) for all z ∈ C and there exists a limit L = L(f) ∈ C such that f(z) → L as | Im(z)| → ∞.

i) Prove that there exists M < ∞ and poles b_{ν} (ν = 1, 2, …, M) of f such that every pole of f is b_{ν} + k for some unique ν ∈ {1, 2, …, M} and k ∈ Z. (Of course if f is analytic then we take M = 0 and interpret “{1, 2, …, M}” as the empty set.)

ii) Prove that ∑_{ν=1}^{M} Res_{z=b_{ν}} f(z) dz = 0. iii) Prove that

f(z) = L(f) + ∑_{k∈Z}[∑_{ν=1}^{M} P_{ν} (\frac{1}{z − (b_{ν} + k)})],

where P_{ν}(1/(z − b_{ν})) is the principal part of f at b_{ν}, and the series converges uniformly in compact subsets of C (with the usual interpretation

at z = b_{ν}). iv) Conclude that A_{0} is the ring of rational functions of e^{2πiz}, say g(e^{2πiz}),

that are regular at 0 and ∞ with g(0) = g(∞).

v) Let A ⊃ A_{0} be the ring of meromorphic functions f on C such that f(z + 1) = f(z) for all z ∈ C and there exist limits L_{±} = L_{±}(f) ∈ C such that f(z) → L_{+} as Im(z) → +∞ and f(z) → L_{−} as Im(z) → −∞. Adapt the argument for (i–iv) to identify A with a suitable ring of rational functions of e^{2πiz}.

A Jensen-style inequality for analytic functions on a half-plane, and an application:

5. i) Let f(z) be a bounded analytic function on the right half-plane Re(z) > 0, with real zeros at x_{k} of multiplicity m_{k}. Show that either f vanishes identically or ∑_{x_{k}>1} m_{k}/x_{k} converges.

ii) Let α(t) be a continuous complex-valued function on the interval [0, 1]. Show that if ∫_{0}^{1} t^{p_{k}}α(t)dt = 0 for distinct p_{k} > 0 with ∑_{k} p_{k}^{−1} = ∞ then α = 0 identically.

In the special case p_{k} = 1, 2, 3, …, this is an easy consequence of Weierstrass’ Approximation Theorem, stating that any continuous function on a closed interval is a uniform limit of polynomials. [Consider 0 ≤ ∫_{0}^{1}(t − t^{2})|α(t)|^{2} dt = ∫_{0}^{1}((t − t^{2})ᾱ(t))α(t) dt, and approximate the ᾱ(t) uniformly by a polynomial.] That Weierstrass result is generalized by a memorable theorem of Müntz: fix 0 = p_{0} < p_{1} < p_{2} < p_{3} < …; then every continuous function on [0, 1] is a uniform limit of linear combinations of the powers t^{p_{k}} if and only if ∑_{k=1}^{∞} p_{k}^{−1} diverges. Indeed, one approach to this theorem uses part (ii) above and related ideas from complex analysis.

This problem set is due Wednesday, October 29, at the beginning of class.