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 P∞
n=−∞ 1/(z3−n3) in closed form.
ii) [Seen in B. Amend’s Foxtrot cartoon, 6.ii.1996] Evaluate in closed form:
P∞
k=1(−1)k+1k2/(k3+ 1).
2. We showed that if fnare analytic functions with fn→funiformly on com-
pact subsets of Ω then fis analytic and f0= limn→∞ f0
nuniformly on
compacta. We noted that as it stands this result requires that Ω ⊆Cand
that fn, f take values in C.
i) Give a definition for “uniform convergence on compacta” for sequences of
maps between any two Riemann surfaces S, S0, and show that your defi-
nition is reasonable in that it agrees with the usual definition for subsets
of Cand does not depend on any choices made in the definition (such as
local coordiante patches).
ii) Do the same for “f0= limn→∞ f0
nuniformly on compacta”.
iii) Now extend “our” theorem to sequences of analytic maps from Sto S0.
[This should be the easy part.]
3. [Suggested by the remark in Ahlfors p.155] For an∈Csuch that |an|<1
for each n, let Pand Sbe respectively the infinite product Q∞
n=1(1 + an)
and the infinite series P∞
n=1 an.
i) Assume P∞
n=1 |an|2<∞. Prove then Pconverges if and only if Scon-
verges.
ii) Without the hypothesis on P∞
n=1 |an|2, show that there exist anfor which
Pconverges but Sdoes not, and anfor which Pdoes not converge but S
does. Can all anbe real?
4. [Prelude to doubly periodic functions] Let A0be the ring of meromorphic
functions fon Csuch that f(z+ 1) = f(z) for all z∈Cand there exists
a limit L=L(f)∈Csuch that f(z)→Las |Im(z)| → ∞.
i) Prove that there exists M < ∞and poles bν(ν= 1,2, . . . , M ) of fsuch
that every pole of fis bν+kfor some unique ν∈ {1,2, . . . , M }and
k∈Z. (Of course if fis analytic then we take M= 0 and interpret
“{1,2, . . . , M }” as the empty set.)
ii) Prove that PM
ν=1 Resz=bνf(z)dz = 0.
iii) Prove that
f(z) = L(f) + X
k∈Z"M
X
ν=1
Pν1
z−(bν+k)#,
where Pν(1/(z−bν)) is the principal part of fat bν, and the series con-
verges uniformly in compact subsets of C(with the usual interpretation
