MAT 542  Assignment No. 8  Due April 12, 2002

(1) Show that

π tan πz = 2z ∑_{n=1}^{∞} \frac{1}{(n − \frac{1}{2})^{2} − z^{2}} .

(2) Let f ∈ H(ℂ) . Assume that

f(z + 1) = f(z)

and

f(z + ı) = f(z),

for all z ∈ ℂ. Show that f must be constant.

(3) a) Prove that the infinite product

(1 + z)(1 + z^{2})(1 + z^{4})(1 + z^{8}) ⋯ = ∏_{n=0}^{∞} (1 + z^{2^{n}})

converges for |z| < 1.

b) Compute its limit; that is lim_{N→∞} ∏_{n=0}^{N} (1 + z^{2^{n}}).

(4) If f is holomorphic in |z| < 1 and |f(z)| < 1 for |z| < 1, prove that

\frac{|f′(z)|}{(1 − |f(z)|^{2})} ≤ \frac{1}{1 − |z|^{2}} for all |z| < 1.

Hint: Compose f with Möbius transformations.

(5) In the previous exercise, prove that equality at one point implies that f is a Möbius transformation ( = fractional linear map).

(6) If f(z) is analytic on ℂ, real on the real axis, and purely imaginary on the imaginary axis, show that f(z) is odd. (I.e., that f(−z) = −f(z).)

(7) Assume that f(z) is analytic on |z| < 1, continuous on |z| ≤ 1 and |f(z)| = 1 for |z| = 1. Show that f(z) is a rational function.

(8) Assume that f(z) is analytic in 0 < |z| < 1 and |f(z)| ≤ |z|^{− \frac{1}{2}} . Show that z = 0 is a removable singularity of f.

(9) Let f ∈ M(ℂ), and assume that

f(z + 1) = f(z) and f(z + ı) = f(z),

for all z ∈ ℂ.

a) Can f have a simple pole at z = 0 (thus, also at z = 1, ı, 1 + ı) and no other poles for z in the parallelogram P, where P = {z ∈ ℂ; 0 ≤ ℜz ≤ 1 and 0 ≤ ℑz ≤ 1.}.

b) Same as a) with double poles. Proof or construction required here!

c) Assume that f is holomorphic on the boundary of the parallelogram P. Compute ∑_{z∈i(P)} Res(f, z).

(10) Show that the polynomials are dense in (H(ℂ), ρ).

(11) Let n ≥ 3 be an integer. Find all entire functions f and g that

solve

f^{n} + g^{n} = 1.