MAT 542  Assignment No 3  Due: Feb. 22, 2002

COMPLEX ANALYSIS

Department of Mathematics The University at Stony Brook

(1) Determine the poles and their orders of the following:

\frac{1}{e^{z} − 1} , \frac{1}{sin z} , (\frac{1}{cos z} )^{2} .

(2) Show that a)

\frac{1}{z^{2}} = 1 + ∑_{n=1}^{∞}(n + 1)(z + 1)^{n}

for |z + 1| < 1; b)

\frac{1}{z^{2}} = \frac{1}{4} + \frac{1}{4} ∑_{n=1}^{∞}(−1)^{n}(n + 1) (\frac{z − 2}{2})^{n}

for |z − 2| < 2.

(3) Prove the representation:

\frac{z}{(z − 1)(z − 3)} = \frac{−1}{2(z − 1)} − 3 ∑_{n=1}^{∞} \frac{(z − 1)^{n−1}}{2^{n+1}} ,

and determine the region in which it is valid.

(4) If f and g have power series expansions at z_{0} and f(z_{0}) = g(z_{0}) = 0, but g′(z_{0}) ≠ 0, show that

lim_{z→z_{0}} \frac{f(z)}{g(z)} = \frac{f′(z_{0})}{g′(z_{0})} .

(5) Show that f(z) = ∑_{n=0}^{∞} z^{2^{n}} is the only power series of the form ∑_{n=1}^{∞} a_{n}z^{n} with a_{0} = 0 which solves the functional equation f(z) = z + f(z^{2}).

(6) Evaluate

∑_{n=1}^{∞} \frac{(−1)^{n+1}}{n} .

Hint: Use Abel’s limit theorem.

(7) Let f and g be meromorphic functions on a domain D. Assume that there is a sequence {z_{n}} ⊂ D of distinct points with z_{0} = lim_{n→∞} z_{n} ∈ D such that f(z_{n}) = g(z_{n}), n = 1, 2, . . .. Show that f = g.

(8) a) Does there exist a meromorphic function f on ℂ such that

f(n) = ∞, n = 1, 2, 3, …?

b) Does there exist a function f having a power series expansion on ℂ such that f(\frac{1}{n}) = \frac{n}{n−1}, n = 2, 3, 4, …?

c) Does there exist an entire function f such that f^{(n)}(0) = 2^{n}n!, n = 0, 1, 2, …?

d) Does there exist an entire function f defined on |z| < δ for

some δ > 0 with f^{(n)}(0) = (n!)^{2}, n = 0, 1, 2, …?

(9) Let f be a meromorphic function on a domain D. Assume that for each z ∈ D either f(z) or f′(z) = 0. What can you conclude about f? Can you generalize the problem?

(10) What is wrong with the statement “z^{\frac{1}{2}} has a zero at z = 0?”