HOMEWORK 2.
Due Friday, February 13.
1. Find the radius of convergence
∞
P
n=0
(n+an)zn, where a∈C.
2. Let R1,R2, and Rdenote the radii of convergence of Panzn,Pbnzn, and
Panbnzn, resp. Prove or disprove: R1R2≤R.
3. Suppose that the radius of convergence of Panznis 1 and that the finite
limit lim
r→1−
Panrnexists. Prove or disprove: the series Panconverges.
4. Suppose that u(x, y) and v(x, y) are continuous and have continuous par-
tial derivatives of first order at z0=x0+iy0. If f=u+iv and the limit
limh→0|f(z0+h)−f(z0)|/|h|exists, then either for ¯
f(= u−iv) has derivative
at z0.
5. Prove or disprove: |sin z| ≤ 1 for all z∈C.
6. Find the error in the following reasoning (ascribed to J. Bernoulli): (−z)2=
z2so 2 log(−z) = 2 log zand therefore log(−z) = log z.
7. Compute (−3+4i)1+i.
8. Determine the analytic function fif |f|= (x2+y2)ex.
