Math 5283, HW Set 2, Fall 2006
Due: Wednesday, September 13, 2006
1. Let h={zโC: Im z > 0}denote the upper half plane in C. Show that
PGL+(2,R) maps hto hand PGLโ(2,R) maps hto the lower half plane.
2. Consider the function defined by
f(x+iy) = p|x| |y|
where x, y โR. Show that fsatisfies the Cauchy-Riemann equations at the
origin, yet fis not holomorphic there.
3. The Bessel function of order r, a positive integer, is defined via
Jr(z) = ๎z
2๎r
โ
X
n=0
(โ1)n
n! (n+r)! ๎z
2๎2n
.
Find its radius of convergence.
4. Page 28, Problem 4 in Ahlfors.
5. (i) Page 37, Problem 3 in Ahlfors.
(ii) Let anโR. Assume that Pโ
n=1 anis convergent but not absolutely (con-
ditional convergence). Prove that one can rearrange the terms in the series in
such a way that it converges to any arbitrary real number.
6. Page 41, Problems 8 and 9 in Ahlfors.
7. Page 47, Problem 7 in Ahlfors.
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