Math 213a: Complex analysis
Problem Set #4 (15 October 2003):
Contour integrals II
1. [Qualifying Exam, Spring 1997] For b > 0, compute R∞
0log x dx/(x2+b2).
2. Evaluate Rπ
0log |a+ cos(θ)|dθ as a function of a≥0. [As you might expect, the
formula will depend on whether a≤1 or a≥1. Can you obtain this formula
without using complex analysis?]
3. Prove that Z∞
0
cos(mx)
cosh(πx)e−mx2dx =1
2e−m/4
for all m > 0. [From the Fall 1998 Qualifying Exam.] Can you evaluate any other
such integrals this way (other than those obtained trivially from this formula by
linear change of variable etc.)?
4. Let fbe a nonconstant analytic function on a neighborhood of z0, and let nbe the
multiplicity of the zero at z0of the function f(z)−f(z0). We have seen that, for any
asufficiently close to f(z0), the equation f(z) = ahas nsolutions (counted with
multiplicity) near z0, call them z1, . . . , znin some order. Prove that the coefficients
of the polynomial Qn
j=1(X−zj) are analytic functions of ain that neighborhood
of z0. [These coefficients are (up to sign) the elementary symmetric functions in
the zj. For n= 1 the claim is equivalent to the existence of an analytic inverse
function. To prove it in general, show that Pn
j=1 zk
jis an analytic function of afor
each k= 1,2,3, . . ..]
5. For any two disjoint circles C1,C2in the Riemann sphere P1(C), define I(C1, C2) as
the value of the double path integral with respect to arc length:
I(C1, C2) := IC1IC2
|dz1| |dz2|
|z1−z2|2.
This may be an improper integral if either C1or C2passes thru ∞(i.e., is a straight
line in C), but even in that case the integral clearly converges.
i) Show that I(C1, C2) is PGL2(C)-invariant; that is, that I(φC1, φC2) = I(C1, C2)
for any fractional linear transformation φ:P1(C)→P1(C).
ii) Determine I(C1, C2) as a function of the radii R1, R2of the circles and the distance d
between their centers. [Check: when d= 0, so C1, C2are concentric, you should
obtain
I(C1, C2) = 4π2R1R2
|R2
1−R2
2|;
more generally, the formula should diverge when d=|R1−R2|.] What is I(C1, C2)
if C1=R?
To b e continued.. .
This problem set is due Wednesday, October 22, at the beginning of class.