Math 213a: Complex analysis Problem Set #4 (15 October 2003): Contour integrals II

1. [Qualifying Exam, Spring 1997] For b > 0, compute ∫_{0}^{∞} log x dx/(x^{2} + b^{2}).

2. Evaluate ∫_{0}^{π} log |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

∫_{0}^{∞} \frac{cos(mx)}{cosh(πx)} e^{−mx^{2}}dx = \frac{1}{2} e^{−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 f be a nonconstant analytic function on a neighborhood of z_{0}, and let n be the multiplicity of the zero at z_{0} of the function f(z)−f(z_{0}). We have seen that, for any a sufficiently close to f(z_{0}), the equation f(z) = a has n solutions (counted with multiplicity) near z_{0}, call them z_{1}, …, z_{n} in some order. Prove that the coefficients of the polynomial ∏_{j=1}^{n}(X − z_{j}) are analytic functions of a in that neighborhood of z_{0}. [These coefficients are (up to sign) the elementary symmetric functions in the z_{j}. For n = 1 the claim is equivalent to the existence of an analytic inverse function. To prove it in general, show that ∑_{j=1}^{n} z_{j}^{k} is an analytic function of a for each k = 1, 2, 3, . . ..]

5. For any two disjoint circles C_{1}, C_{2} in the Riemann sphere P^{1}(C), define I(C_{1}, C_{2}) as the value of the double path integral with respect to arc length:

I(C_{1}, C_{2}) := ∮_{C_{1}} ∮_{C_{2}} \frac{|dz_{1}| |dz_{2}|}{|z_{1} − z_{2}|^{2}} .

This may be an improper integral if either C_{1} or C_{2} passes thru ∞ (i.e., is a straight line in C), but even in that case the integral clearly converges.

i) Show that I(C_{1}, C_{2}) is PGL_{2}(C)-invariant; that is, that I(ϕC_{1}, ϕC_{2}) = I(C_{1}, C_{2}) for any fractional linear transformation ϕ : P^{1}(C) → P^{1}(C).

ii) Determine I(C_{1}, C_{2}) as a function of the radii R_{1}, R_{2} of the circles and the distance d between their centers. [Check: when d = 0, so C_{1}, C_{2} are concentric, you should obtain

I(C_{1}, C_{2}) = 4π^{2} \frac{R_{1}R_{2}}{|R_{1}^{2} − R_{2}^{2}|} ;

more generally, the formula should diverge when d = |R_{1} − R_{2}|.] What is I(C_{1}, C_{2}) if C_{1} = R? To be continued…

This problem set is due Wednesday, October 22, at the beginning of class.