Masters Exam Complex Analysis January 1989

Instructions: Do any five of the following problems.

1. Determine whether

u(x, y) = x^{3} − 3xy^{2} + x^{2} − y^{2}

is the real part of an entire function. If so, construct the entire function.

2. Determine whether or not an entire function f(z) exists satisfying the following conditions. If there is, give an example. If not, explain why.

(a) f(0) = 1, f(i) = 0, and |f(z)| ≤ 6 for all z.

(b) f(z) = 0 for all z = nπi, n = 0, ±1, ±2, ±3, ....

(c) |f(i)| = 2 and |f(z)| = 1 for all z with |z| = 2.

3. For each of the following functions determine whether z = 0 is a removable singularity, an essential singularity, or a pole. If it is a pole, give its order. In all cases, compute the residue at z = 0.

(a) \frac{(1 + z) sin(z)}{z^{2}}  (b) cosh(\frac{3}{z^{2}} )  (c) \frac{cosh(z^{2}) − 1}{z^{3}}

4. Find a series expansion (Taylor or Laurent) for the given function centered at z_{0}, valid in a neighborhood of z_{0} (possibly with z_{0} removed). Determine the domain of convergence of the series.

(a) \frac{1}{(z − 1)^{3}z} centered at z_{0} = 1.  (b) sinh(z) centered at z_{0} = \frac{πi}{2}.

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5. Compute one of the following two integrals:

(a) ∫_{−∞}^{∞} \frac{cos(x)}{(x^{2} + 1)(x^{2} + 4)} dx  (b) ∫_{−∞}^{∞} \frac{x^{2}}{(x^{2} + 9)(x^{2} + 4)} dx

6. (a) Suppose that f(z) is analytic on and inside the circle of radius R centered at z_{0}. Let

M(R) = {max \atop |z − z_{0}| = R}|f(z)|

be the maximum of f on this circle. Use the Cauchy integral formulas to show that

|f^{(n)}(z_{0})| ≤ \frac{n! M(R)}{R^{n}}.

(b) Show that if f(z) is an entire function which satisfies |f(z)| ≤ |z|^{M} for a fixed positive integer M then f(z) must be a polynomial of degree at most M.

7. (a) Write down the linear fractional transformation T mapping D = {z : |z| < 1} to H = {w : Im(w) > 0} and normalized so that T (0) = i, T (i) = ∞, and T (−i) = 0.

(b) What is the image of H under this mapping?

(c) Is this mapping conformal on D? Explain why or why not.