Mathematics 132/2 Terence Tao Second Midterm, May 14, 1997 Try to do all five problems; they are all of equal value. There is plenty of working space, and a blank page at the end. You should supply brief reasons to justify all your answers. Good luck!

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Problem 1. Problem 2. Problem 3. Problem 4. Problem 5.

Total:

Problem 1. Let Γ be the contour traversing the circle |z| = 2 twice in the anti-clockwise direction.

(a) Find a parameterization of Γ.

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(b) Compute ∫_{Γ} |z| dz.

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(c) Compute ∫_{Γ} \frac{z + e^{z}}{z(z − 4)} dz.

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Problem 2. Let Γ_{1}, Γ_{2}, and Γ_{3} be parameterized by z_{1}(t) = (1 + i)t 0 ≤ t ≤ 2

z_{2}(t) = (1 − i)t 0 ≤ t ≤ 2
z_{3}(t) = 2 + it − 2 ≤ t ≤ 2

(a) Sketch Γ_{1}, Γ_{2}, and Γ_{3} in the complex plane.

(b) Let f be a function in the complex plane. Give a condition (or set of conditions) on f which is sufficient to ensure that ∫_{Γ_{1}} f(z) dz − ∫_{Γ_{2}} f(z) dz − ∫_{Γ_{3}} f(z) dz = 0.

(c) Let f = \frac{1}{z − 1}. Compute ∫_{Γ_{1}} f(z) dz − ∫_{Γ_{2}} f(z) dz − ∫_{Γ_{3}} f(z) dz.

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Problem 3. Let γ be the contour that traverses the line segment from 1 − \frac{i}{2} to 1 + \frac{i}{2} once in the upward direction.

(a) Find a branch of the multi-valued function log(2z − 1) which is analytic in a domain that contains γ.

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(b) Based on your answer to (a), or otherwise, evaluate the contour integral ∫_{γ} \frac{2}{2z − 1} dz.

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Problem 4. Let Γ be the circle {|z| = 10} traversed once in the positive (anti-clockwise) direction.

(a) Compute ∫_{Γ} \frac{e^{z^{2}}}{(2z − 1)^{2}} dz.

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(b) Compute ∫_{Γ} \frac{sin z}{(z − 1)(z − 3)} dz.

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(c) Estimate the size of the integral ∫_{Γ} \frac{e^{iRe(z)}}{z^{2}} dz.

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Problem 5. Let f be the power series

f(z) = ∑_{n=0}^{∞}(n + 1)\frac{(z − 1)^{n}}{2^{n}} = 1 + 2\frac{z − 1}{2} + 3\frac{(z − 1)^{2}}{2^{2}} + 4\frac{(z − 1)^{3}}{2^{3}} + …

(a) Find the disk of convergence of this power series.

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(b) Find a power series expansion for \frac{2}{3 − z} around the point z_{0} = 1. Where does this power series converge?

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(c) Based on your answer to (b), or otherwise, find a closed form expression for f. (Hint: use differentiation).

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