Mathematics 132/2 Terence Tao Final Examination, Jun 13, 1997 The test consists of nine problems of varying difficulty and value, adding up to 200 points. Note that all numerical answers must be either in Cartesian or polar form in order to obtain full credit. When asked to classify a singularity or a zero, give the type of singularity (removable, pole, essential), and (if a pole or zero) give its order. If a removable singularity has a zero of a certain order after the singularity is removed, indicate it as such. Good luck!

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Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9.

Total

Math. 132/2, Final Examination, Spring 1997. page 2

Problem 1. (25pts) Let w = f(z) be the Möbius transform that sends the three points 0, 1, −1 to 0, 1, ∞ respectively.

(a) Find an explicit formula for f. Verify that f(0) = 0, f(1) = 1, and f(−1) = ∞. Ans.

(b) Find the image of {z : |z| < 1} under f.

Ans. Problem 1 continues on the next page.

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Problem 1 continued.

(c) Compute f^{−1}(i).

Ans.

(d) Classify the singularities of f at −1 and at ∞. Ans.

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Problem 2. (25pts) Let f be the function f(z) = \frac{1}{z^{3}(z + 1)}.

(a) Find all the Laurent series expansions of f(z) around z_{0} = 0, and state the regions in which each expansion is valid. (Ignore any issues concerning the boundary of these regions).

Ans.

(b) Classify the singularities of f in the complex plane, and compute the residues at these singularities.

Problem 2 continues on the next page.

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Problem 2 continued.

(c) Evaluate the integral of f(z) on the contour {z : |z| = 2}, traversed once in the anti-clockwise direction.

Math. 132/2, Final Examination, Spring 1997. page 6

Problem 3. (15pts) Consider the function f = u + iv, where

u(x + iy) = x^{2} + 2xy + y^{2}, v(x + iy) = x^{2} − 2xy − y^{2}.

[Warning: be careful of sign errors in this question.]

(a) Where is f complex differentiable? Where is f analytic?

Ans.

(b) Does there exist an entire function g whose real part is equal to u? If so, provide an example of such a function.

Ans.

(c) Does there exist an entire function h whose imaginary part is equal to v? If so, provide an example of such a function.

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Problem 4. (20 pts)

(a) What are all the possible values that the multi-valued function log(i − z) can take when z = 1?

Ans.

(b) Let f(z) be the principal branch of log(i − z). Where is f analytic?

Ans.

(c) Find a branch of log(i − z) which takes the value of \frac{1}{2} ln2 − \frac{5πi}{4} at 1.

Ans.

Problem 5. (10pts) Suppose f is an entire function and M is a positive number such that |f(z)| > M for all complex numbers z. Prove that f must be a constant function. (Hint: look at the reciprocal of f).

Problem 6. (25pts) Evaluate p.v. ∫_{−∞}^{∞} \frac{cos(x + 1)}{x^{2} + 9} dx. Show your reasoning.

Problem 7. (25pts) Evaluate ∫_{0}^{π} \frac{1}{2 + sin2θ} dθ. Show your reasoning.

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Problem 8. (20pts) Classify the zero or singularity of the following functions at the indicated point(s). If it is a singularity, compute its residue.

(a) \frac{cos(z)}{2z − π} at z = π/2 and z = −π/2.

Ans.

(b) e^{\frac{1}{z}} (1 − z) at z = 0.

Ans.

(c) \frac{Log^{2}(z)}{(z − 1)e^{z}} at z = 1.

Ans.

(d) \frac{z^{2} + 1}{z^{3} − z^{5}} at z = 0. (Hint: find the first few terms of the Laurent series.)

Problem 9. (35 pts) Let f be the function f(z) = \frac{L_{−π/2}(z)}{z^{2} + 4}, and let R > 2 > ε > 0 be positive numbers. Let γ_{1} be the straight line from −R to −ε, γ_{2} be the upper semicircle of radius ε around the origin, oriented clockwise, γ_{3} be the straight line from ε to R, and γ_{4} be the upper semicircle of radius R around the origin, oriented anti-clockwise.

(a) Compute the integral

∫_{γ_{1}+γ_{2}+γ_{3}+γ_{4}} f(z) dz.

Ans.

(b) Show that

∫_{γ_{2}} f(z) dz → 0 as ε → 0.

(c) Show that

∫_{γ_{4}} f(z) dz → 0 as R → ∞.

Problem 9 continues on the next page.

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Problem 9 continued.

(d) Show that

∫_{γ_{3}} f(z) dz = ∫_{ε}^{R} \frac{lnx}{x^{2} + 4} dx

and

∫_{γ_{3}} f(z) dz = ∫_{ε}^{R} \frac{lnx}{x^{2} + 4} dx + ∫_{ε}^{R} \frac{πi}{x^{2} + 4} dx.

(e) Based on your answers to (a-d) or otherwise, compute the integrals p.v. ∫_{0}^{∞} \frac{ln(x)}{x^{2} + 4} dx and ∫_{0}^{∞} \frac{dx}{x^{2} + 4}. (Hint: take the real and imaginary parts of your answer to (a)).

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