Mathematics 132/2 Terence Tao First Midterm, April 23, 1997 Try to do all five problems; they are all of equal value. Read each question in its entirety before attempting any portion of it, as later parts of a question may assist you with earlier parts. There is plenty of working space, and a blank page at the end. Good luck!

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

Total:

Problem 1. Consider the identity

|z + w|^{2} = |z|^{2} + |w|^{2} + 2Re(z\overline{w}).  (∗)

(a) Verify (*) when z = 1 − i and w = 2i.

(b) Show that (*) holds for all complex z and w.

Problem 2. Let u be the function

u(x + iy) = cos(x) cosh(y)

defined on the entire complex plane.

(a) Show that u is harmonic, by whatever means you wish.

(b) Find an entire function f whose real part is equal to u.

Ans.

Problem 3. Let f be the Möbius transform

f(z) = \frac{z}{z + 1}.

(a) What is the image of the closed unit disk {|z| ≤ 1} under this transformation? Ans.

(b) What point maps to the complex number 2i under this transformation?

Ans.

(c) Find a Möbius transformation that sends the right semi-circle {|z| = 1, Rez ≥ 0} to the interval {Re(z) = 0, −1 ≤ Im(z) ≤ 1}.

Ans.

Problem 4.

(a) Compute all values of (−1)^{i}.

Ans.

(b) Find a branch f of the multi-valued function z^{i} which is continuous on the left half-plane {Rez < 0}.

Ans.

(c) Repeat question (b), but with the additional requirement that f(−1) = e^{5π}. Ans.

Problem 5. Let f be the function

f(x + iy) = y^{2} + ixy.

(a) Find all points in the complex plane where f is differentiable. Justify your answer. Ans.

(b) Find all points in the complex plane where f is analytic.

Ans.

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