Be sure that this examination has 2 pages.
The University of British Columbia
Final Examinations - December 2010
Mathematics 305
R. Froese, M. Ward
Closed book examination. No notes, texts, or calculators allowed. Time: 21
2hours
Special Instructions: No notes, book, or calculator allowed
Marks
[40] 1. Identify whether each of the following statements are true or false. You must give
reasons for your answers.
(i) Arg(z1z2) = Arg(z1) + Arg(z2).
(ii) Re (i/¯z) = −Im(z)/|z|2.
(iii) sin(nθ) = Im{(cos θ+isin θ)n}where nis a positive integer.
(iv) f(z) = |z|2is analytic at z= 0 but not at any other point.
(v) u=rncos(nθ) is a harmonic function, where nis a positive integer, r2=x2+y2
and tan θ=y/x.
(vi) If f(z) = u+iv is an entire function, then u2−v2is a harmonic function.
(vii) Let M= max(|eiz2|) over the disk |z| ≤ 2. Then, M= 1.
(viii) |sin(z)|is bounded as |z| → ∞.
(ix) the equation √z+ (1 −i) = 0, where √zis the principal branch of the square
root function, has no solution.
(x) |ez2| ≤ e|z|2for all z.
(xi) log(ez) = z.
(xii) RCz−1/2sin(√z)dz = 0 where Cis the simple closed curve |z|= 1 oriented
counterclockwise, and √zis the principal branch of the square root function.
[15] 2. Consider the function f(z) defined by
f(z) = z
z2−z−2,
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