# Quiz 2 Summer 2011

1. Find all complex numbers z satisfying
$\cos z=\frac{13}{5}$
Use only real-valued elementary functions of real variable in your answer (e.g. arccos (13/5) is not a valid answer, as arccos x is not real when |x|>1)

Solution:
$\frac{e^{iz}+e^{-iz}}{2}=\frac{13}{5}$
is the same as
$e^{iz}+1/e^{iz}=26/5$
Let t=e^{iz}. Then
$t+1/t=26/5$
or
$t^2+1=26/5 t$
Solving this quadratic equation for t we find t=5, 1/5 so that e^{iz}=5,1/5 or
$iz=\ln 5 + 2 \pi i k, \ln 1/5 + 2 \pi k$
or
$z=\pm \ln 5 + 2 \pi k$
for any integer k.

2. Find all functions f of variable z=x+iy that are complex-differentiable for all complex z and whose imaginary part is Im f(z)=2xy+x

Solution: Let u and v denote the real and imaginary parts of f. Then Cauchy-Riemann equations imply
$u_x=v_y=2 x$
and hence u=x^2+c(y). The other equation in Cauchy-Riemann equations then says that
$u_y=c'(y)$
must be the same as
$-v_x=2y+1$
so that c(y)=-y^2-y+c. Finally
$f(x+iy)=x^2-y^2-y+c+i(2xy+x)$
or
$f(z)=z^2+iz+c$
with real c.

3. Find the radius of convergence of the series
$\sum_{n=0}^{\infty} \left(2i\cos \frac{2\pi n}{3} \right)^{2n} z^n$

Solution:
$R=\liminf \frac 1 {\left|2i\cos \frac{2\pi n}{3} \right|^{2}}=\liminf \frac 1 {\left|2\cos \frac{2\pi n}{3} \right|^{2}}=\frac 1 4$
(since
$\left|2\cos \frac{2\pi n}{3} \right|$
is the sequence 2,1,1,2,1,1,2,1,1,...)