Quiz+2+Summer+2011

1. Find all complex numbers z satisfying math \cos z=\frac{13}{5} math 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: math \frac{e^{iz}+e^{-iz}}{2}=\frac{13}{5} math is the same as math e^{iz}+1/e^{iz}=26/5 math Let t=e^{iz}. Then math t+1/t=26/5 math or math t^2+1=26/5 t math Solving this quadratic equation for t we find t=5, 1/5 so that e^{iz}=5,1/5 or math iz=\ln 5 + 2 \pi i k, \ln 1/5 + 2 \pi k math or math z=\pm \ln 5 + 2 \pi k math 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 math u_x=v_y=2 x math and hence u=x^2+c(y). The other equation in Cauchy-Riemann equations then says that math u_y=c'(y) math must be the same as math -v_x=2y+1 math so that c(y)=-y^2-y+c. Finally math f(x+iy)=x^2-y^2-y+c+i(2xy+x) math or math f(z)=z^2+iz+c math with real c.

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

Solution: math 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 math (since math \left|2\cos \frac{2\pi n}{3} \right| math is the sequence 2,1,1,2,1,1,2,1,1,...)