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:

is the same as

Let t=e^{iz}. Then

or

Solving this quadratic equation for t we find t=5, 1/5 so that e^{iz}=5,1/5 or

or

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

and hence u=x^2+c(y). The other equation in Cauchy-Riemann equations then says that

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:

is the same as

Let t=e^{iz}. Then

or

Solving this quadratic equation for t we find t=5, 1/5 so that e^{iz}=5,1/5 or

or

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

and hence u=x^2+c(y). The other equation in Cauchy-Riemann equations then says that

must be the same as

so that c(y)=-y^2-y+c. Finally

or

with real c.

3. Find the radius of convergence of the series

Solution:

(since

is the sequence 2,1,1,2,1,1,2,1,1,...)