Complex Analysis Answers

Show that u(x,y)=e^xcosy and v(x,y)=e^xsiny are harmonic for all values of (x,y).

Show that the function f(z) = x^2 + y^2 + i 2xy has a derivative only at points that lie on

the x-axis.

   On the first day of the month, 4 customers come to a restaurant. Afterwards, those 4 customers come to the same restaurant once in 2,4,6 and 8 days respectively.

a)     On which day of the month, will all the four customers come back to the restaurant together?

b)     Briefly explain the technique you used to solve (a).

Which of the following sets are closed in C. (Justify your answer)

a) A={z∈C:Rez<1}∪{z∈C:Imz≥1}

b) B={z∈C:-∞<x≤3}

"\\intop" 1/(3-2cos"\\theta" +sin"\\theta" ) using contour integration limit 0 to 2pi

expand f(z)=z/(z-1)(2-z) in a laurent series valid 1<|z|<2

find the residue of f(z) =1/(z²+1)² at z=i

Find the laurent series about the indicated singularity for the function e^2z/(z-1)³ at z=1

Evaluate the integral "\\oint" 1/(z²+1)(z²-4)dz where |z|=1.5