Question

Find all soluctions to e^z=-c, where c>0 (c is positive real number) and z=x+iy.

Answer 1

Find the Laurent Series expansion of the function f(z)= 1/sin(z) that converges in a deleted neighborhood of the origin.

Answer 2

Find all soluctions to e^z=-c, where c>0 (c is positive real number) and z=x+iy.

Answer 3

Find u(x,y) and v(x,y) such that f(z)=u(x,y) + jv(x,y)= tanz.

Determine where f(z) is defined and show that u(x,y) and v(x,y) satisfy the Cauchy- Riemann equations at these points (x,y).

Answer 4

Evaluate the integral |z^2|dz where is the line segment from -j to 1.

Answer 5

Consider the function f(z)= e^(iz)/(z^2+1)(z^2+4).

Find the poles of f(z).
Compute the residues at each of the poles of f(z)/
Let C be a large semicircle centered at the origin in the upper half plane
(Im(z)\[\ge\]0).
Integral f(z) around C to show that cos(x)/(x^2+1)(x^2+4)dx =Pi/6e^2 (2e=1).