Complex integrals

(1) let f:C----R be an analytic function such that f(1)=1. Find the value of f(3) (2) Evaluate the integral over & of dz/ z^2 -1 where & is the circle |z-i|=2 (3)Evaluate the integral over & of (z-1/z) dz where & is the line path from 1 to i (4) Evaluate the integral between 2pi and 0 of e^-i@ . e ^e^i@ d@ ...continues

Complex function: Decomposition into its real and imaginary part.

Given a function w = f(z) = z^2 then decompose f(z) into its real and imaginary parts.

Find the existence of a limit in a complex value function

Find the existence of a limit in a complex value function. Please see the attached file.

Harmonic complex conjugates

Given u = y^3 - 3x^2y, find f(z) = u + iv such that f(z) is analytic. The solution is detailed and well presented.

Complex Numbers: Logarithms

Given: z1 = i and z2 = -1-i Show that: Log (z1 z2) = Log z1 + Log z2

Complex Number Logarithms

Given: z1 = i and z2 = -1+i Show that: Log (z1 z2) ≠ Log z1 + Log z2

Complex Logarithms: Discontinuity

Show that Log (z) is discontinuous along the negative real axis.

Working with complex numbers.

Show that if z0 is an nth root of unity(z0 is not equal to 1) then 1+z0+z0^2+...+z0^(n-1)=0. Hence show that cos(2Pi/1998)+cos(4Pi/1998)+...+cos(2Pi*1997/1998)=-1

Calculating the residue at pole.

Find Res{Sin(z^3)/z^100, z=0}?

Finding a conformal map to fit certain restrictions.

Find a conformal mapping which maps the upper half plane Im(z)>0 to |w|<1 such that w(a)=0 where a=a_x+ia_y and Im(a)=a_y>0.