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.