For each of the following functions determine all complex nu

For each of the following functions, determine all complex numbers for which the function is holomorphic. If you run into a logarithm, use the principal value unless otherwise stated. Z^2 sin z/z^3 + 1 Log (z - 2i + 1) where Log(z) = ln |z| + i Arg (z) with 0 lessthanorequalto Arg (z)

Solution

(c)

The singularity of Log(z) is z=0

so, the singularity of Log(z-2i+1) is z-2i+1=0 or z=2i-1

that is the function Log(z-2i+1) is holomorphic on complex plane except z=-1+2i

(f)

By definition of complex power function we have (i)^z-3=(z-3)log(i)

The singularities of  (i)^z-3 are nothing but the singularities of (z-3)log(i)

but (z-3 )log(i)has no singularities and so as (i)^z-3

that is the function (i)^z-3 is holomorphic throughout the complex plane