Thread: complex analytic functions

hi to all can i please see the answer for this to check what i have been doing wrong thanks heaps.

(a) Where is the function

f(z) = z/(1+iz)^4 analytic?

(b) Find f(-i).

a) Plug f(z) in the Cauchy-Riemann equations and see where they are satisfied
b) Replace z by -i

Originally Posted by sandy

hi to all can i please see the answer for this to check what i have been doing wrong thanks heaps.

(a) Where is the function

f(z) = z/(1+iz)^4 analytic?

(b) Find f(-i).

Originally Posted by vincisonfire

a) Plug f(z) in the Cauchy-Riemann equations and see where they are satisfied
b) Replace z by -i

What vincisonfire said is true, but surely you have proved that any rational function in is analytic everywhere it's denominator is non-zero, right?

Originally Posted by vincisonfire

a) Plug f(z) in the Cauchy-Riemann equations and see where they are satisfied
b) Replace z by -i

can i please have a hint of how to break it up
do i substitute z by x+iy but then its to big
what do i do?

It is a bit messy.
It may be simpler in this form :

Originally Posted by vincisonfire

It is a bit messy.
It may be simpler in this form :

thanks so much for your help i done it and found that cauchy riemann equations are satisfied every where so do we say f(z) is analytic everywhere on C??