# complex analytic functions

#### 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).

#### vincisonfire

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

#### Drexel28

MHF Hall of Honor
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
What vincisonfire said is true, but surely you have proved that any rational function in $$\displaystyle \mathbb{C}$$ is analytic everywhere it's denominator is non-zero, right?

#### sandy

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?

#### vincisonfire

It is a bit messy.
It may be simpler in this form :
$$\displaystyle f(z) = \frac{z}{(1+iz)^4} = \frac{r_1e^{i\theta_1}}{(r_2e^{i\theta_2})^{4}} = \frac{r_1e^{i\theta_1}}{r_2^4e^{4i\theta_2}}$$

#### sandy

It is a bit messy.
It may be simpler in this form :
$$\displaystyle f(z) = \frac{z}{(1+iz)^4} = \frac{r_1e^{i\theta_1}}{(r_2e^{i\theta_2})^{4}} = \frac{r_1e^{i\theta_1}}{r_2^4e^{4i\theta_2}}$$
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??