Where is the Function f(z) = (z+1)/(z-i) Differentiable on the Complex Plane?

[nk735]

Homework Statement

Find where;

f(z) = (z+1)/(z-i)

is differentiable on the complex plane and find the formulas for f'

Homework Equations

CR equations;

if f(z) = u(x,y) + iv(x,y)

u_x - v_y = 0
v_x + u_y = 0

if function is differentiable

The Attempt at a Solution

My problem is splitting this into its real and imaginary components. Once I have it in re, I am parts I know how to use the CR equations to find whether it's differentiable and then find the derivative.

 

[Cyosis]

To simplify the problem slightly write your fraction as:

$$\frac{z+1}{z-i}=\frac{(z-i)+i+1}{z-i}=1+\frac{1+i}{z-i}$$

Now use the definition of z that is $z=x+iy$. Then multiply top and bottom by the complex conjugate of the denominator.

 

[HallsofIvy]

If the problem were
$$\frac{x+1}{x-1}$$
with x a real number you could differentiate it using the quotient rule couldn't you?

Well, the rules for differentiating a function of complex numbers are just the same as for functions of real numbers! Differentiate the above and then replace x by z.