- #1

Redwaves

- 134

- 7

- Homework Statement
- Find real and imaginary parts of a complex function

- Relevant Equations
- ##f(z)= \frac{1}{1+z}##

Hi,

I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}##

First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}##

Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}##

thus, ##\frac{df}{dz} = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{-(1+x)^2 + y^2}{((1+x)^2 + y^2)^2} + \frac{i2y(1+x)}{((1+x)^2 + y^2)^2}##

However, it doesn't match with ##\frac{1}{(1+z)^2}##,where is my error.

I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}##

First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}##

Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}##

thus, ##\frac{df}{dz} = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{-(1+x)^2 + y^2}{((1+x)^2 + y^2)^2} + \frac{i2y(1+x)}{((1+x)^2 + y^2)^2}##

However, it doesn't match with ##\frac{1}{(1+z)^2}##,where is my error.