Derivative of complex function using definition

• Apr 9th 2011, 04:40 AM
Glitch
Derivative of complex function using definition
The question

Show that the definition of $\displaystyle \frac{d}{dz}(\frac{1}{1 - z}) = \frac{1}{(1 - z)^2}$

My attempt
I tried using:

$\displaystyle \lim_{\delta z \to 0} \frac{f(z + \delta z) - f(z)}{\delta z}$

But having fractions in the numerator in this case appears difficult to simplify. I tried substituting z = x + iy, but I ended up with a massive mess which still results in a division by 0. Is there a good method of solving this problem?

Thanks.
• Apr 9th 2011, 04:48 AM
Plato
Quote:

Originally Posted by Glitch
Show that the definition of $\displaystyle \frac{d}{dz}(\frac{1}{1 - z}) = \frac{1}{(1 - z)^2}$
$\displaystyle \lim_{\delta z \to 0} \frac{f(z + \delta z) - f(z)}{\delta z}$
I ended up with a massive mess

WHY?

$\displaystyle \dfrac{\frac{1}{1-z-dz}-\frac{1}{1-z}}{dz}=\dfrac{(1-z)-(1-z-dz)}{dz[(1-z)(1-z-dz)}$
• Apr 9th 2011, 04:49 AM
chisigma
Quote:

Originally Posted by Glitch
The question

Show that the definition of $\displaystyle \frac{d}{dz}(\frac{1}{1 - z}) = \frac{1}{(1 - z)^2}$

My attempt
I tried using:

$\displaystyle \lim_{\delta z \to 0} \frac{f(z + \delta z) - f(z)}{\delta z}$

But having fractions in the numerator in this case appears difficult to simplify. I tried substituting z = x + iy, but I ended up with a massive mess which still results in a division by 0. Is there a good method of solving this problem?

Thanks.

With simple algebraic steps You obtain...

$\displaystyle \displaystyle \frac{f(z + \delta)- f(z)}{\delta} = \frac{1}{\delta} \ \{\frac{1}{1-z-\delta} - \frac{1}{1-z}\} =$

$\displaystyle \displaystyle = \frac{1}{(1-z-\delta)\ (1-z)}$ (1)

Now what is the $\displaystyle \lim \delta \rightarrow 0$ of the expression (1)?...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Apr 9th 2011, 04:55 AM
Glitch
Quote:

Originally Posted by Plato
WHY?

$\displaystyle \dfrac{\frac{1}{1-z-dz}-\frac{1}{1-z}}{dz}=\dfrac{(1-z)-(1-z-dz)}{dz[(1-z)(1-z-dz)}$

I'm such an idiot sometimes. Thanks.