I understand how the equations work but i can't seem to be able to determine whether "1 / (1 + z)" satisfies the cauchy-riemann equations and hence determine whether f ' (z) exists.

Any help would be greatly apprichiated

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- Dec 21st 2009, 03:44 AMjezzyjezCauchy-Riemann equations
I understand how the equations work but i can't seem to be able to determine whether "1 / (1 + z)" satisfies the cauchy-riemann equations and hence determine whether f ' (z) exists.

Any help would be greatly apprichiated - Dec 21st 2009, 03:49 AMmr fantastic
- Dec 21st 2009, 04:13 AMjezzyjez
Yeh cool i get that now thanks, am i missing a trick with the last part of the question

"determine whether f ' (x) exists"

or does it literally exist when the equations satisfy the C-R equations?? - Dec 21st 2009, 04:23 AMtonio

You can evaluate the partial derivatives with the function as Mr. Fantastic showed you, or else realize (most books have it) that the CR-equations

are equivalent to $\displaystyle \frac{\partial f}{\partial\,\overline{z}}=0$ , where we have that $\displaystyle \frac{\partial}{\partial\,\overline{z}}=\frac{1}{2 }\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right)$ , so putting $\displaystyle z=x+iy$, we get:

$\displaystyle f(z)=f(x,y)=\frac{1}{1+x+iy}\Longrightarrow \frac{\partial f}{\partial x}=-\,\frac{1}{(1+x+iy)^2}\,,\,\,\frac{\partial f}{\partial y}=-\,\frac{i}{(1+x+iy)^2}$.

Check the above and conclude your function is analytic in $\displaystyle \mathbb{C}\setminus \{1\}$

Tonio - Dec 21st 2009, 07:13 PMmr fantastic
A thread of related interest: http://www.mathhelpforum.com/math-he...erivation.html

(And a correction of a minor typo in red). - Dec 21st 2009, 08:14 PMBruno J.