# How to use Cauchy Riemann

• May 4th 2010, 10:01 AM
zizou1089
How to use Cauchy Riemann
I have reached a step in the question where I have to apply the Cauchy Riemann formulas to:

$(-6x-2By)^2+(-12x+4By)i$

How would I apply the Cauchy Riemann formula to answer the rest of the question?

And how would I know if it is analytic?

The cauchy-riemann formula:

The function f(z)= g(x,y)+ h(x,y)i is differentiable with respect to z= x+ iy only if it satisfies the "Cauchy-Riemann equations":

$\frac{\partial g}{\partial x}=\frac{\partial h}{\partial y}$

and

$\frac{\partial g}{\partial y}=-\frac{\partial h}{\partial x}$
• May 4th 2010, 10:38 AM
piglet
Quote:

Originally Posted by zizou1089
I have reached a step in the question where I have to apply the Cauchy Riemann formulas to:

$(-6x-2By)^2+(-12x+4By)i$

How would I apply the Cauchy Riemann formula to answer the rest of the question?

And how would I know if it is analytic?

The cauchy-riemann formula:

The function f(z)= g(x,y)+ h(x,y)i is differentiable with respect to z= x+ iy only if it satisfies the "Cauchy-Riemann equations":

$\frac{\partial g}{\partial x}=\frac{\partial h}{\partial y}$

and

$\frac{\partial g}{\partial y}=-\frac{\partial h}{\partial x}$

You do precisely what you've wrote down. let $g$ = real part of your function i.e. $(-6x-2By)^2$ and let $h$ = imaginary part of your function i.e. $(-12x+4By)$

Now it's just straight-forward partial differentiation

Find $\frac{\partial g}{\partial x}$. This means you need to differenciate g w.r.t x.

so $\frac{\partial g}{\partial x}$ = $-12(-6x-2By)$

and as $\frac{\partial h}{\partial y}$ is the partial derivative of h w.r.t y

we get $\frac{\partial h}{\partial y} = 4B$

Clearly $\frac{\partial g}{\partial x}\ne \frac{\partial h}{\partial y}$

A function is analytic in the complex plane if the BOTH caucy riemann equations are satisfied. So since the first C-R equation i checked is not satisfied you can say that the function is not analytic