Complex Function Differentiability

First off, sorry if this is in the wrong section it could be in calculus I guess, but my unit is titled complex analysis so I thought it might go in here.

So I have just started Complex analysis for the first time and just needed some help.

When I am required to find all the points in the complex plane at which a complex function of several variables (i.e f(x+iy)=u(x,y)+iv(x,y)) is differentiable, what steps do i take in determining the answer. When dealing with real numbers i would usually find the answer via inspection or asymptotes ect. But i am having trouble picturing it on the complex plane. Do i use the Cauchy-Riemann Equations to test for analyticity?

For example f(x+iy)= 2(x^2-y^2) -i4xy , has the answer differentiable at z=0. but how do i determine this.

Any help would be greatfully appreciated :) as my lecturer has not covered this yet but trying to wrap my head around it

Re: Complex Function Differentiability

Quote:

Originally Posted by

**olski1** First off, sorry if this is in the wrong section it could be in calculus I guess, but my unit is titled complex analysis so I thought it might go in here.

So I have just started Complex analysis for the first time and just needed some help.

When I am required to find all the points in the complex plane at which a complex function of several variables (i.e f(x+iy)=u(x,y)+iv(x,y)) is differentiable, what steps do i take in determining the answer. When dealing with real numbers i would usually find the answer via inspection or asymptotes ect. But i am having trouble picturing it on the complex plane. Do i use the Cauchy-Riemann Equations to test for analyticity?

For example f(x+iy)= 2(x^2-y^2) -i4xy , has the answer differentiable at z=0. but how do i determine this.

Any help would be greatfully appreciated :) as my lecturer has not covered this yet but trying to wrap my head around it

I think you should read here:

Cauchy

Re: Complex Function Differentiability

Quote:

Originally Posted by

**olski1** First off, sorry if this is in the wrong section it could be in calculus I guess, but my unit is titled complex analysis so I thought it might go in here.

So I have just started Complex analysis for the first time and just needed some help.

When I am required to find all the points in the complex plane at which a complex function of several variables (i.e f(x+iy)=u(x,y)+iv(x,y)) is differentiable, what steps do i take in determining the answer. When dealing with real numbers i would usually find the answer via inspection or asymptotes ect. But i am having trouble picturing it on the complex plane. Do i use the Cauchy-Riemann Equations to test for analyticity?

For example f(x+iy)= 2(x^2-y^2) -i4xy , has the answer differentiable at z=0. but how do i determine this.

Any help would be greatfully appreciated :) as my lecturer has not covered this yet but trying to wrap my head around it

A complex variable function $\displaystyle f(z)= u(x,y) + i\ v(x,y)$ is differenziable in a point $\displaystyle z=z_{0}= x_{0}+i\ y_{0}$ only if there the 'Cauchy-Riemann equations' are satisfied...

$\displaystyle \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}\ ,\ \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$ (1)

In particular for $\displaystyle u(x,y)= 2\ (x^{2}-y^{2})$ and $\displaystyle v(x,y)= 4\ x\ y$ the (1) are satisfied in the whole complex plane... in Your example however is $\displaystyle u(x,y)= 2\ (x^{2}-y^{2})$ and $\displaystyle v(x,y)= - 4\ x\ y$ and the (1) are satisfied only for x=y=0...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

Re: Complex Function Differentiability

Thanks Guys, thats what i was sort of thinking. Got it now! Man i love this forum :D