Determing where function is differentiable (Complex Analysis)

[scothoward]

Homework Statement

Determine where the function f has a derivative, as a function of a complex variable:

f(x +iy) = 1/(x+i3y)

The Attempt at a Solution

I know the cauchy-riemann is not satisfied, so does that simply mean the function is not differentiable anywhere?

 

[Dick]

No. Where Cauchy-Riemann is satisfied, the function is differentiable. There could be some places. Are there?

 

[scothoward]

Hmm I was thinking, maybe I could take the partials of U and V (Cauchy-Riemann) equate them, and find out what x,y have to be in order for the equations to be satisfied. Am I on the right track?

 

[Dick]

Yes, if CR is satisfied at a point, then the function is differentiable at that point. That's where CR comes from.

 

[scothoward]

Alright, so doing the partials and equating I get

x = 3y and
xy = 3xy

But, I am a little confused on what to do from here.

Thanks for the help

 

[scothoward]

Thinking about it a little more, would this mean when x = 3y, the function is differentiable and when either x=0, y=0, the function is differentiable? Or is my thinking wrong?

 

[Dick]

IF that's right then putting the first equation into the second gives 3y^2=9y^2, so y=0. If y=0 then x=0. So differentiable only at 0+0i.

 

[scothoward]

Thanks for the help Dick!

Just to clarify one more thing. Presuming that I did calculate the paritals right and that leads to 0 + i0, wouldn't that lead to the original function being undefined? As a result, wouldn't the function be differentiable no where?

 

[Dick]

Very astute, scothoward. I was on my way to missing that.