Complex analysis conjugation help

[CrazyCalcGirl]

Homework Statement

If f(z) is analytic at a point Zo show that the Conjugate(f(z conjugate)) is also analytic there. (The bar is over the z and the entire thing as well.)

The Attempt at a Solution

I know if a function is analytic at Zo if it is differentiable in some neighborhood of Zo. I also know the Cauchy Riemann equations would hold there. I also know that the partial with respect to Z conjugate is zero. I guess I am having trouble with the double conjugation here and what kind of formal argument to make.

 

[Dick]

Write z=x+iy and f(z)=u(x,y)+iv(x,y). If conjugate(f(conjugate(z))=U(x,y)+iV(x,y) can you write U and V in terms of u and v? You know the Cauchy-Riemann equations hold for u and v, can you show they also hold for U and V?

 

[Office_Shredder]

Write z=x+iy and f(z)=u(x,y)+iv(x,y). If conjugate(f(conjugate(z))=U(x,y)+iV(x,y) can you write U and V in terms of u and v? You know the Cauchy-Riemann equations hold for u and v, can you show they also hold for U and V?

I'd just like to point out that proving U and V satisfy the Cauchy-Riemann equations isn't enough to show that f is holomorphic (you need continuous partials which you might not have).

Go directly to the definition of a derivative. Look at the difference quotient and see if you can manipulate the conjugates to make it look like the derivative of f. It won't be exact, but you can turn it into a limit that you know exists

 

[Dick]

I'd just like to point out that proving U and V satisfy the Cauchy-Riemann equations isn't enough to show that f is holomorphic (you need continuous partials which you might not have).

Go directly to the definition of a derivative. Look at the difference quotient and see if you can manipulate the conjugates to make it look like the derivative of f. It won't be exact, but you can turn it into a limit that you know exists

Why would you think continuity of partial derivatives would be a problem? f is given to be analytic in a neighborhood of z0. It has derivatives of all orders.

 

[CrazyCalcGirl]

Why would you think continuity of partial derivatives would be a problem? f is given to be analytic in a neighborhood of z0. It has derivatives of all orders.

I did get it to satisfy the Cauchy Riemann conditions. I think since f(z) is given to be analytic that we already know it has continuous first partials at U and V. The conjugate will obviously also have this as well.