Show that f is entire when u is harmonic and v=

[zardiac]

Homework Statement

Assume that u is harmonic everywhere in R^2, and let
$$v(x,y)=\int_0^y u_x'(x,t)dt - \int_0^x u_y'(s,0)ds$$
show that f=u+iv is entire analytic.

Homework Equations

Maybe Cauchy Riemann: $$u_x'=v_y'$$ and $$u_y'=-v_x'$$

The Attempt at a Solution

I have only tried to see what happens if I use the Cauchy Riemann equations, but I get stuck right away. I am not sure how to use the fact that u is harmonic either.

Any hints would be very appreciated.

 

[HallsofIvy]

u_x' is very strange notation. I think you mean just the derivative of u with respect to x- but that is just u_x. There is no need for the ' here.

I have only tried to see what happens if I use the Cauchy Riemann equations, but I get stuck right away.

Good. Show what you did and where you got stuck.

I am not sure how to use the fact that u is harmonic either.

What does "harmonic" mean?