Complex Analysis: Finding an Analytic Function for Re(z)=1-x-2xy

[john88]

hi


I want to find an analytic funktion if Re(z) = 1 - x - 2xy

My initial thought was to set U(x,y) = 1 - x - 2xy and then solve for V(x,y) through
du/dx = dv/dy but it doesn't seem to go as far as I am concernd.

Then I thought about the fact that Re(z) = (z + zbar)/2 and then work from there but I can't figure out how.

My book says: 1 - z + iz^2 + iC, CeR

 

[HallsofIvy]

Riemember the Riemann conditions: if f(x+ iy)= u(x,y)+ iv(x,y) then
$$\frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}$$
$$\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}$$

If Re(f(z))= u(x,y)= 1- x- 2xy, then
$$\frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}= -1- 2y$$
$$\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}= -2x$$
You can find v from that. There are many correct answers.

 

[john88]

ok I got it! ty...I was alittle confused by the answer.