Writing a function of x and y as a function of z

[ericm1234]

Is there a general method for this?
I remember in complex analysis going the other direction; that is, a function of z becoming a function of x and y.
But I need to turn a complex function in a function of just z, for the purpose of integrating with respect to z.
In particular, the function is
x^2/2-cx-y^2/2+ i(x-c)y where c is a constant.

 

[SteamKing]

When you say z, are you referring to z, as in the third coordinate of an x,y,z system or the complex variable z, as in z= x + iy?

 

[ericm1234]

complex variable z, as in z=x+iy.

 

[ericm1234]

In fact I now see how to write this in terms of z but it is from recognizing the form. Is there a more general method?
Please no one reply with "well if you recognize the form in this particular equation.", because again, I already see it.
General method please?

 

[Bacle2]

Hi, try:

x= (z+ z^)/2 , where z^ is the conjugate, i.e. if z=x+iy, then z^= x-iy

y=( z- z^)/2i

It comes from z=x+iy

 

[ericm1234]

Thanks

 

[Mute]

Be aware that you cannot necessarily write a function f(x,y) solely as a function of $z = x + i y$. You may find that the complex conjugate $z^\ast = x - iy$ does not cancel out of your expression.

You will only be left with a function of z (and not z*) if your function is analytic in the complex plane.