1. ## analytic function

I have that f(z) = u(x,y) + iv(x,y) is analytic.
Under what circumstances will g(z) = u(x,y) - iv(x,y) be analytic?

Using the Cauchy-Riemann Equations,

du/dx = dv/dy and du/dy = -dv/dx

and for g to be analytic,

du/dx = -dv/dy and du/dy = dv/dx

but these equations put together imply,

dv/dy = -dv/dy, du/dy = -du/dy, etc.

It seems to me that u(x,y) and v(x,y) must be constants.

Is this true?

Note all derivatives are partial derivatives obviously.

2. Hello,
Originally Posted by PvtBillPilgrim
I have that f(z) = u(x,y) + iv(x,y) is analytic.
Under what circumstances will g(z) = u(x,y) - iv(x,y) be analytic?

Using the Cauchy-Riemann Equations,

du/dx = dv/dy and du/dy = -dv/dx

and for g to be analytic,

du/dx = -dv/dy and du/dy = dv/dx

but these equations put together imply,

dv/dy = -dv/dy, du/dy = -du/dy, etc.

It seems to me that u(x,y) and v(x,y) must be constants.

Is this true?

Note all derivatives are partial derivatives obviously.
Yes, looks like it is true
(constants with respect to both x and y)