Simple complex function question

[John O' Meara]

Find f(z)= u(x,y) + iv(x,y), given U = x^2 - 2xy - y^2 \\ and check for analyticity.
We have to find v(x,y) as follows:
u_x = v_y and u_y = -v_x Cauchy-Riemann equations
u_x = 2x - 2y and u_y = -(2x+2y) \\.
Thereforev_y = 2x - 2y...(i)
and v_x = 2x + 2y \\ ...(ii), integrating (i) with respect to y and then differentiating it with respect to x , we v=2xy - y^2 +h(x) and v_x = 2y + \frac{dh}{dx} \\ on comparision with (ii) \frac{dh}{dx} = 2x therefore h(x)= x^2+c \\ Therefore v = 2xy - y^2 +x^2+c\\. Question. How can h(x) be a constant of integration, I thought the constant of integration could only be a pure number?

 

[Dick]

If you are integrating and differentiating wrt y, then sure, h(x) can be considered a constant of integration. It's derivative wrt y is 0. Of course, it can't wrt x. It's all relative.