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 $$u_y = -(2x+2y) \\$$.
Therefore$$v_y = 2x - 2y$$...(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.