An analytic function of a complex variable z = x + iy is

An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) + i v(x, y) where i = √-1. If u = xy, the expression for v should be

Given u = x,y
For analytic function

	∂u	=	∂v
	∂x		∂y

and	∂u	= -	∂v
	∂y		∂x

By Milne Thomson method
Let w = u + zv

∴	dw	=	∂y	+ i	∂v
	dz		∂x		∂x

=	∂u	- i	∂u
	∂x		∂y

or	dw	= y - ix
	dz

Replacing x by z and y by 0, we get

	dw	= 0 - iz
	dz

where , z = x + iy
∴ dw = - iz dz

Integrating , w = -i	z²	+ C
	2

where C is a constant ,

∴ V = Im		-i	z²	+ C
			2

= Im		-i	(x² - y² + 2ixy)	+ C
			2

or V =	y² - x²
	2

(x + y)²	+ k
2

(x - y)²	+ k
2