$f(z)=u(x,y)+iv(x,y)$ is an analytic function of the complex variable $z=x+iy$, where $i=\sqrt{-1}$. If $u(x,y)=2xy$, then $v(x,y)$ may be expressed as

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$-x^2+y^2+\text{constant}$. For an analytic function, the Cauchy–Riemann equations give $u_x=v_y$ and $u_y=-v_x$. Since $u=2xy$, we have $u_x=2y$ and $u_y=2x$, so $v_y=2y$ and $v_x=-2x$, which integrates to $v=-x^2+y^2+C$.

\(f(z)=u(x,y)+iv(x,y)\) is an analytic function of the complex variable \(z=x+iy\), where \(i=\sqrt{-1}\). If \(u(x,y)=2xy\), then \(v(x,y)\) may be expressed as

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