Exercise 2185Calculus › Multivariable calculus › Partial derivativesM1Added 3 Nov 2026

Harmonic conjugate

#harmonic conjugate#Cauchy-Riemann#complex

Statement

For $u(x,y)=x^2-y^2$ , find a harmonic conjugate $v$ such that $f=u+iv$ satisfies Cauchy–Riemann.

Hint

Use $u_x=v_y$ , $u_y=-v_x$ .

Show solutionHide solution

CR equations. $u_x=2x=v_y$ , so $v=2xy+g(x)$ .

Second equation. $u_y=-2y=-v_x=-2y-g'(x)$ , so $g'(x)=0$ , $g$ constant.

Result. $v(x,y)=2xy$ .

Conclusion. $u+iv=(x+iy)^2=z^2$ is holomorphic, and $v$ is the harmonic conjugate of $u$ .