Harmonic conjugate

If $u(x,y)$ is a harmonic function on a simply connected domain $\Omega$, a harmonic conjugate of $u$ is a harmonic function $v(x,y)$ such that

$f(z)=u(x,y)+iv(x,y)$

is analytic on $\Omega$.

Equivalently, $u$ and $v$ satisfy the Cauchy–Riemann equations $u_x=v_y$ and $u_y=-v_x$.

The harmonic conjugate exists (on simply connected domains) and is unique up to an additive constant. The construction is the same recipe as finding a potential function for a conservative field.

Recipe

Given harmonic $u$:

1. From $v_y=u_x$, integrate w.r.t. $y$: $v(x,y)=\int u_{x}dy+g(x)$.
2. From $v_x=-u_y$, get an equation for $g^{\prime }(x)$.
3. Integrate to find $g$.

Worked example

$u(x,y)=x^2-y^2+x$.

Check harmonic: $u_{xx}=2$, $u_{yy}=-2$, sum $=0$. ✓

Find $v$: $v_y=u_x=2x+1$, integrate: $v=2xy+y+g(x)$. Differentiate: $v_x=2y+g^{\prime }(x)$, set equal to $-u_y=2y$: $g^{\prime }(x)=0$, $g=0$. So $v=2xy+y$.

Combined: $f(z)=(x^2-y^2+x)+i(2xy+y)=(x^2+2ixy-y^2)+(x+iy)=z^2+z$.

So $u=x^2-y^2+x$ is the real part of $f(z)=z^2+z$.

Why it exists on simply connected domains

The C–R equations $v_y=u_x$, $v_x=-u_y$ define $\nabla v=⟨-u_y,u_x⟩$. This vector field has zero curl: $(-u_y{)}_y-(u_x{)}_x=-u_{yy}-u_{xx}=-{\nabla }^{2}u=0$ (using $u$ harmonic). So $⟨-u_y,u_x⟩$ is conservative on simply connected $\Omega$, and there’s a potential function $v$ with $\nabla v=⟨-u_y,u_x⟩$. That $v$ is the harmonic conjugate.

On non-simply-connected domains, the C–R equations may not admit a single-valued $v$. The example: $u(x,y)=\log \sqrt{x^2+y^2}=\ln r$ on $\Omega ={\mathbb{R}}^2\setminus \{(0,0)\}$. The “conjugate” is the polar angle $\theta =\arctan (y/x)$, but it’s multi-valued — going around the origin adds $2\pi$. Multivaluedness comes from non-simply-connectedness. This is the same phenomenon as the complex logarithm being multi-valued on $\mathbb{C}\setminus \{0\}$.

Geometric content

$u$ and its conjugate $v$ have:

• Mutually orthogonal level curves. The level sets $\{u=c_1\}$ and $\{v=c_2\}$ meet at right angles wherever they cross. See Conformality for the proof.
• Mean-value, max-principle properties inherited from $u$ being harmonic.

The combined $f=u+iv$ is analytic, so all the rigid properties of analytic functions apply.

In context

The harmonic conjugate is what lets you recover a complex-analytic function from its real part alone. In physics applications: given a steady-state potential $u$ (say, electrostatic potential in 2D), constructing its harmonic conjugate $v$ gives a stream function — the level curves of $v$ are the field lines of $\nabla u$. The complex potential $f=u+iv$ packages both into a single analytic function.

This is the bridge between physical 2D potentials and the complex-analytic machinery (Cauchy’s theorem, conformal mapping).