Explain the geometrical interpretation a pair of harmonic function conjugated each other.

Question

Explain the geometrical interpretation a pair of harmonic function conjugated each other.

Could you help me? I am wondering how to draw it but unfortunately my abstract imagination can't cope with it.

Answer 1

Two functions are harmonic conjugate if they can be written as real and imaginary parts of a holomorphic function on some open domain.

An example involves the trigonometric functions $\sin$ and $\cos$. They are harmonic conjugate to each other, because we can write \begin{equation*} \cos(\theta)+i\sin(\theta)=e^{i\theta} \end{equation*} which is just the unit circle.

Keep in mind that this holomorphic function will have a zero, which is a complex number that will annihilate the function at that point. The harmonic conjugates can be thought of as trajectories that are orthogonal to each away from this point(s) (by definition, they are at right angles to each other). You could think about this from an argand diagram with the real and imaginary axis at right angles to each other.

Answer 2

Let $u$ and $v$ be two harmonic functions defined in some region $\Omega\subset{\mathbb R}^2$. Then both are $C^1$ to begin with. Therefore both $u$ and $v$ have gradients $\nabla u:=(u_x,u_y)$, $\>\nabla v:=(v_x,v_y)$, which are vector fields defined in $\Omega$. The function $v$ is harmonic conjugate to $u$ if at each point $(x,y)\in\Omega$ the vector $\nabla v(x,y)$ is obtained by turning the vector $\nabla u(x,y)$ counterclockwise by $90^\circ$.

In terms of the components of $\nabla u$ and $\nabla v$ this means that we have $$v_x(x,y)=-u_y(x,y), \quad v_y(x,y)=u_x(x,y)\qquad\forall\>(x,y)\in\Omega\ ,$$ and this is equivalent to the condition that $f(x+iy):=u(x,y)+iv(x,y)$ is an analytic function of $z=x+iy$ in $\Omega$.