harmonic conjugate function

Two harmonic functions $u$ and $v$ from an open (http://planetmath.org/OpenSet) subset $A$ of $ℝ\times ℝ$ to $ℝ$, which satisfy the Cauchy-Riemann equations

$u_x=v_y,u_y=-v_x,$(1)

are the harmonic conjugate functions of each other.

• The relationship between $u$ and $v$ has a geometric meaning:  Let’s determine the slopes of the constant-value curves  $u(x,y)=a$  and  $v(x,y)=b$  in any point  $(x,y)$  by differentiating these equations.  The first gives  $u_{x}dx+u_{y}dy=0$,  or

  $${\frac{dy}{dx}}^{(u)}=-\frac{u_x}{u_y}=\tan \alpha ,$$

  and the second similarly

  $${\frac{dy}{dx}}^{(v)}=-\frac{v_x}{v_y}$$

  but this is, by virtue of (1), equal to

  $$\frac{u_y}{u_x}=-\frac{1}{\tan \alpha }.$$

  Thus, by the condition of orthogonality, the curves intersect at right angles in every point.

• If one of $u$ and $v$ is known, then the other may be determined with (1):  When e.g. the function $u$ is known, we need only to the line integral

  $$v(x,y)={\int }_{(x_0,y_0)}^{(x,y)}(-u_{y}dx+u_{x}dy)$$

  along any connecting  $(x_0,y_0)$  and $(x,y)$  in $A$.  The result is the harmonic conjugate $v$ of $u$, unique up to a real addend if $A$ is simply connected.

• It follows from the preceding, that every harmonic function has a harmonic conjugate function.

• The real part and the imaginary part of a holomorphic function are always the harmonic conjugate functions of each other.

Example.  $\sin x\cosh y$  and  $\cos x\sinh y$  are harmonic conjugates of each other.