Two harmonic functions u and v from an open subset A of ℝ × ℝ to ℝ, which satisfy the Cauchy-Riemann equations

are the harmonic conjugate functions of each other.

• The relationship between u and v has a simple 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_xdx + u_ydy = 0, or
  

  and the second similarly

  

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

  

  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 calculate the line integral
  

  along any path connecting (x₀, y₀) 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.