The harmonic conjugate of uV is -vU, and the harmonic conjugate of vU is uV. Therefore, the harmonic conjugate of uV + vU is -vU + uV, which simplifies to (u - v)V.
Let's consider the given harmonic functions u, v, U, and V, with v being the harmonic conjugate of u and V being the harmonic conjugate of U.
Recall that a function f(z) = u + iv is harmonic if and only if its real part u and imaginary part v satisfy the Laplace's equation: ∇^2u + ∇^2v = 0, where ∇^2 is the Laplacian operator.
Since v is the harmonic conjugate of u, we have ∇^2u + ∇^2v = 0. Similarly, for U and V, we have ∇^2U + ∇^2V = 0.
Now, let's consider the function w = uV + vU. To show that w is harmonic, we need to demonstrate that ∇^2Re(w) + ∇^2Im(w) = 0, where Re(w) and Im(w) are the real and imaginary parts of w, respectively.
The real part of w is Re(w) = uV - vU, and the imaginary part is Im(w) = vV + uU.
Now, compute the Laplacians:
∇^2Re(w) = ∇^2(uV - vU) = (∇^2u)V + 2∇u · ∇V + u∇^2V - (∇^2v)U - 2∇v · ∇U - v∇^2U
∇^2Im(w) = ∇^2(vV + uU) = (∇^2v)V + 2∇v · ∇V + v∇^2V + (∇^2u)U + 2∇u · ∇U + u∇^2U
Add these two expressions, and notice that the terms involving Laplacians of u, v, U, and V cancel out pairwise, resulting in ∇^2Re(w) + ∇^2Im(w) = 0.
Therefore, uV + vU is harmonic.
To find the harmonic conjugate of uV + vU, consider the real and imaginary parts of the expression separately.
The harmonic conjugate of uV is -vU, and the harmonic conjugate of vU is uV.
Therefore, the harmonic conjugate of uV + vU is -vU + uV, which simplifies to (u - v)V.