Final answer:
To show that the product uv - vu is harmonic, one must apply the Laplacian to this product and verify that it equates to zero. The harmonic conjugate would typically be found using the Cauchy-Riemann equations. Without specific functions provided, an exact proof or conjugate cannot be derived.
Step-by-step explanation:
If we have two harmonic functions u and v, and v is a harmonic conjugate of u, and vice-versa, we want to show that the product uv - vu is harmonic and determine its harmonic conjugate. A function is harmonic if it satisfies Laplace's equation, ∂²f/∂x² + ∂²f/∂y² = 0, where ∂²f/∂x² and ∂²f/∂y² are the second partial derivatives of the function with respect to x and y, respectively. For the product of two functions to be harmonic, the Laplacian operator applied to the product must be equal to zero.
However, to accurately respond to this task, we would need to apply the product rule for differentiation to the Laplacian of the product uv - vu. The Laplace's equation for the harmonic conjugates and their products involves calculating mixed partial derivatives of second order and ensuring that the resulting equation equates to zero.
Without the specific details of u and v, we can't derive the exact harmonic conjugate or provide a rigorous proof. However, typically in complex analysis, the harmonic conjugate can be found using the Cauchy-Riemann equations, which relate the partial derivatives of u and v.