11. The latter is the integral occurring in Dirichlet's principle for harmonic functions . 12. Thus, there is no simple connection between frequency ratios and harmonic function . ]] 13. The spherical harmonics are derived from the approach of looking for harmonic functions of the form 14. One important use of these inequalities is to prove existence of harmonic functions with particular properties. 15. In particular, the ultrahyperbolic equation satisfies an analog of the mean value theorem for harmonic functions 16. Since the mean of a harmonic function over a circle is its value at the centre. 17. Since the mean of the normal derivative of a harmonic function over a circle is zero. 18. Traditionally, harmonic functions are associated with the real and imaginary parts of an analytic function. 19. The spherical harmonic functions form a complete orthonormal set of functions in the sense of Fourier series. 20. In fact, something should be written about harmonic functions in the minimal surface article . ..
