31. More recent background field removal methods directly or indirectly exploit the fact that the background field is a harmonic function . 32. The definition of harmonic functions and 1-forms is intrinsic and only relies on the underlying Riemann surface structure. 33. His works include theory of functions of complex variables, theory of approximation, and theory of potential and harmonic functions . 34. Any smooth complex-valued function on a sphere can be written as a weighted sum of the spherical harmonic functions . 35. Let " u " be a real harmonic function on ? extending to a smooth function on its closure. 36. This mean value property immediately implies that a non-constant harmonic function cannot assume its maximum value at an interior point. 37. Third, one can use conformal transforms to map harmonic functions in one disk to harmonic functions on a half-plane. 38. Third, one can use conformal transforms to map harmonic functions in one disk to harmonic functions on a half-plane. 39. Just as harmonic functions in 2 variables are closely related to complex analytic functions, so are biharmonic functions in 2 variables. 40. The eigenvalue, E _ { e } ( X _ n ) is often approximated by an harmonic function for simplification.
