Fourier's Series. It was first shown by Fourier in his researches into the Conduction of Heat in 1812 that under certain very general conditions concerning the continuity of f 27x f(X) =- 2 1 bi cosCOS b: cos rx + b, cos + . . . ax 2irx 3irx + am a, sin a, sin ..., (3) fc mitx where an, - f (x) sin dx, c 1 mrx and bus = - f (x) cos dx, -6 for all values of x between -c and c.
If f(-x)=--f (x), that is, if f (x) is an odd function. (3) reduces to . rx + 2rx f (x) sin - a, sin a, sin - 2c , mrx where a,,, f kx,, sin dx.
0 If that is, if f (x) is an even function, (3) reduces to 1 2rx f (x) =--- 2 b, + bs cos r + b, cos + b, cos 3 irx c + . . . , (5) 2 nirx where b,,, = f (x) cos dx.
0 If the development need hold good merely for values of x between 0 and c, any one of the forms given above may be employed.
Harmonic Laplace's Equation alVa'V arV + + --= 0, (II: ax' ay' in the numerous forms it assumes in different systems of co-ordinates plays a larger part in the various branches of mathematical physics than any other differential equation, and the harmonic analysis is required in a large propor tion of the physical problems that obey the law it expresses.
A function which together with its first space derivatives is continuous within a speci fied region and which satisfies Laplace's equa tion at every point within the region is said to be harmonic in the region in question.
The form to which a harmonic function re duces on one of the level surfaces of the appro priate co-ordinate system is called a Surface Harmonic.
Zonal The coefficient of sl in the development of (1 - 2ps in as cending powers of sr, where ΅ = cos 0, is rep resented by and is called a Sudan Zonal Harmonic of the mth degree, or, a Le gendre's Coefficient or Legendrian.
It can be shown that V --= 0) ant 1 P m(cos 0) are particular solutions o Laplace's equation in spherical co-ordinates a tt) 1 a (sin 8 a01 a' V ars + sin aft They are called Solid Zonal Harmonics. The first form is harmonic within the sphere whose centre is at the origin of co-ordinates and whose radius is unity, and the second form is harmonic in all space outside of that sphere. They are appropriate functions to use in solvingproblems where a solution of (III) is required, if it is evident from considerations of symmetry that the solution is independent of the co-ordinate 1.3.5.... m(m-1)
Pns(11) 123...m 2 (2m- i) fc m (m-1 ) (m-2) (m-3) 2.4(2m-1) (2m-3) whence Po(p)=-- 1, =--- f 1), - + 3), A(΅) = f 7 -I- 15/2).
A very important property of the Surface Zonal Harmonic P,,,(#) which follows readily from its definition is Pm(1) 1. That is, the function reduces to unity at all points on the polar axis.
If, in a problem where V must satisfy La place's Equation and there is symmetry about the polar axis, the value of V on the axis is represented by a convergent series a. +ail + ad+ . . . , a being the distance of the point from the origin, then the series formed by writing rniP,0(cos 0) instead of sni in the given series gives the value of V at any point in space at which the new series is convergent. If the value of V on the axis is represented by a convergent series . . . , then the series formed from the given series by + re sns 1 , tin placing by P,,,(cos 0) gives the value of any point in space at which the new series is convergent.
For instance, if a charge M of statical elec tricity be placed on a conductor in the form of a thin circular disc of radius a, it is known that the charge will so distribute itself that the surface density a at any point of the disc at the distance s from its centre will be M a 4aaV a' - st If the axis of the disc is taken as the polar axis, the value of the Potential Function V due to the charge, at a point of the axis at the dis M tance x from the centre is V= -- cos -
+ at 2a This can be developed into the series M a
a x
- - 5xs * .. 1 if x
Hence V= M- 2 a
- P, (cos e? + I P,
e ) 5 1 0 Ps (cos 0) + . . .1 if r
and M r - - 3 1 a 24 g 0 1 a -0 6 -P2 (cos + P, (cos . . .1 a r if r>a.
If, in a problem where V must satisfy La place's Equation and there is symmetry about an axis, the value of V on the surface of the sphere r-a is given and can be expressed as a sum or as a series of Surface Zonal Har monics, the value of V at a point not on the sphere will be obtained by replacing the Surface Zonal Harmonics by the appropriate Solid Zonal Harmonics.