139, 140] TRIGONOMETRIC SERIES. 389 



so that a circular harmonic is a solution of this and Laplace's Equa- 

 tion simultaneously. The homogeneous function of degree n 



a n x n + dn-ix"- 1 H a^xy n ~ l -f a y n 



contains n + 1 terms, the sum of its second derivatives is a homo- 

 geneous function of degree n 2 containing n 1 terms, and if this 

 is to vanish identically each of its n 1 coefficients must vanish, 

 consequently there are n 1 relations between the n + 1 coefficients 

 of V ny or only two are arbitrary. Accordingly all harmonics of 

 degree n can be expressed in terms of two independent ones. The 

 theory of functions of a complex variable 1 ) tells us that the real 

 functions u(x, y), v(x, y) in the complex variable u -f- iv f which is a 

 function of the complex variable x -\- iy, are harmonic functions of 

 x, y, and making use of Euler's fundamental formula, 



65) x -f iy = Q {cos co -f i sin co} = ge ita , 

 and raising to the n ih power, we have 



66) (x 4- iy) n = Q n e in(a = $ n (cos no + i sin no). 



Accordingly we have the two typical harmonic functions 



rr\ w * 



It may be at once shown that these functions are harmonic by 

 substitution in Laplace's equation in polar coordinates, equation 47). 

 Accordingly the general harmonic of degree n is 



/^ Q \ T 7" 47 ( A | TO * \ m /TT 



We may call the trigonometric factor T n , which is the value of the 

 harmonic on the circumference of a circle of radius unity, the 

 peripheral harmonic of degree n. 



If a function which is harmonic in a circular area can l>e 

 developed in an infinite trigonometric series 



69) V(x, y} 



on the circumference of the circle of radius R, the solution of 

 Dirichlet's Problem for the interior of the circle is given by the series 



For every term is harmonic, and therefore the series, if convergent, 

 is harmonic. At the circumference Q = JR, and the series takes the 

 given values of V. The absolute value of every term is less than 

 the absolute value of the corresponding term in the series 69), in 



1) See 197. 



