302-304] 



Two-dimensional Problems 



255 



Fourier s Theorem. 



304. The value of any function F of position on the circumference of a 

 circle can be expressed, at every point of the circumference at which the 

 function is continuous, as a series of sines and cosines, provided the function is 

 single-valued, and has only a finite number of discontinuities and of maxima 

 and minima on the circumference of the circle. 



Let P(f, a) be any point outside the circle, then if R is the distance 

 from P to the element ds of the circle p ,, 



(a, 0) we have 



j = l. 



This result can easily be obtained by inte- 

 gration, or can be seen at once from physical 

 considerations, for the integrand is the charge 

 induced on a conducting cylinder by unit line 

 charge at P. 



Let us now introduce a function u defined by 



-^ds 



FIG. 82. 



.(226). 



Then, subject to the conditions stated for F we find, as in 240, that on 

 the circumference of the circle, the function u becomes identical with F. Also 

 we have 



f 2 + a 2 - 2a/cos (6 - a) 

 1 



f* _ & \f- 



f 



a - 



Hence 



1 r6 = 2ir 1 QO f n \n / 



=- Fdd + -^(^) 

 2?r J e=o TT ! \t ] J 



