2.^6 BELL SYSTEM TECHNICAL JOURNAL 



II. Fourier Series Method 



The second method is of interest because it obtains the same results 

 as the only previously known solution,^ which is in terms of infinite 

 series involving Bessel functions. The fact that the results agree is a 

 check on the validity of certain doubtful rearrangements of multiple 

 series necessary in the process by which these results were originally 

 obtained. Furthermore by comparison with the corresponding results 

 of the first method we can sum the infinite series in terms of complete 

 elliptic integrals; a number of interesting mathematical theorems are 

 thus proved, which have been made the basis of a paper by the author 

 in the December, 19v32 issue of the Bulletin of the American Mathe- 

 matical Society. 



By expanding the function : 





2' 



(23) 



in a Fourier series in n, we may verify that: 



4 "^2 Tr^h{2r- 1)2^°^ c - "' " 



If we let u = P cos x -\- Q cos y, the left hand member of (24) is equal 

 to /(x, 3') provided \P\ -\- \Q\ < c. With this restriction on c, we 

 may substitute the resulting expression for f(x, y) in the integrand of 

 (6), and no change in the limits of integration are required. Term by 

 term integration of the series may be justified without difficulty, and 

 making use of well known definite integrals, we obtain finally: 



4c 



m+n+2 



TT' r=l (2r — 1)2 



where m + n is an even integer. When m + w = 0, the extra term 

 c/4 must be added. When m + n is odd and greater than one, the 

 value of A„,„ is zero; when m -\- n = 1, the values are Aw = P/2, 

 Aoi = (2/2. 



Peterson and Keith obtained the above result ^ by substituting 



' Peterson and Keith, "Grid Current Modulation," Bell Svston Technical Journal, 

 \"ol. 7, pp. 138 9, January, 1928. 



