THE CALCULATION OF MODULATION PRODUCTS 237 



u = P COS X -\- Q cos y in the left hand member of (24), applying 

 Jacobi's expansions in series of Bessel coefficients, and rearranging the 

 resulting triple series. It appears that it is much more difficult to 

 justify the series rearrangement than term by term integration. From 

 the results obtained by the first method it follows that the series in 

 (25), which might be termed a generalized Schlomilch series,* is sum- 

 mable in terms of elliptic integrals. 



III. Trigonometric Integral Method 



Following a suggestion of Mr. S. O. Rice, we may make use of the 

 following relation: 



u , u r* sin u\ ,. _ "I 



2+ Jo "1^^^ = "' ^'^^ (26) 



= 0, 7^ < O.J 



Evidently if we substitute u = P cos x + Q cos y, the left hand member 

 of (26) represents the function /(.v, y) and may be substituted in the 

 integrand of (6) without change in the limits. Interchange of the 

 order of integration may then be justified without difficulty and the 

 following result is obtained in terms of a special case of the integral of 

 Weber and Schafheitlin : 





n.+n+2 ^»/,,^(px)/„((2X) 



X2 



d\, (27; 



where m + w is even and greater than zero. When m + h = 0, the 

 above integral should be replaced by an infinite contour integral taken 

 along the real axis except for an indentation to avoid the origin and 

 with all other quantities remaining the same except for a division by 

 two. For all even order modulation products it may now be deduced '" 

 that: 



(-) . +-r( '" + "-' )fr-f 



A — 



2.r(» + i)r( "'-;' + -^ ) 



Xf( "' + ''-K n-,,,-l , ^_^j. ^,, (28) 



The case of w? + w = requires a special investigation, which shows 

 that (28) holds for this case also. 



* Cf. Watson, "Theory of Bessel Functions," Chapter XIX. 

 ^ Watson, "Theory of Bessel Functions," p. 401. 



