Our — Extensions of Fourier'' s and the Bessel-Fourier Theorems. 217 



Making use of this result, then, and taking 



e^''Jn{\r)d\\ J„(Xp)p^{p)dp, (46) 



on inverting the order of integration, it becomes 



In finding the limit of this as t diminishes indefinitely, we may evidently 

 use the approximate expression, derivable from (17), (18), of the / function 

 for large values of the variable, viz. : — 



/„(p;) =(27ra;)-i.e^ 



And on doing so we obtain, when t is diminished indefinitely, 



Lt. 





(1 - e^"''*) 



or, writing p - r = 2thi, 



(1 - e'"'^^) ^^- " TT-i . c-"'(l + 2tiur-^)i<l>{r + 2tH)du] 

 J p=« 



and evidently this is the same as 



or. 



(1 - e^'"^') 7r"a {</> (r - e) e-''^du + (^(r + e) 

 i(l -e^«-0!«^(^-O+ '/'O' + O)- 



6"^'"f?i<.i, 



Art. 9. Extensions of Equations of Arts. 3-6, the K's being replaced hy 

 Differential Coefficients of any Orders. 



If we denote d/ldxP \F{x)] by pF{x), we have for large values of x, 



pK,(o^ = {-y(wl2x)h-^, 



while arg. x > - 37r/2 and < 37r/2 ; and 



pKn{x) = (7r/2«)H(-)Pe-^ + 2icosmr.e''\, 



when arg. x > - 7r/2 and < 57r/2.* Evidently, then, the preceding arguments 

 equally establish the equations : — 



Lt. ''' 



\.pK,n(-iXr)d\ ,Kn{iXp)p<p{p)dp = {-)p^^ .^ ^{r - ,), (47) 



h la ^ 



* The forms of these approximate equations are obtainable from (17), (20) by differentiation. 

 A simple and satisfactory way of establishing them is by the aid of equations of the type of (44) 

 and others deducible from it by successive differentiation. 



