522 Mr. 0. Heaviside. [Feb. 2 



By changing n to n + 1 we find that 



so that when r = oo, 



or \n2)JW'JJV~~- L J' (W) 



When n = 0, (54) gives /(o) = 1. Then (58) gives f(n) for any 

 integral n. Thus /(l) = 2, /(2) = f, /(3) = ^ /(*) = gjjjj. &c. 



And on the negative side we have /( |) = 0, /( I) = J, 

 /(-1J) = o,/(-2) = f,/(-2|) = 0,/(-3) =; -V-.&c. The curve 

 is similar to that of the inverse factorial, but with a much bigger 

 hump on the positive side, near n = 1. But I have not found 

 any use for this cosine split, and we may now return to the other 

 one. 



The Exponential Theorem Generalized. 



20. Although we cannot, owing to its limited applicability, use 

 Euler's integral to express \n generally, we may employ it when 

 found convenient, within its own range, and supplement the informa- 

 tion it gives by other means. Thus, we know that 



when n is over 1. Now the indefinite integral may be exhibited in 

 two different ways, say 



Ce~ x x n , / x n+l 



j (tossr-h -- +-. - 4- +.. .) = Wl , (60) 



J C. 



in ascending powers of # multiplied by the exponential function, 

 and by 



fc~ x x n 



J-- 



c~ x x n , x x 



-li. _^ 



These are true for all values of n. Subtracting (61) from (60) we 

 see that the function w t + w? 2 , or w say, must have the same value at 

 any two finite limits we may choose for the integral. That is, the 

 value of w is independent of the value of x. 



Or we may proceed thus, and determine the value. Let n be 

 greater than -1, and divide the integral (59) into two, one going 



