Definite Integrals, with Physical Applications 405 



The last terms in <p{x) seem to be of an improper form as involving 

 positive powers of x, the first terms are also snch that <p(x) apparently 

 becomes infinite for values of x between and oo . All this however is 

 merely in appearance, for if we put for h. I. ^1 its value 



- 1 -— -— - -JL 1 1 



a 2a- 3« 3 "' xa* (x + l) M r+l (* + «).«'+» ~~ &c - 



we see that the real value of <j> (x) is 



1 L_ _ 1 



a.(x + l) a*.(x + 2) a 3 .(x + 3)~ &C - 



which is evidently of the proper form, and gives 



« « « *-« 



A still more striking instance of an apparently improper form of <b(x) 

 is the following : r ' 



Given <j>(x) = l -x+x.(x- l)-x.(x-l).(x-2) 



+ --±x.{x-l).{x-2).3.2Tx.(x-\).(x-2)...2.\ .(l- I) 



This function appears, at first sight, to consist of only positive powers 



of x; while in reality the function consists strictly of negative powers 



and converges to as x approaches ». To see this, put for - or .-», 

 its value, 



1.2 i.a.s- ± i.a...« + ra...(* + i ) &c - 



1.2 1.2.3 "- 1.2.. 

 and we get <£(#) in the proper form; viz. 



d>(x)= -i 1 . ! 



x + l (* + l).(* + 2) (x + l).(x + 2).(x + 3) 



3F2 



