62 Mr QlaisheVy On expressions for the [May 7, 



§ 2, At first sight it seems as if the summation could be at 

 once effected by means of the integral 



/, 



but, putting q — e-'^"', the right-hand side of (1) becomes 



9 r* 



-^ I e-^- (1 — cos2a:cos2ai + cos4a;cos4a« — &c.) Ji •••(2), 



and the value of the quantity in brackets, under the integral sign, 

 is indeterminate, so that the method does not give an expression 

 for the series as a definite integral. In point of fact, as was shown 

 by Cauchy. the result to which (2) does lead, when evaluated, is 

 the transfurmation (3) used further on. 



§ 3. The difficulty was avoided by Kummer as follows. In 

 order to sum the series 1 ± ^ + 5* ± 2^ + &c., he starts with the 

 summation 



cos 2/3ai + y cos 2 (/3 - 1) at + v" cos 2 (/3 - 2) ai + &c. 



_ cos 2^at - V cos 2 {/3 + V)at 

 ~ 1 - 2u cos 2at ~+v'^ ~ ' ^ 



whence, q being as before equal to e~"''^, we find 



_ _2_ r _^^ cos 2/3q^ - vcos 2(/3+])a^ 

 \j7rjQ 1 — 2v C08 2at -\- v^ 



The general term of the series is 2;"g(^-«)'^ = 'y"^^'+«=-2^» which 

 = z^ 2^'+«'- \{ z = vq~'^^, and therefore 



1 + zq + z'^q^ + zY + &c. 



^ 2q-^' r _ ^, cos 2/3at - zq^^ cos 2 (^Q + 1) at 

 '^ir }^^ 1 - 2zq^^ cos 2at + z^ q^^ ' 



in which /S is arbitrary, subject only to the condition that zq^^ 

 must be less than unity. Kummer puts /3 = |-, so that, taking 

 z = l, 



1 , ,4,9.0 2 r" ,2 COS at — q cos oat ,, 



1 + 2 + ^M- 5' + &c. = -T— - e-^ z ^~- -, dt. 



^ q^^/TTJo 1 - 2q cos 2at + q"" 



