1877.] theta functions as definite integrals. 63 



§ 4. On applying a similar method to the tlieta series, I find 

 that 



= A- -)s= r -i» f cos 2/3at + g^^^ cos {2(/3 + l)at + 2x] 

 ~ Vtt ^ J ^ i 1 + ^-^^ cos (2a^ + 2x) + q'^* 



c os 2/3a^ + q^^i cos (2 (/3 + 1) g^ - 2a; }] , 

 ■^ 1 + 2^2^* cos (2a^ - 2x) + ^^^^ J ' 



ttK' 



where /3 is arbitrary, and q = e~"'^ = e~ k . 



Thus, putting /3 = |^, 



1 — e~^' cos 2ic + e~**^ cos 4a; — e"'-'*^ cos %x + &c. 



_ J_ _j^2 p _^, f cos at + 6-^°'' cos (3a^ + 2x) 

 "Vtt^ Jo^ [1 + 2e-«'^ cos (2a« + 2a;) +7-^«^^ 



c os gj^ + e ~ ^'^ cos (3a^ - 2a;) ) 



"^ r+"2^«^^ cos (2gi - 2ci;) + e'^^^^ 



§ 5. In virtue of the formula 



V^r- f. , o -- 27ra; , _ J± A-kx , „ .^z' ^-rrx , „ 



■^^ — -{1 + 26 a^ cos 1- le a"- COS h 2e a2 cos H &c. 



a ( g a a 



it is clear that a summation of the series 



e-^'+ e-<«-^*^+e-<«+^'' + &c. 



gives rise to an expression for the theta function. 



By the use of Kummer's method, it will be found that, if a and 

 h have the same sign, 



__2_ (^,_i)^2 f"" _ ,. cos 2rat - g- ^ (^-D «& cos (2m + 2/>) t 

 "Vtt^ Jo ^ l-2e-2<'-i)«^cos2i^ + e-*('-i)«'-' ' 



g-a^ + e-(a+6)= + e-(«+2&)- + &c. 



2 (^o_j- 2 r ^, cos 2ra^ -e-2(>-+^)«5 cos (2m + 27.)^ 

 "V'^^*' Jo^" l-2e-2<''+')«*cos26« + 6-^<'-+^»"* ' 



