( 350 ) 

 Writing- it = 1 — u' we may also conclude from the preceding that 



i r(2w') , r cos (2V/), it 



— ; = — - — - (e 5 ™ — 2 cos 2mt! + «-»««) 



mod* T (1 — u' + iv) .-r s J («« -J- e~ ( ) ïB 



o 



or, because 



r(u' + ic) r(l— m' + ii>) = 



morf' r(»' + ir) == 4T(2 



"if 



•sin jr (»<' -f- iv) 

 CO* (2itf) <ft 



(e* + e-') 2 "' 



which formula holds not only for <[ Ü' <^ 1, but also for "' > 1 

 Introducing in this equation, the in finite product, we have 



ƒ cos (2vt) dt 



r 3 («') 



4T(2m') co 



o / l> 2 \ 



a formula which enables us to write the integral in a finite form in 



1 

 two cases viz. u = n and u = n — -. If u' = n= positive number 



n( 



s = V 



with 



this gives 



2 



i -\ = n l + 



-s = n H 



"- 1 / i> 2 \ 



/7 [ 1 + - | 



rcos (2vt) dt _ xv r 2 (n) s== i v «V 



2T(2«) <>™ — e-™ 



1 



It //' = n , we have 



2 



II 1 + - = 77 1 + - 



s= oV (» - i + «)V s =»-iV (ï + «)' 



which gives with 



«™ + e- 11 " » / i> 2 \ 



2 t=0 ^ r tH*)V 



this result 



'°° co* (2 »«) (ft / 2 (n — h) s =oV (è + s ) 



( - 



n i + 



_|_ e _,)2«_i 2T(2w - 1) e™ -f e-" u 



