( 202 ) 



o 



u du 



and lieiice 



00 00 00 X- 3 



. , , , / — (( m — 1 , i — «£* , \ / i 4« *> 7 



r{)ii)/ {.v,m) =■ Ie a dule cos xt at :=:i ^ y trt \ e u ~ du. 



o o o 



From the latter integral a simple functional relation is derix'able. 



Changing the variable u into -- we may write 



00 X- 1 

 — Mi 



2 



1 /,,.\2//i— 1 r—v 



dv^ 



2w— 1 



r{l - >n)f{,v, 1 — m) 



and so it follows that the function 



fx \"» 2cos:x m V 2 7 V 2 



w 



+ 22'«-i i^ r(m) ( - J Ji(..,;/i) 



remains unaltered, if in is replaced bv 1 — in. 



Now obviously the series L and J/ are connected by the relation 

 L {x, 1 — in) = M (.r, ?/i), 

 hence we must have 



jr /I 



22«t-i B r{m) = -\ r - 



2 cos St m \^2 



n ru) /n2'«-i 



2 cos Jt m r{m)\2j 

 and therefore 



r cosM jr r(4) ( /aA2'«-i ) 



J (l-fi-)'« zc'o.s- :t?/i F(?ri)( ^2/ ) 







Now it will be observed, that the series L{x,ni) and M{x,iii) 

 converge for all values of ./' and ///, and so we must conclude, that 

 the function ƒ(,(■, in) exists over the whole .t'-plane, that its only 

 singularities are ,?■ = and .ii=cc, and that therefore the integral, 

 we started with, represents tlie function in a very incomplete manner. 



