277 
BEC Inr + (—De-1} 906) gree (0) =(— It ange (0) 
According to the deduction, this formula holds only true for 
O<t<1 but it is to be seen that for all real values of ¢ it 
is still correct. If A and £ differ both from 1, the coefficient of 
g,(@ on the right hand side will be equal to zero. Both sides of the 
equation are therefore continuous functions, with the period 1, which 
functions must coincide everywhere because they are equal to each 
other in the interval (0,1). 
We arrive at the same conclusion, if / and / are both equal to 
1, and should we have h=1, & unequal to 1, the continuity for 
k odd remains on both sides, while for & even the two sides show 
quite the same discontinuity for ¢=0O and for t= 1. In this case 
too we conclude to the equality of the two sides for all real values 
of ¢, and the equation has therefore general validity. 
This equation now gives the means to calculate p(«,3) directly, 
if a+ 8—= M is odd. 
Let @ be odd, 3 even, then we have 
M—1 
——_ 
4(—1) 2 (dt"=~ sin2a nt "=" cos 2a mt 
ea an nS , 
— BEI Mi . a EE EN nt te 
t ni nr m=l mi? 
ie galt) 
0 0 
For @ >1 the integrand is continuous, integration by terms of 
the product of the two series is allowed, and the following result 
is in this way found for the integral: 
ee 
Tin oe SOL 1 1 1 
RE Si ets »2, EE PE a 7 2 + OS ato pees tS 5 - = Saad 
(2zr)M RER 22 (n— 1)? ni? 
ae 
4(—1) 2 a | 8 } 
at 5 i (a. 8) — 436 (My . 
In exactly the same way we find 
M—1 
7 Im —oy ()9,, (9) ; A(—1) 7? , Dy) et sin Za nt 
at zE “(2 p>; : = 
f t (2)™ | tm AMW 
0 0 
M—1 
4(—1) : ‘one M—292 ) 
any a te 
and 
M41 
