( D&A 
Mathematics. — „On Integrals containing functions of Bessel”. 
An extract out of a letter of Prof. L. GEGENBAUER at 
Vienna communicated by Prof. W. Kapreyn. 
In your interesting paper ,Sur quelques intégrales définies con- 
tenant des fonctions de Bessel” in the „Archives Néerlandaises des 
sciences exactes et naturelles” you have determined the value of 
the integrals 
go 
if 1"(0) cal £0) f In( sin (tsin g) 5 
t 
0 : 0 
for entire positive values of #. I shall take the liberty to communicate 
to you in the following lines a generalisation of your noteworthy 
results. 
If we multiply the relation 
2/—1 | 
giteosocosp T° (t sin p sin w) == 22 [MI (v—1)]? (sin p sin w) ‘ 
gie (Atv) H(A) . E ie 
= Sf PE Cy hosp) Celeste 
V a2: IT (4k + 2 y—1) ) beau) (cos HG 26 
given by me in the tenth volume of the „Monatshefte fiir Mathematik 
und Physik” (,Notiz über die Bessen’schen Functionen erster Art.”), 
in which relation C* («) is the coefficient of a” in the development of 
(l—2aer4 @?)-? 
according to ascending powers of @, with 
[»+m (t) dt 
cha LOI: (m > 0) 
Vt 
and if we integrate according to t from 0 to oo, we obtain by 
making use of the well-known value of the integral 
oo 
f Pr (t) f+") dt 
t 
0 
i a 
