( 201 ) 



Mathematics. — Prof. Kluyver presents a paper : "Evaluation oj 

 two definite integrals.'' 



Supposing .c to be real the iiitegrals 



00 



f COS ,vt f sin .i 

 f Lv, m) = I — iU and (f (tC, in) =: I 



dt 



will have a detinite meaning, if only the real part of the parameter 

 m be positive. In what follows we will show how to expand these 

 integrals into rapidly converging power series. 



The tirst of them, the integral f{.v,m), is a particular solution 

 of the linear equation 



d^y dy 



CO 2 (m — 1) xy := 0, 



dx^ dx 



the primitive of which, involving two arbitrary constants A and B, 

 may be written in the form 



y = AL {x, m) -f- ^A'2'«-i M (.i-, m), 



where 



.2A 



LU 



(x. m) = \ 



^ ^ Z^h! r(— 



6) 



(— m + 3 _|_ /,) ' 



A=o 



2h 



A=:oo 



M(x,7n) = V — 



and the constants A and B must now be determined so that y 

 represents the function ƒ (.x', m). To find A, we suppose ???/> — and 

 put X equal to zero. In tiiat case we have 



dt r(è) r{m—\) , r r. . -'^ 



r dt ra) r{m—h) . ^ ^ 



/(O, m) — 1 — —^ ^ ~ z=z A L (0, m) 







and hence 



2 cos jr7n r{m) 



For the deduction of the constant B it is convenient to consider 

 first the function /{,v,)i)) in another form. Let the real part of in 

 still be positive, then we have 



