﻿Airy^s 
  Integral 
  to 
  the 
  Bessel 
  Functions. 
  11 
  

  

  The 
  Evaluation 
  of 
  some 
  Definite 
  Integrals. 
  

  

  Some 
  further 
  integrals 
  which 
  present 
  themselves 
  in 
  con- 
  

   nexion 
  with 
  the 
  Bessel 
  functions 
  of 
  order 
  ^ 
  will 
  now^ 
  be 
  

   considered. 
  The 
  symbols 
  Ji, 
  J— 
  i, 
  Ii, 
  I_4 
  will 
  be 
  used 
  for 
  

  

  the 
  corresponding 
  Bessel 
  functions 
  of 
  argument 
  -rrx 
  / 
  o 
  • 
  

   AVritino- 
  

  

  Jo 
  

   then 
  if 
  accents 
  denote 
  differentiations 
  with 
  respect 
  to 
  cr, 
  

  

  2(" 
  — 
  a'"=—li—Ua{u 
  + 
  tv), 
  

   so 
  that 
  

  

  u" 
  + 
  l(TV 
  = 
  0, 
  

  

  whence 
  

  

  (u 
  + 
  vy'+l-(r(u 
  + 
  r)=-:^^ 
  

  

  (u-v)"-^^ct{u^v) 
  = 
  -. 
  

  

  Treating 
  the 
  second 
  equation 
  first, 
  we 
  note 
  that 
  a 
  particular 
  

   integral 
  of 
  

  

  is 
  given 
  bv 
  

  

  .CO 
  

  

  - 
  = 
  I 
  dw 
  . 
  e-«-H<7"; 
  

  

  

  

  Adding 
  the 
  complementary 
  function, 
  it 
  appears 
  that 
  

   Similarly, 
  if 
  

  

  dw 
  . 
  e~'v5~^n'^ 
  

  

  Then 
  

  

  w 
  + 
  r 
  +,?/ 
  = 
  Co-^ 
  Ji 
  -f 
  Do-* 
  J_i. 
  

   It 
  is 
  found 
  at 
  once 
  that 
  

  

  