﻿Airi/'s 
  Integral 
  to 
  the 
  Bessel 
  Functions. 
  7 
  

  

  When 
  these 
  functions 
  of 
  fractional 
  order 
  are 
  tabulated 
  for 
  a 
  

   small 
  argument, 
  the 
  results 
  may 
  be 
  readily 
  applied 
  to 
  the 
  

   rapid 
  tabulation 
  of 
  all 
  Bessel 
  functions 
  of 
  very 
  high 
  order, 
  

   if 
  the 
  order 
  and 
  argument 
  do 
  not 
  differ 
  widely. 
  

   Airy's 
  integral 
  may 
  be 
  defined 
  by 
  the 
  formula 
  

  

  dw 
  .GOri{lC^ 
  + 
  pi0), 
  . 
  . 
  . 
  . 
  (1) 
  

  

  and 
  it 
  will 
  be 
  convenient 
  to 
  define 
  also 
  two 
  associated 
  integrals 
  

   in 
  the 
  forms 
  

  

  Mp) 
  = 
  f 
  ° 
  

  

  dw 
  .sm(^w^' 
  + 
  pw) 
  .... 
  (2) 
  

   and 
  

  

  dice-^-^+P" 
  (3) 
  

  

  Jo 
  

  

  Stokes 
  has 
  studied 
  the 
  properties 
  of 
  fi(p) 
  in 
  detail, 
  and 
  

   more 
  particularly 
  its 
  asymptotic 
  expansion 
  when 
  p 
  is 
  not 
  

   small. 
  He 
  has 
  also 
  given 
  a 
  differential 
  equation 
  satisfied 
  by 
  

   the 
  integral, 
  but 
  appears 
  to 
  have 
  overlooked 
  a 
  transformation 
  

   reducing 
  it 
  to 
  a 
  form 
  which 
  was 
  already 
  well 
  known. 
  

  

  If 
  

  

  u 
  = 
  \ 
  dio.e-"''-'p"=fi-Lf2, 
  

   Jo 
  

  

  it 
  may 
  be 
  shown 
  at 
  once 
  that 
  

  

  u"-.lpu 
  = 
  Lf, 
  

  

  where 
  the 
  accent 
  denotes 
  differentiation 
  with 
  respect 
  to 
  p. 
  

   Accordingly, 
  isolating 
  the 
  real 
  and 
  imaginary 
  portions, 
  

  

  fi"-yA 
  = 
  o, 
  (4) 
  

  

  Moreover, 
  

  

  /V'-J«ri 
  = 
  -J 
  (5) 
  

  

  c'o 
  

  

  SO 
  that 
  

  

  (.h+fz)"-y{h+fz)=0; 
  . 
  . 
  . 
  . 
  (0) 
  

  

  and 
  fz+fs 
  therefore 
  satisfies 
  the 
  same 
  differential 
  equation 
  

   as 
  t\. 
  If/ 
  denote 
  /i 
  or 
  /, 
  + 
  /*,, 
  

  

  