﻿Airy^s 
  Integral 
  to 
  the 
  Bessel 
  Functions. 
  13 
  

  

  not 
  siiitall, 
  

  

  j»«=iGy.«p> 
  (22) 
  

  

  J-„(-)= 
  -( 
  - 
  n/,(/») 
  cos 
  mr+f^{p) 
  sin 
  n-n-\ 
  

  

  IT 
  \Z 
  ' 
  

  

  ITT 
  nr 
  «T 
  ITT 
  

  

  ■¥l.(^^\"''Ann-,r[e^ 
  hip^' 
  ') 
  +^" 
  V.(P«" 
  ^') 
  } 
  (23) 
  

  

  [iiul 
  if 
  n 
  be 
  an 
  integer, 
  

  

  Y„(--) 
  = 
  -(3' 
  [^ 
  V. 
  {pe 
  '^) 
  + 
  e-^f,{pe-^) 
  \ 
  . 
  (24) 
  

  

  The 
  functions 
  /i, 
  /2 
  have 
  the 
  same 
  significance 
  as 
  before, 
  

   /i 
  being 
  an 
  Airv's 
  integral 
  of 
  argument 
  p. 
  

   Y„ 
  {z) 
  is 
  Hankers 
  function 
  detined 
  by 
  

  

  Y,.(,) 
  = 
  -^-i 
  J-"-''°^""J" 
  } 
  . 
  . 
  . 
  (25) 
  

  

  ^ 
  ' 
  I 
  Sin 
  WTT 
  J 
  ,i=integer. 
  

  

  or, 
  in 
  series 
  form, 
  

  

  Y,.(-)=2J,.(^){logir 
  + 
  -557}-g)"{»-l! 
  + 
  '*-^gJ 
  + 
  ..| 
  

  

  where 
  

  

  S„=H-i+ 
  ...+-. 
  . 
  . 
  . 
  . 
  (27) 
  

   n 
  

  

  An 
  alternative 
  expression 
  for 
  J_„ 
  (z) 
  is 
  

  

  J-„(^) 
  = 
  ^;y{/Krtcos„.+/,(rtsin..} 
  +@'?^J°<;«>e--+^" 
  (28) 
  

  

  ^vhere 
  the 
  latter 
  integral 
  has 
  already 
  been 
  denoted 
  by 
  /s 
  (p) 
  . 
  

   In 
  all 
  these 
  results 
  p 
  stands 
  for 
  

  

  and 
  is 
  positive 
  if 
  n 
  is 
  greater 
  than 
  z. 
  

  

  These 
  expressions 
  are 
  asymptotic, 
  but 
  may 
  be 
  verified 
  to 
  

   three 
  places 
  of 
  decimals 
  when 
  n 
  = 
  z 
  = 
  %. 
  

  

  "When 
  n 
  differs 
  from 
  z^ 
  it 
  is 
  necessary 
  for 
  p 
  to 
  be 
  very 
  

   small 
  in 
  comparison 
  with 
  z-'\ 
  the 
  error 
  then 
  introduced 
  being 
  

  

  