﻿of 
  Equal 
  Argument 
  and 
  Order. 
  27 
  ?> 
  

  

  when 
  n 
  is 
  integral. 
  The 
  one 
  selected 
  is 
  to 
  some 
  extent 
  a 
  

   matter 
  of 
  convention, 
  so 
  far 
  as 
  an 
  additive 
  multiple 
  of 
  J„(z) 
  

   is 
  concerned. 
  We 
  shall 
  choose 
  Hankel's 
  function 
  *_, 
  defined 
  

  

  by 
  

  

  y 
  (z)= 
  <T? 
  J 
  " 
  -r- 
  V'-- 
  J 
  ~"T 
  

  

  ?i 
  = 
  integer 
  

   __ 
  fJ_„. 
  — 
  COS 
  717T. 
  J 
  n 
  l 
  

  

  \ 
  Sill 
  717T 
  J 
  ' 
  

  

  (3) 
  

  

  (4) 
  

  

  Sill 
  717T 
  

  

  n— 
  integer 
  

  

  where 
  n 
  is 
  made 
  integral 
  after 
  the 
  general 
  expressions 
  above 
  

   have 
  been 
  differentiated. 
  

  

  Expressed 
  in 
  series 
  form, 
  a 
  little 
  reduction 
  shows 
  that 
  

  

  Y,<--)-2J,W 
  .{ 
  7+ 
  logi}-(*)"{„-l 
  I 
  +*=p(j)' 
  

  

  ^W 
  + 
  4-@^>.'-rT^(>«Xl)'->> 
  

  

  where 
  

  

  S»=l+g+ 
  •••+-' 
  

  

  indicating 
  the 
  behaviour 
  of 
  the 
  function 
  when 
  z 
  is 
  small. 
  

   When 
  z 
  is 
  positive 
  (or, 
  if 
  complex, 
  when 
  its 
  real 
  part 
  is 
  

   positive), 
  it 
  may 
  be 
  shown 
  that 
  f 
  

  

  e 
  .dd 
  .(6) 
  

  

  (?) 
  

  

  J 
  n 
  (z) 
  = 
  ~ 
  f'cos 
  sin 
  6-nd)d0- 
  ^L^ 
  

  

  77 
  JO 
  7T 
  J, 
  

  

  for 
  all 
  values 
  of 
  w. 
  

  

  Accordingly, 
  it 
  is 
  also 
  true 
  that 
  

  

  J_„(2)=-| 
  cos 
  (* 
  sin 
  6 
  + 
  n0)dd+ 
  e 
  .dd. 
  

  

  The 
  functions 
  so 
  defined 
  are 
  those 
  possessing 
  the 
  asymptotic 
  

   expansions, 
  when 
  z 
  is 
  very 
  large 
  in 
  comparison 
  with 
  n 
  J, 
  

  

  J„(,)(f)'=U„ 
  S 
  in( 
  2 
  -f 
  + 
  |)+V„co 
  S 
  (,-f 
  + 
  f)| 
  

  

  * 
  Hankel, 
  Math. 
  Ann. 
  i. 
  1869. 
  

  

  f 
  Vide 
  e. 
  g. 
  Whittaker, 
  ' 
  Modern 
  Analysis/ 
  p. 
  281. 
  

  

  % 
  Hankel, 
  I. 
  c. 
  

  

  