﻿272 
  Dr. 
  J. 
  W. 
  Nicholson 
  on 
  Bessel 
  Functions 
  

  

  rrr 
  

  

  the 
  range 
  hto—, 
  the 
  asymptotic 
  expansion 
  does 
  not 
  converge 
  

  

  even 
  in 
  its 
  first 
  three 
  terms. 
  The 
  ratio 
  of 
  successive 
  terms 
  

   it 
  o£ 
  order 
  ah, 
  where 
  a 
  is 
  large. 
  But 
  unfortunately, 
  Lorenz's 
  

   argument 
  subsequently 
  compels 
  him 
  to 
  choose 
  the 
  (i 
  small 
  

  

  quantity 
  h 
  " 
  of 
  order 
  -, 
  so 
  that 
  ah 
  is 
  comparable 
  with 
  unity. 
  

  

  This 
  vitiates 
  the 
  whole 
  argument, 
  and 
  the 
  only 
  apparent 
  

   means 
  of 
  avoiding 
  the 
  difficulty 
  is 
  to 
  divide 
  the 
  range 
  of 
  

   integration 
  into 
  three 
  parts, 
  of 
  which 
  the 
  intermediate 
  one 
  

  

  passes 
  between 
  the 
  limits 
  h 
  and 
  k, 
  where 
  h 
  is 
  of 
  order 
  - 
  as 
  

  

  before, 
  and 
  h 
  is 
  such 
  that 
  ak 
  is 
  really 
  of 
  high 
  order. 
  The 
  

   consideration 
  of 
  this 
  intermediate 
  portion, 
  which 
  must 
  be 
  

   proved 
  negligible, 
  is 
  very 
  arduous, 
  for 
  no 
  asymptotic 
  value 
  

   of 
  the 
  Bessel 
  function 
  may 
  be 
  continuously 
  used 
  throughout 
  

   a 
  range 
  of 
  this 
  character. 
  

  

  But 
  since 
  all 
  these 
  troubles 
  may 
  be 
  avoided 
  by 
  a 
  more 
  

   direct 
  investigation, 
  it 
  seems 
  desirable 
  to 
  obtain 
  the 
  expan- 
  

   sions 
  from 
  the 
  ordinary 
  definite 
  integrals 
  for 
  the 
  Bessel 
  

   functions. 
  The 
  results 
  may 
  then 
  be 
  found 
  to 
  any 
  desired 
  

   order 
  of 
  approximation. 
  Moreover, 
  they 
  may 
  be 
  expressed 
  

   in 
  terms 
  of 
  well-known 
  transcendents 
  whose 
  tabulation, 
  

   originally 
  made 
  for 
  other 
  purposes, 
  is 
  fairly 
  complete. 
  

  

  The 
  functions 
  to 
  be 
  treated 
  are 
  usually 
  defined 
  in 
  the 
  

   following 
  manner 
  : 
  — 
  

  

  Whether 
  n 
  be 
  an 
  integer 
  or 
  not, 
  the 
  Bessel 
  function 
  

   J„(.~) 
  of 
  the 
  first 
  kind 
  is 
  given 
  by 
  

  

  ( 
  z 
  2 
  - 
  : 
  4 
  -) 
  

  

  J»0) 
  = 
  2TO 
  + 
  1) 
  ' 
  1 
  1_ 
  2 
  2 
  . 
  n 
  + 
  1 
  . 
  1 
  ! 
  + 
  2Kn 
  + 
  l 
  .n 
  + 
  2. 
  2 
  l~~' 
  ' 
  ' 
  j 
  ' 
  (1) 
  

  

  where 
  T(n 
  + 
  1) 
  is 
  a 
  gamma 
  function 
  becoming 
  identical 
  with 
  

   n 
  ! 
  if 
  n 
  be 
  an 
  integer. 
  

  

  With 
  this 
  is 
  associated., 
  when 
  n 
  is 
  not 
  an 
  integer, 
  the 
  

   function 
  J_ 
  ? 
  /~), 
  differing 
  from 
  the 
  above 
  by 
  a 
  change 
  of 
  

   sign 
  of 
  n 
  throughout, 
  or, 
  

  

  2"r(V).sinn7r 
  f 
  z 
  2 
  z* 
  y 
  J 
  

  

  J 
  ~^~ 
  7T~" 
  I 
  2 
  2 
  . 
  1-71 
  . 
  IT 
  2M-?2 
  . 
  2-n 
  . 
  2 
  !""• 
  * 
  j' 
  W 
  

  

  When 
  n 
  is 
  integral, 
  the 
  first 
  n 
  terms 
  of 
  this 
  series 
  vanish 
  bv 
  

   the 
  factor 
  sin?27r. 
  An 
  evanescent 
  factor 
  then 
  appears 
  in 
  the 
  

   subsequent 
  denominators, 
  and 
  evaluating 
  the 
  indeterminate 
  

   form 
  presented, 
  J_ 
  W 
  (V)=:(— 
  ) 
  n 
  J 
  n 
  (z) 
  in 
  this 
  special 
  case. 
  

   Another 
  function 
  must 
  therefore 
  be 
  associated 
  with 
  J 
  n 
  (z) 
  

  

  