﻿274: 
  Dr. 
  J. 
  W. 
  Nicholson 
  on 
  Bessel 
  Functions 
  

  

  when 
  n 
  is 
  not 
  integral, 
  and 
  

  

  Y„(:) 
  .(£)*— 
  U. 
  cos 
  (--f 
  + 
  £) 
  + 
  V.*f-!J 
  + 
  J 
  ), 
  (9) 
  

  

  when 
  n 
  is 
  integral, 
  provided 
  

  

  4;i 
  2 
  -l 
  2 
  . 
  4/i 
  2 
  - 
  3 
  2 
  4ji 
  2 
  -1 
  2 
  . 
  4k 
  2 
  -3 
  2 
  . 
  4?t 
  g 
  -5 
  2 
  . 
  4» 
  2 
  -7 
  2 
  > 
  

   2! 
  (8*) 
  2 
  4! 
  (&)* 
  -| 
  (10) 
  

  

  4k 
  2 
  - 
  l 
  2 
  _ 
  4tt 
  a 
  -l 
  a 
  .4n 
  2 
  -3 
  2 
  .4n 
  2 
  -5 
  2 
  J 
  

  

  n= 
  1!8*" 
  ~ 
  " 
  3 
  ! 
  (8*) 
  3 
  

  

  If 
  -/i-f 
  ^ 
  be 
  written 
  for 
  n, 
  their 
  relations 
  to 
  the 
  forms 
  

   frequently 
  used 
  in 
  physical 
  problems 
  concerning 
  wave- 
  

   motion 
  in 
  or 
  about 
  spheres, 
  become 
  apparent. 
  We 
  proceed 
  

   to 
  obtain 
  an 
  expression 
  for 
  J 
  n 
  (z), 
  when 
  n 
  and 
  z 
  are 
  nearly 
  

   equal, 
  and 
  each 
  is 
  fairly 
  large. 
  The 
  more 
  definite 
  limitation 
  

   required 
  by 
  the 
  last 
  statement 
  will 
  appear 
  later. 
  

  

  Jf 
  z 
  (or 
  more 
  generally, 
  its 
  real 
  part) 
  be 
  positive, 
  by 
  (6) 
  

  

  t/\ 
  if 
  77 
  / 
  • 
  a 
  n\ 
  m 
  sin 
  nir 
  f 
  °°-"0-~smli0 
  

   J 
  n 
  (z) 
  = 
  -\ 
  cos 
  (zsm 
  6 
  — 
  nd)dQ 
  1 
  e 
  .dO 
  

  

  ^Jo 
  ir 
  Jo 
  

  

  1 
  T 
  sin?27r 
  r 
  f 
  . 
  , 
  t1 
  v 
  

  

  7T 
  7T 
  

  

  Ij 
  is 
  the 
  real 
  part 
  of 
  I 
  3 
  (if 
  z 
  be 
  real), 
  where 
  

   I 
  3 
  = 
  pexp. 
  {t-(sin 
  6-0)- 
  i. 
  n-z 
  . 
  0\ 
  dO 
  

  

  -(")*. 
  ( 
  d 
  " 
  1 
  '''.!». 
  «sii.-.{«"-f,(j)* 
  .A... 
  * 
  P»}, 
  (12) 
  

  

  where 
  fi 
  x 
  

  

  * 
  = 
  *-"■(*)'' 
  

  

  and 
  the 
  variable 
  of 
  integration 
  has 
  been 
  changed 
  by 
  the 
  

   substitution 
  

  

  e 
  = 
  (J)**. 
  

  

  When 
  n 
  and 
  ^ 
  are 
  nearly 
  equal, 
  and 
  so 
  nearly 
  that 
  n—z 
  is 
  

   small 
  in 
  comparison 
  with 
  zs, 
  then 
  the 
  portion 
  of 
  the 
  exponent 
  

   (in 
  the 
  integrand) 
  involving 
  w 
  is 
  of 
  less 
  importance 
  than 
  

  

  