﻿368 
  Prof. 
  G. 
  N. 
  Watson 
  on 
  Bessel 
  Functions 
  

  

  Now 
  6 
  + 
  sin 
  6 
  cos 
  — 
  20 
  2 
  cot 
  # 
  vanishes 
  when 
  — 
  and 
  has 
  

   the 
  positive 
  derivate 
  2(cos# 
  — 
  6 
  cosec 
  #) 
  2 
  ; 
  hence 
  it 
  is 
  positive 
  

   when 
  < 
  6 
  < 
  ir. 
  The 
  function 
  

  

  7r-tan- 
  1 
  {2rsm0/{l-r 
  2 
  )} 
  

  

  therefore 
  has 
  a 
  negative 
  derivate, 
  and 
  the 
  desired 
  result 
  is 
  

   proved. 
  

  

  Taking 
  F(0) 
  as 
  a 
  new 
  variable, 
  F, 
  and 
  writing 
  </>(F) 
  for 
  

   the 
  function 
  ir 
  — 
  tan 
  -1 
  {2r 
  sin 
  0/(1 
  — 
  r 
  2 
  )}, 
  we 
  have 
  

  

  Vj 
  n 
  (nx)dx= 
  i 
  f" 
  <j>(F)e- 
  n 
  *dF, 
  

   Jo 
  """Jo 
  

  

  where 
  0(F) 
  is 
  a 
  positive 
  decreasing 
  function 
  of 
  F 
  such 
  that 
  

  

  Hence, 
  since 
  the 
  integral 
  on 
  the 
  right 
  is 
  uniformly 
  con- 
  

   vergent 
  for 
  large 
  values 
  of 
  n 
  *, 
  

  

  Lim 
  \n\ 
  J 
  n 
  (na)dx\ 
  = 
  Lim 
  — 
  I 
  </>(iij 
  'n)e~ 
  u 
  du 
  

  

  = 
  *(0)/ir=j. 
  

   That 
  is 
  to 
  say, 
  

  

  J 
  n 
  (nx)dxco 
  5-, 
  

   - 
  on 
  

  

  1 
  

  

  which 
  is 
  the 
  result 
  stated. 
  

   From 
  the 
  formula 
  

  

  J 
  n 
  (nx)dx=- 
  \ 
  6(u/n)e~ 
  u 
  du 
  

   ^Jo 
  

  

  it 
  is 
  evident 
  that 
  the 
  function 
  on 
  the 
  left 
  steadily 
  increases 
  

   (for 
  all 
  positive 
  values 
  of 
  71) 
  as 
  n 
  increases. 
  

  

  3. 
  When 
  n 
  is 
  an 
  odd 
  integer 
  we 
  can 
  express 
  the 
  integral 
  

   as 
  a 
  sum 
  of 
  Bessel 
  functions, 
  since 
  we 
  have 
  

  

  n\ 
  J 
  n 
  (nx)dx=\ 
  J 
  n 
  {y)dy 
  

   Jo 
  Jo 
  

  

  and 
  j 
  m 
  (y)=j„ 
  1 
  _ 
  2 
  (» 
  / 
  )-2j:_ 
  1 
  (y), 
  

  

  whence 
  we 
  find 
  

   n 
  \ 
  J»(n#) 
  dx 
  

  

  =JjJi(3/)- 
  2 
  J2'(y)-2J 
  4 
  '(y)-...-2X_ 
  1 
  (^)}^ 
  

   = 
  l-Uo(n) 
  + 
  2J 
  2 
  (n) 
  + 
  2J 
  n 
  (n) 
  + 
  ...+2J 
  B 
  . 
  1 
  (w)}. 
  

   * 
  Bromwich, 
  'Infinite 
  Series,' 
  pp. 
  434, 
  436. 
  

  

  