#! /usr/bin/env python
#
def t_moment ( e ):

#*****************************************************************************80
#
## T_MOMENT: integral ( -1 <= x <= +1 ) x^e dx / sqrt ( 1 - x^2 ).
#
#  Discussion:
#
#    Set 
#      x = cos ( theta ), 
#      dx = - sin ( theta ) d theta = - sqrt ( 1 - x^2 ) d theta
#    to transform the integral to
#      integral ( 0 <= theta <= pi ) - ( cos ( theta ) )^e d theta
#    which becomes
#      0 if E is odd,
#      (1/2^e) * choose ( e, e/2 ) * pi if E is even.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    14 July 2015
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer E, the exponent of X.
#    0 <= E.
#
#    Output, real VALUE, the value of the integral.
#
  import numpy as np
  from r8_choose import r8_choose

  if ( ( e % 2 ) == 1 ):

    value = 0.0

  else:

    value = r8_choose ( e, e // 2 ) * np.pi / 2.0 ** e

  return value

def t_moment_test ( ):

#*****************************************************************************80
#
## T_MOMENT_TEST tests T_MOMENT.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    14 July 2015
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'T_MOMENT_TEST:' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  T_MOMENT returns the value of' )
  print ( '  integral ( -1 <=x <= +1 ) x^e / sqrt ( 1 - x^2 ) dx' )
  print ( '' )
  print ( '   E       Integral' )
  print ( '' )
  for e in range ( 0, 11 ):
    value = t_moment( e )
    print ( '  %2d  %14.6g' % ( e, value ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'T_MOMENT_TEST:' )
  print ( '  Normal end of execution.' )
  return

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( )
  t_moment_test ( )
  timestamp ( )