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

#*****************************************************************************80
#
## U_MOMENT: integral ( -1 <= x <= +1 ) x^e sqrt ( 1 - x^2 ) dx.
#
#  Discussion:
#
#     E    U_MOMENT
#    --    --------------  
#     0         pi /    2 
#     2         pi /    8
#     4         pi /   16
#     6     5 * pi /  128
#     8     7 * pi /  256
#    10    21 * pi / 1024
#    12    33 * pi / 2048
#
#  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_gamma import r8_gamma

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

    value = 0.0

  else:

    arg1 = 0.5 * float ( 1 + e )
    arg2 = 2.0 + 0.5 * float ( e )
    value = 0.5 * np.sqrt ( np.pi ) * r8_gamma ( arg1 ) / r8_gamma ( arg2 )

  return value

def u_moment_test ( ):

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

  print ( '' )
  print ( 'U_MOMENT_TEST:' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  U_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 = u_moment ( e )
    print ( '  %2d  %14.6g' % ( e, value ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'U_MOMENT_TEST:' )
  print ( '  Normal end of execution.' )
  return

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