original_docstring: 
      Compute the logarithmic derivative by upwards recurrence.
  
      Args:
          z: function argument
          N: order of Ricatti-Bessel function
  
      Returns:
          All the Ricatti-Bessel function values for orders from 0 to N for an
          argument z using the upwards recurrence relations.
      
docstring: Compute the logarithmic derivative by upwards recurrence.
docstring_tokens: Compute the logarithmic derivative by upwards recurrence .
code: def _D_upwards(z, N):
      """
      Compute the logarithmic derivative by upwards recurrence.
  
      Args:
          z: function argument
          N: order of Ricatti-Bessel function
  
      Returns:
          All the Ricatti-Bessel function values for orders from 0 to N for an
          argument z using the upwards recurrence relations.
      """
      D = np.zeros(N, dtype=complex)
      exp = np.exp(-2j * z)
      D[1] = -1 / z + (1 - exp) / ((1 - exp) / z - 1j * (1 + exp))
      for n in range(2, N):
          D[n] = 1 / (n / z - D[n - 1]) - n / z
      return D
code_tokens: def _D_upwards ( z , N ) : D = np . zeros ( N , dtype = complex ) exp = np . exp ( - 2j * z ) D [ 1 ] = - 1 / z + ( 1 - exp ) / ( ( 1 - exp ) / z - 1j * ( 1 + exp ) ) for n in range ( 2 , N ) : D [ n ] = 1 / ( n / z - D [ n - 1 ] ) - n / z return D
short_docstring: Compute the logarithmic derivative by upwards recurrence.
short_docstring_tokens: Compute the logarithmic derivative by upwards recurrence .
comment: """
      Compute the logarithmic derivative by upwards recurrence.
  
      Args:
          z: function argument
          N: order of Ricatti-Bessel function
  
      Returns:
          All the Ricatti-Bessel function values for orders from 0 to N for an
          argument z using the upwards recurrence relations.
      """