code: def _tau_frank_expansion(x):
      x = np.asarray(x)
      # expansion derived using wolfram alpha
      # agrees better with R copula for x<=1, maybe even for larger theta
      tau = (
          x / 9
          - x**3 / 900
          + x**5 / 52920
          - x**7 / 2721600
          + x**9 / 131725440
          - x**11 * 691 / 4249941696000
      )
      return tau
  
imports: import sys
  import numpy
  from scipy import stats, integrate, optimize
  from .None import transforms
  from .copulas import Copula
  from statsmodels.tools.rng_qrng import check_random_state
label_desc: Use image node_id 4 for calling a global function with example usage: _tau_frank_expansion(x) and returns: tau