class_docstr: Class for the Hypersphere Metric.
docstr: Compute the normalization factor - even dimension.
docstr_tok: ["Compute","the","normalization","factor","-","even","dimension","."]
code: def _normalization_factor_even_dim(self, variances):
      """Compute the normalization factor - even dimension."""
      dim = self._space.dim
      half_dim = (dim + 1) / 2
      area = 2 * gs.pi**half_dim / math.gamma(half_dim)
  
      def summand(k):
          exp_arg = -((dim - 1 - 2 * k) ** 2) / 2 / variances
          erf_arg_1 = (dim - 1 - 2 * k) * 1j / gs.sqrt(2 * variances)
          erf_arg_2 = (
              gs.pi * variances - (dim - 1 - 2 * k) * 1j
          ) / gs.sqrt(2 * variances)
          sign = (-1.0) ** k
          comb = gs.comb(dim - 1, k)
          erf_terms = gs.imag(gs.erf(erf_arg_2) + gs.erf(erf_arg_1))
          return sign * comb * gs.exp(exp_arg) * erf_terms
  
      half_dim_2 = int((dim - 2) / 2)
      if half_dim_2 > 0:
          sum_term = gs.sum(
              gs.stack([summand(k)] for k in range(half_dim_2))
          )
      else:
          sum_term = summand(0)
      coef = (
          area
          * (-1.0) ** half_dim_2
          / 2 ** (dim - 2)
          * gs.sqrt(gs.pi / 2 / variances)
      )
  
      return coef * sum_term
  
imports: import logging
  import math
  from itertools import product
  from scipy.stats import beta
  import geomstats.algebra_utils
  import geomstats.backend
  from geomstats.geometry.base import LevelSet
  from geomstats.geometry.euclidean import Euclidean
  from geomstats.geometry.riemannian_metric import RiemannianMetric
  from geomstats.vectorization import get_batch_shape, repeat_out
label_desc: Use image node_id 12 for calling the HypersphereMetric obj's underlying member method code with example usage: obj._normalization_factor_even_dim(variances) and returns: unknown, unknown