code: def _a1inv(x):
      """Compute inverse function.
  
      Inverse function of the ratio of the first and
      zeroth order Bessel functions of the first kind.
  
      Returns the value k, such that a1inv(x) = k, i.e. a1(k) = x.
      """
      if 0 <= x < 0.53:
          return 2 * x + x**3 + (5 * x**5) / 6
      elif x < 0.85:
          return -0.4 + 1.39 * x + 0.43 / (1 - x)
      else:
          return 1 / (x**3 - 4 * x**2 + 3 * x)
  
imports: import warnings
  import numpy
  from scipy.fftpack import fft
  from scipy.optimize import brentq
  from scipy.signal import convolve, convolve2d, gaussian
  from scipy.sparse import coo_matrix
  from scipy.special import ive
  from ..utils import _cov, _dot, _stack, conditional_jit
label_desc: Use image node_id 8 for calling a global function with example usage: _a1inv(x) and returns: unknown, unknown, unknown