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# scipy.special.hyperu

scipy.special.hyperu(a, b, x, out=None) = <ufunc 'hyperu'>

  Confluent hypergeometric function U.

  It is defined as the solution to the equation

  \[x \frac{d^2w}{dx^2} + (b - x) \frac{dw}{dx} - aw = 0\]

  which satisfies the property

  \[U(a, b, x) \sim x^{-a}\]

  as \(x \to \infty\). See [DLMF] for more details.

  Parameters:
    a, b array_like

      Real-valued parameters

    x array_like

      Real-valued argument

    out ndarray, optional

      Optional output array for the function values

  Returns:
    scalar or ndarray

      Values of U

  References

  [DLMF]

  NIST Digital Library of Mathematics Functions https://dlmf.nist.gov/13.2#E6

  Examples

      >>> import numpy as np
      >>> import scipy.special as sc

  It has a branch cut along the negative x axis.

      >>> x = np.linspace(-0.1, -10, 5)
      >>> sc.hyperu(1, 1, x)
      array([nan, nan, nan, nan, nan])

  It approaches zero as x goes to infinity.

      >>> x = np.array([1, 10, 100])
      >>> sc.hyperu(1, 1, x)
      array([0.59634736, 0.09156333, 0.00990194])

  It satisfies Kummer’s transformation.

      >>> a, b, x = 2, 1, 1
      >>> sc.hyperu(a, b, x)
      0.1926947246463881
      >>> x**(1 - b) * sc.hyperu(a - b + 1, 2 - b, x)
      0.1926947246463881