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# scipy.special.gamma

scipy.special.gamma(z, out=None) = <ufunc 'gamma'>

  gamma function.

  The gamma function is defined as

  \[\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt\]

  for \(\Re(z) > 0\) and is extended to the rest of the complex plane by analytic continuation. See [dlmf] for more details.

  Parameters:
    z array_like

      Real or complex valued argument

    out ndarray, optional

      Optional output array for the function values

  Returns:
    scalar or ndarray

      Values of the gamma function

  Notes

  The gamma function is often referred to as the generalized factorial since \(\Gamma(n + 1) = n!\) for natural numbers \(n\). More generally it satisfies the recurrence relation \(\Gamma(z + 1) = z \cdot \Gamma(z)\) for complex \(z\), which, combined with the fact that \(\Gamma(1) = 1\), implies the above identity for \(z = n\).

  References

  [dlmf]

  NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/5.2#E1

  Examples

      >>> import numpy as np
      >>> from scipy.special import gamma, factorial

      >>> gamma([0, 0.5, 1, 5])
      array([         inf,   1.77245385,   1.        ,  24.        ])

      >>> z = 2.5 + 1j
      >>> gamma(z)
      (0.77476210455108352+0.70763120437959293j)
      >>> gamma(z+1), z*gamma(z)  # Recurrence property
      ((1.2292740569981171+2.5438401155000685j),
       (1.2292740569981158+2.5438401155000658j))

      >>> gamma(0.5)**2  # gamma(0.5) = sqrt(pi)
      3.1415926535897927

  Plot gamma(x) for real x

      >>> x = np.linspace(-3.5, 5.5, 2251)
      >>> y = gamma(x)

      >>> import matplotlib.pyplot as plt
      >>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)')
      >>> k = np.arange(1, 7)
      >>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6,
      ...          label='(x-1)!, x = 1, 2, ...')
      >>> plt.xlim(-3.5, 5.5)
      >>> plt.ylim(-10, 25)
      >>> plt.grid()
      >>> plt.xlabel('x')
      >>> plt.legend(loc='lower right')
      >>> plt.show()