package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

gammaln(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True, signature, extobj)

gammaln(x, out=None)

Logarithm of the absolute value of the gamma function.

Defined as

.. math::

\ln(\lvert\Gamma(x)\rvert)

where :math:`\Gamma` is the gamma function. For more details on the gamma function, see dlmf_.

Parameters ---------- x : array_like Real argument out : ndarray, optional Optional output array for the function results

Returns ------- scalar or ndarray Values of the log of the absolute value of gamma

See Also -------- gammasgn : sign of the gamma function loggamma : principal branch of the logarithm of the gamma function

Notes ----- It is the same function as the Python standard library function :func:`math.lgamma`.

When used in conjunction with `gammasgn`, this function is useful for working in logspace on the real axis without having to deal with complex numbers via the relation ``exp(gammaln(x)) = gammasgn(x) * gamma(x)``.

For complex-valued log-gamma, use `loggamma` instead of `gammaln`.

References ---------- .. dlmf NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/5

Examples -------- >>> import scipy.special as sc

It has two positive zeros.

>>> sc.gammaln(1, 2) array(0., 0.)

It has poles at nonpositive integers.

>>> sc.gammaln(0, -1, -2, -3, -4) array(inf, inf, inf, inf, inf)

It asymptotically approaches ``x * log(x)`` (Stirling's formula).

>>> x = np.array(1e10, 1e20, 1e40, 1e80) >>> sc.gammaln(x) array(2.20258509e+11, 4.50517019e+21, 9.11034037e+41, 1.83206807e+82) >>> x * np.log(x) array(2.30258509e+11, 4.60517019e+21, 9.21034037e+41, 1.84206807e+82)

Returns the log of multivariate gamma, also sometimes called the generalized gamma.

Parameters ---------- a : ndarray The multivariate gamma is computed for each item of `a`. d : int The dimension of the space of integration.

Returns ------- res : ndarray The values of the log multivariate gamma at the given points `a`.

Notes ----- The formal definition of the multivariate gamma of dimension d for a real `a` is

.. math::

\Gamma_d(a) = \int_A>0 e^

tr(A)

}

|A|^a - (d+1)/2 dA

with the condition :math:`a > (d-1)/2`, and :math:`A > 0` being the set of all the positive definite matrices of dimension `d`. Note that `a` is a scalar: the integrand only is multivariate, the argument is not (the function is defined over a subset of the real set).

This can be proven to be equal to the much friendlier equation

.. math::

\Gamma_d(a) = \pi^d(d-1)/4 \prod_=1^d \Gamma(a - (i-1)/2).

References ---------- R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in probability and mathematical statistics).