scipy.special.spence

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

Spence’s function, also known as the dilogarithm.

It is defined to be

\[\int_1^z \frac{\log(t)}{1 - t}dt\]

for complex \(z\), where the contour of integration is taken to avoid the branch cut of the logarithm. Spence’s function is analytic everywhere except the negative real axis where it has a branch cut.

Parameters:

zarray_like

Points at which to evaluate Spence’s function

outndarray, optional

Optional output array for the function results

Returns:

sscalar or ndarray

Computed values of Spence’s function

Notes

There is a different convention which defines Spence’s function by the integral

\[-\int_0^z \frac{\log(1 - t)}{t}dt,\]

which is equivalent to spence(1 - z).

Array API Standard Support

spence has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	✅	✅
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. spence does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

Examples

Try it in your browser!

>>> import numpy as np
>>> from scipy.special import spence
>>> import matplotlib.pyplot as plt

The function is defined for complex inputs:

>>> spence([1-1j, 1.5+2j, 3j, -10-5j])
array([-0.20561676+0.91596559j, -0.86766909-1.39560134j,
       -0.59422064-2.49129918j, -1.14044398+6.80075924j])

For complex inputs on the branch cut, which is the negative real axis, the function returns the limit for z with positive imaginary part. For example, in the following, note the sign change of the imaginary part of the output for z = -2 and z = -2 - 1e-8j:

>>> spence([-2 + 1e-8j, -2, -2 - 1e-8j])
array([2.32018041-3.45139229j, 2.32018042-3.4513923j ,
       2.32018041+3.45139229j])

The function returns nan for real inputs on the branch cut:

>>> spence(-1.5)
nan

Verify some particular values: spence(0) = pi**2/6, spence(1) = 0 and spence(2) = -pi**2/12.

>>> spence([0, 1, 2])
array([ 1.64493407,  0.        , -0.82246703])
>>> np.pi**2/6, -np.pi**2/12
(1.6449340668482264, -0.8224670334241132)

Verify the identity:

spence(z) + spence(1 - z) = pi**2/6 - log(z)*log(1 - z)

>>> z = 3 + 4j
>>> spence(z) + spence(1 - z)
(-2.6523186143876067+1.8853470951513935j)
>>> np.pi**2/6 - np.log(z)*np.log(1 - z)
(-2.652318614387606+1.885347095151394j)

Plot the function for positive real input.

>>> fig, ax = plt.subplots()
>>> x = np.linspace(0, 6, 400)
>>> ax.plot(x, spence(x))
>>> ax.grid()
>>> ax.set_xlabel('x')
>>> ax.set_title('spence(x)')
>>> plt.show()

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