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Python/numba implementation of Wigner D Matrices, spin-weighted spherical harmonics, and associated functions

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Spherical Functions Status of automatic build and test suite MIT license

NOTE: This package will still be maintained, but active development has moved to the spherical package. While this package works well for ℓ (aka ell, L, j, or J) values up to around 25, errors start to build rapidly and turn into NaNs around 30. The spherical package can readily handle values up to at least 1000, with accuracy close to ℓ times machine precision. —Mike

Python/numba package for evaluating and transforming Wigner's 𝔇 matrices, Wigner's 3-j symbols, and spin-weighted (and scalar) spherical harmonics. These functions are evaluated directly in terms of quaternions, as well as in the more standard forms of spherical coordinates and Euler angles.1

The conventions for this package are described in detail on this page.


The only true requirements for this code are python and the python package numpy, as well as my accompanying quaternion package (installation of which is shown below).

However, this package can automatically use numba, which uses LLVM to compile python code to machine code, accelerating most numerical functions by factors of anywhere from 2 to 2000. It is possible to run the code without numba, but the most important functions are roughly 10 times slower without it.

The only drawback of numba is that it is nontrivial to install on its own. Fortunately, the best python installer, anaconda, makes it trivial. Just install the main anaconda package.

If you prefer the smaller download size of miniconda (which comes with no extras beyond python), you'll also have to run this command:

conda install pip numpy numba


Assuming you use conda to manage your python installation (like any sane python user), you can install this package simply as

conda install -c conda-forge spherical_functions

This should automatically download and install the package quaternion, on which this package depends.

Alternatively, if you prefer to use pip (whether or not you use conda), you can also do

pip install git+git://
pip install git+git://

Or, if you refuse to use conda, you might want to install inside your home directory without root privileges. (Anaconda does this by default anyway.) This is done by adding --user to the above commands:

pip install --user git+git://
pip install --user git+git://

Finally, there's also the fully manual option of just downloading both code repositories, changing to the code directory, and issuing

python install

This should work regardless of the installation method, as long as you have a compiler hanging around.


First, we show a very simple example of usage with Euler angles, though it breaks my heart to do so:1

>>> import spherical_functions as sf
>>> alpha, beta, gamma = 0.1, 0.2, 0.3
>>> ell,mp,m = 3,2,1
>>> sf.Wigner_D_element(alpha, beta, gamma, ell, mp, m)

Of course, it's always better to use unit quaternions to describe rotations:

>>> import numpy as np
>>> import quaternion
>>> R = np.quaternion(1,2,3,4).normalized()
>>> ell,mp,m = 3,2,1
>>> sf.Wigner_D_element(R, ell, mp, m)

If you need to calculate values of the 𝔇(ℓ) matrix elements for many values of (ℓ, m', m), it is more efficient to do so all at once. The following calculates all modes for ℓ from 2 to 8 (inclusive):

>>> indices = np.array([[ell,mp,m] for ell in range(2,9)
... for mp in range(-ell, ell+1) for m in range(-ell, ell+1)])
>>> sf.Wigner_D_element(R, indices)

Finally, if you really need to put the pedal to the metal, and are willing to guarantee that the input arguments are correct, you can use a special hidden form of the function:

>>> sf._Wigner_D_element(R.a, R.b, indices, elements)

Here, R.a and R.b are the two complex parts of the quaternion defined on this page (though the user need not care about that). The indices variable is assumed to be a two-dimensional array of integers, where the second dimension has size three, representing the (ℓ, m', m) indices. This avoids certain somewhat slower pure-python operations involving argument checking, reshaping, etc. The elements variable must be a one-dimensional array of complex numbers (can be uninitialized), which will be replaced with the corresponding values on return. Again, however, there is no input dimension checking here, so if you give bad inputs, behavior could range from silently wrong to exceptions to segmentation faults. Caveat emptor.


I very much appreciate Barry Wardell's help in sorting out the relationships between my conventions and those of other people and software packages (especially Mathematica's crazy conventions).

This code is, of course, hosted on github. Because it is an open-source project, the hosting is free, and all the wonderful features of github are available, including free wiki space and web page hosting, pull requests, a nice interface to the git logs, etc.

Every change in this code is auomatically tested on Travis-CI. This is a free service (for open-source projects like this one), which integrates beautifully with github, detecting each commit and automatically re-running the tests. The code is downloaded and installed fresh each time, and then tested, on python 2.7, 3.4, and 3.5. This ensures that no change I make to the code breaks either installation or any of the features that I have written tests for.

Finally, the code is automatically compiled, and the binaries hosted for download by conda on This is also a free service for open-source projects like this one.

The work of creating this code was supported in part by the Sherman Fairchild Foundation and by NSF Grants No. PHY-1306125 and AST-1333129.

--- ###### 1 Euler angles are awful

Euler angles are pretty much the worst things ever and it makes me feel bad even supporting them. Quaternions are faster, more accurate, basically free of singularities, more intuitive, and generally easier to understand. You can work entirely without Euler angles (I certainly do). You absolutely never need them. But if you're so old fashioned that you really can't give them up, they are fully supported.

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