I am working on a project and I would like to make sure I use Legendre function correctly. I've made a simple comparision between mpmath and Mathematica and the results are different:

for mpmath in python: import mpmath print(mpmath.legenp(-0.5, 1, 1.04018069)) the outcome is: (1.323429779732253753588079e-36 + 0.03499697660646475313011583j) And for Mathematica: LegendreP[-1/2, 1, 1.04018069] and the result is: -2.14295*10^-18 + 0.034997 I It seems they have the same imaginary part but different real part. I want to know where is the issue and how should I use the Legendre function of first and second kind, of degree one and half integer order.

I am working on a project and I would like to make sure I use Legendre function correctly. I've made a simple comparision between mpmath and Mathematica and the results are different:

for mpmath in python: import mpmath print(mpmath.legenp(-0.5, 1, 1.04018069)) the outcome is: (1.323429779732253753588079e-36 + 0.03499697660646475313011583j) And for Mathematica: LegendreP[-1/2, 1, 1.04018069] and the result is: -2.14295*10^-18 + 0.034997 I It seems they have the same imaginary part but different real part. I want to know where is the issue and how should I use the Legendre function of first and second kind, of degree one and half integer order.

• python
• wolfram-mathematica
• mpmath

Share Improve this question Follow asked Mar 22 at 19:55 user29990275user29990275 1

• I'll hazard a guess that it is trying to approximate 0+iy and that mpmath has by sheer good luck got a considerably better cancellation than Mathematica. Both are actually rather good approximations to zero in double precision arithmetic but 1e-36 is much better. Throw it at a symbolic algebra system to get confirmation of this hypothesis (guess). – Martin Brown Commented Mar 22 at 20:13

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Firstly, I would welcome you to the issue of computer precision in doing calculations. As commented by Martin Brown below your question, 1e-36 is fairly close to 0 as far as double precision computing goes.

Secondly, I would suggest you consider implementing a common practice in many languages where some "tolerance" is used to decide if something is sufficiently close to zero that we consider it as such. mpmath.chop(x, tol=None) is a function used to do just this.

Furthermore, mpmath has documentation which discusses setting or changing the precision which is worth a read.

Lastly, to address your question of "how to use Legendre function correctly", this really depends on your use case, to which you would need to provide more detail. You may only need 3 or 4 decimal places, in which case the mpmath.chop(x, tol=1e-4) route would work well enough.