I am using the Python library Sympy to calculate the partial fraction decomposition of the following rational function:

Z = (6.43369157032015e-9*s^3 + 1.35203404799555e-5*s^2 + 0.00357538393743079*s + 0.085)/(4.74334912634438e-11*s^4 + 4.09576274286244e-6*s^3 + 0.00334241812250921*s^2 + 0.15406018058983*s + 1.0)

I am calling the method apart() like this:

Z_f = apart(Zth.expand(), full=True).doit()

And this is the result:

-1.30434068814287e+23/(s + 85524.0054884464) + 19145168.0/(s + 774.88576677949) + 91.9375/(s + 40.7977016133126) + 2.59375/(s + 7.79746609204661)

As you can see from the result, these are the residuals:

• -3.60418953263334e+22
• -4789228.25000000
• 11.0468750000000
• 0.787109375000000

The problem is that by using other tools I get different residuals.

By using GNU Octave’s residue() I get these residuals:

• 133.6
• 1.0776
• 0.39501
• 0.56426

Via Wolphram Alpha I get the same residuals as Octave see here the results (please wait until you see the partial fraction expansion) .

Why isn’t the Sympy library not working as expected?

The poles are always correct. Only the residuals are not as I expect.

Thanks for any suggestion.