#4132 partfrac() problem with complex fraction

[Dr. Thorsten Wolterink]

Hello,
today I found a (small) problem with the use of partfrac() for complex numbers.

I have tried some examples similar to that in the documentation.

---

/ First example works /
1/(z+%i) + (1)/(z+1);
ratsimp(%)$
partfrac(%,z);

/ Second example doesn't work /
1/(z+%i) + (1)/(z+1+%i);
ratsimp(%)$
partfrac(%,z);

/ Third example works again /
1/(z+1) + (1)/(z+1+%i);
ratsimp(%)$
partfrac(%,z);

---

Some more information can be found within the attached file.

Best regards,
Thorsten

[Barton Willis]

For a workaround, call gfactor before partfrac. For example

(%i5)   1/(z+%i) + (1)/(z+1+%i);
(%o5)   1/(z+%i+1)+1/(z+%i)

(%i6)   ratsimp(%);
(%o6)   (2*z+2*%i+1)/(z^2+(2*%i+1)*z+%i-1)

(%i7)   partfrac(gfactor(%),z);
(%o7)   1/(z+%i+1)+1/(z+%i)

The user documentation says that partfrac "does a complete partial fraction decomposition." In this context, I'm not sure that "complete" has a clear meaning, so I don't consider this to be a bug.

We could insert a call to gfactor into the partfrac code.

[Stavros Macrakis]

This is not a bug, but the intended behavior.

The typical use case for partfrac is to calculate the real integrals of real integrands.
Thus for example partfrac(1/(x^4-1),x) yields
(-1/(2*(x^2+1)))-1/(4*(x+1))+1/(4*(x-1)),
which can be integrated as (-log(x+1)/4)-atan(x)/2+log(x-1)/4.
Most users do not want the result
(-(%i*log(x+%i))/4)+(%i*log(x-%i))/4-log(x+1)/4+log(x-1)/4,
even though it is of course equivalent -- though messy to simplify to the first form.

As was mentioned above, you can use partfrac(gfactor(1/(x^4-1)),x).

Similarly, most users will not want partfrac(1/(x^2-2),x) to yield 1/(2^(3/2)*(x-sqrt(2)))-1/(2^(3/2)*(x+sqrt(2)))....