What does the second argument in apart() do?

[Hugh]

Hello,

There is this block of code in the tutorial:

>>> l = []
>>> frac = apart(frac, a0)
>>> frac
                a₂⋅a₃⋅a₄ + a₂ + a₄
a₀ + ───────────────────────────────────────
     a₁⋅a₂⋅a₃⋅a₄ + a₁⋅a₂ + a₁⋅a₄ + a₃⋅a₄ + 1
>>> l.append(a0)
>>> frac = 1/(frac - a0)
>>> frac
a₁⋅a₂⋅a₃⋅a₄ + a₁⋅a₂ + a₁⋅a₄ + a₃⋅a₄ + 1
───────────────────────────────────────
           a₂⋅a₃⋅a₄ + a₂ + a₄

I've read the documentation on apart() and factor() but I still don't understand the purpose of putting a0 as the second argument to the call to apart(). Also, leaving out a0 produces a NotImplementedError: multivariate partial fraction decomposition error. and replacing a0 with other symbols like a1, a2, etc works fine. But I still don't understand what's going on. Please help.

[Sartaj Singh]

Here, frac is multivariate, i.e it depends on more than one variable. So, you need to explicitly tell the apart function for which variable should it perform the decomposition.

Eg. No need to give x as the second argument

In [21]: apart(1/((x+1)*(x+2)))
Out[21]:
    1       1  
- ───── + ─────
  x + 2   x + 1

x is required here (multivariate)

In [22]: apart(y/((x+1)*(x+2)), x)
Out[22]:
    y       y  
- ───── + ─────
  x + 2   x + 1

[Hugh]

Thanks Sartaj for the clarification.