>
> Partial fractions. What precisely should be taught? I ask because I
> had not realised until I wrote a paper last year that they are not
> unique. There is an identity:
>
> 0 = (a-b)/(f-a)(f-b) + (b-c)/(f-b)(f-c) + (c-a)/(f-c)(f-a)
That problem only arises if you stop too soon. (a-b)/(f-a)(f-b) can
be decomposed into partial fractions as 1/(f-a) - 1/(f-b) and of course
one has to do that when integrating. One must go on with partial fractions
until one has a sum of terms like P(1/q(f)) where P is a polynomial in
the variable 1/q, and q is a linear function of f, say f-k. See for
example Ahlfors Complex Analysis.
In elementary calculus one may not wish to tolerate the possibility of
complex k, of course (try the partial fraction expansion of
1/((f^2+1)(f^2+2*f+2)^2) for example), and then one uses those more
elaborate rules about irreducible quadratic factors.
John Harper, School of Mathematical and Computing Sciences,
Victoria University, Wellington, New Zealand
e-mail john.harper@vuw.ac.nz phone (+64)(4)463 5341 fax (+64)(4)463 5045