RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

356 Chapter 19. The Paris memoir
ABEL and the method of Lagrange interpolation. Without any specific reference,
ABEL employed a result to the effect that
n
∑
k=1
p (xk) χ ′ (xk) =
�
0if deg p < n − 1
1if deg p = n − 1
for any normed polynomial p where x1, . . . , xn are the roots of the polynomial equation
χ (x) = 0, i.e.
χ (x) =
n
∏ (x − xk) . (19.14)
k=1
The tool behind this result is known today as Lagrange interpolation, and — in various
forms — it played central roles in ABEL’S arguments in the Paris memoir. Lagrange
interpolation is used to demonstrate that for any polynomial such as (19.14),
1
χ (x) =
n
∑
k=1
1
(x − xk) χ ′ . (19.15)
(xk) Expansion into partial fractions using Lagrange interpolation. The method of ex-
panding a quotient of polynomials into partial fractions was well established in the
18 th century once it was known that the denominator could be decomposed into a
product of linear and quadratic terms. The generality of the method thus rested es-
sentially on the Fundamental Theorem of Algebra, and the proof of the latter theorem was
often seen mainly as a prerequisite in rigorously founding this established practice (cf.
GAUSS).
The central trick in expanding a quotient into partial fractions is closely related to
the method of Lagrange interpolation. If the polynomials are
f2 (x) =
f1 (x) and
n
∏ (x − xk) ,
k=1
where the roots of f2 are distinct, Lagrange interpolation (19.15) yields
f1 (x)
f2 (x) =
n
∑
k=1
f1 (x)
(x − x k) f ′ 2 (x k) .
Thus, when applied in the integral calculus, this formula reduces the integration of
a fraction to the integration of n fractions, the denominator of each of which only
contains a first degree polynomial.