Unknown

406 Notes on Explorations

In dealing with polynomial divisors with more than one distinct zero,
the divided difference technique of Exploration E.55 may be used. Al-
ternatively, the remainder upon division of a polynomial p(t) by (t - ai)
(t - us)... (t - a,) is the Lagrange polynomial of degree less than m which
assumes the value p(ai) at ai (see Exercise 7.1.5). In a similar way, the re-
mainder for division by (t - u)‘(t - b)S can be identified as that polynomial
of degree less than r + s for which the kth derivative at Q (resp. b) agrees
withthatofpata(resp.b)forO<k<r-l(resp.O<kss-1).

E.22. The nth derivative of p o q is a sum of terms of the form

where Ciai = n, Coi = k, and the coefficient is positive. The determination
of the coefficients is an interesting combinatorial problem solved by Faa
di Bruno in the middle of the last century. For a recent treatment and
bibliography, consult Steven Roman, The formula of Fa& di Bruno, Amer.
Math. Monthly 87 (1980), 805-809.

E.23 & 24. Partial derivatives are studied in a second calculus course
and are discussed in any textbook. The equation Cxidfldxi = kf for a
homogeneous polynomial of degree k is called Euler’s equation.

E.25. A natural generalization of polynomials is the class of complex-valued
functions f(z) of a complex variable which satisfy the differentiability con-
dition

jimo f (’ + h, - f (‘)

h

exists

(*I

for each point .z in the complex plane. Students who are familiar only with
the calculus of real-valued functions will not appreciate the strength of this
condition. In contrast to the real case, in which h can tend to 0 from one of
two real directions, in the complex plane h is permitted to tend to 0 in any
way over a two dimensional neighborhood of 0. As a result, the condition
implies that the functions (known as entire) have derivatives of all higher
orders and can be represented as the sum of an infinite convergent power
series a0 + ~1% + ~2%~ + 03%~ + .. for every complex Z. These functions
share many properties with polynomials.
As for polynomials, we can write f(z) = u(x, y) + iv(x, y) and discover
that () will be valid if and only if the Cauchy-Riemann conditions

au/ax= avlay aulay= -avlax

hold, where now the partial derivatives are defined through limits.

A pleasant introduction to the theory is George Polya & Gordon Latta,

Complex Variables, (Wiley, 1974). Another source which will richly reward

the patient reader is the set of five short volumes by Konrad Knopp, pub-

lished by Dover, New York: Elements of the Theory of Functions; Theory