2.4. Graphing Polynomials 71realized through the medium of limits). A second order differential equation
satisfied by the real and imaginary parts of these functions is one that has
an important part to play in physics. Partly for this reason, functions of a
complex variable have a useful role in this science.
E.26. The Legendre Equation. In applied mathematics, the differential
equation
(1 - x’)y” - 2x4 + n(n + 1)y = 0... (1)
(where y is a function of x) plays an important role. What sort of solutions
does this equation have?
To find out, differentiate the equation r times to obtain
(1 - x’)y(‘+‘) - 2(r + l)xy(‘+‘) + (n - r)(n + 1 + r)ycr) = 0.... (2)Thus, if n is a positive integer and r = n, z = y(‘+l) satisfies the equation
(1 - 2’)~ - 2(r + 1)zz = 0.... (3)Show that (3) is not satisfied by any nonzero polynomial z in the variable
z. Deduce that any polynomial solution y of (1) has degree not exceeding
n.
Equation (1) in fact does have a polynomial solution y = P,(x). Check
this for n = 1,2,3,4. To find it more generally, observe that, from (2),
PC+“)(O) = -(n - r)(n + 1 + r)Pc)(O).... (4)It follows from (4) that ~~+2’(0) = 0 for r = n, n+2, n+4,.... Suppose we
ensure that pik’(0) = 0 for k > n + 1 by setting P?-“(O) = Ppv3’(0) =
... = pp-w(q =... = 0. Then, by Taylor’s Theorem, we have that
P*(x) = ;Pp(o)x” + -ppqqp-2 +....
(nT2)!Suppose that Pp)(O) = n!. Use (4) to obtain the remaining coefficients.
Verify that the polynomial so obtained is a solution of (1).2.4 Graphing Polynomials
One picture is worth a thousand words. This adage is especially true in
mathematics in dealing with the behaviour of functions. The graphs of real
polynomials can provide at a glance valuable information about their zeros
and degrees. It can be a useful tool in analyzing results about polynomials in
such areas as the theory of approximation of functions by polynomials. This
section will require knowledge from a first course in differential calculus.
Let us review some of the terminology required: