1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 51

20. (a) If n is a positive integer, show that

cos nB = Pn( cos B)

where Pn(x) is a polynomial of the nth degree in x (the so-called

Chebyshev polynomial).

(b) Prove that

Pn(x) =^1 / 2 [(x + Jx2=It + (x - Jx2=ltJ

(c) Compute P1(x), P2(x), and Pa(x).

1.13 POWERS WITH FRACTIONAL EXPONENTS

As in elementary mathematics, the notation zm/n is introduced here as a

matter of convenience.

Definition 1. 7 If m denotes any integer and n a positive integer greater

than 1, we define

(1.13-1)

where the right-hand side means the mth power of each of the numbers in
the set * 'i{Z, assuming that z =/:- 0 if m < 0. Thus for z = reiO =/:-O, we have

zmfn = {rmfnei(m0+2km11")/n: k = 0, 1, ... 'n - 1}

Examples

1. i2/3 = ( *$)2 = { -1, %(1 + iv'3),1/2(1-iv's)}

2. (l i)-1/2 = (*Vf=i)-1 = { !_ ei11"/B 1 e9i11"/8}

~ '~

(1.13-2)

We point out that if m' /n' is the irreducible fraction equivalent to m/n,

the number of distinct values in (1.13-2) is n' = n/(m,n), where (m,n)
denotes the greatest common divisor of m and n. In fact, we have the
property

(1.13-3)

Example i4/6 = i2/3

However, the usual laws of radicals (or of fractional exponents) do not
hold under the interpretations (1.12-7) and (1.13-2). First, we note that
in general

(1.13-4)