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any hyperbola of the series by keeping s fixed and varying r from ∞ to∞, passing from cos s to +∞, У from -through o to +∞. The other branch of the same hyperbola is then obtained by changing s into + s and causing to vary again from ∞ to +∞. If then, in addition to the variation of r, s be given all values between s。 and s+, the morphosis is completed. To the z-plane in this case are co-ordinated two parallel bands, a a and bb of Fig. 43, having a width equal to mπ and extending to infinity in both directions, and separated from one another by a third band of like dimensions.

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III. Agenda. Problems in Orthomorphosis. scribe the graphical transformation of u+iv into each of the following functions :

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(5). Prove that the graphical transformation of u+ i v into cos (iv), or into any of the other cyclic functions is an orthomorphosis.

(6). Perform upon e(u+iv)/K, sin, (u+ iv), cos, (u+iv) and tank (uiv), the transformation of Art. 101 and trace, upon the surface of the sphere, the path of each of these functions when the affix of u+ iv moves upon a straight line in the plane.

CHAPTER VI.

PROPERTIES OF POLYNOMIALS.

XVIII. ROOTS OF COMPLEX QUANTITIES.

112. Definition of an nth Root. The nth root of a given quantity is defined to be such another quantity as, when multiplied by itself n-1 times (used n times as a factor), will produce the given quantity. An nth root of w is denoted by win.

Throughout the discussion concerning roots, whether of quantities or equations, n is supposed to be an integer, and unless statement be made to the contrary, a positive integer.

113. Evaluation of nth Roots. Every complex quantity has n nth roots of the form

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in which r1/" is a tensor, (2k′′ +0)/n an amplitude, and k has one of the values o, I, 2,

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N — I.

Let the complex quantity, whose roots are to be investigated, be denoted by r cis 0. Since r is a real positive magnitude, /" has one real positive value (Art. 23). How to find this value we do not here enquire.

By Demoivre's theorem (Art. 74), for all integral values

of k,

2απ + θ

1/n cis }" = r cis (2kπ + 0) = r cis 8,

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are all different. Hence each of them is a distinct nth root of r cis 0, and there are ʼn of them. Thus r cis has n nth roots, which was to be proved.

No additional values are derived from 1/" cis (2kπ+6)/n by giving to k any values other than those contained in the series o, I, 2, . . n - 1, and r1/" has only one real

positive value.

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Hence

A complex quantity has only n nth roots.

114. Agenda. Examples in the Determination of th Prove the following:*

Roots.

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(6). Find the fifth roots of +1 and the sixth roots

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(7). If w be one of the complex cube roots of + 1,

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Show that I w2 is one of the twelfth roots of +1.

(8). The twentieth roots of I are the successive powers, from the first to the twentieth inclusive, of

{√(10+2 √5) + i (√ 5 − 1 )}.

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even powers of σ are the tenth roots of + 1.

XIX.

POLYNOMIALS AND EQUATIONS.

115. Definition of Polynomial. An algebraic expression of the form

a+a, z + a2 z2 + . . . . + anz",

in which a, a,, . . . an are any quantities not involving z, and in which the exponents of z are all integers, is called a rational, integral polynomial in z. In what follows it will be sufficient to speak of it more briefly as a polynomial, the qualifying adjectives being understood. The highest exponent of a contained in it is its degree.

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116. Roots of Equations. The investigation of Art. 113 solves the problem of finding what are called the roots of the equation

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in which w is a known complex quantity, and shows that such an equation, which would commonly present itself in the binominal form

has exactly n roots.

azn+b=0,

[b/a =

= -W w]

If additional terms containing powers of z lower than the nth be introduced into this equation, the problem of its solution becomes at once difficult, or impossible. In fact, the so-called algebraic solution of a general algebraic equation of a degree higher than the fourth, that is, a solution involving only radicals and having a finite number of terms, is known to be impossible.*

A discussion of the methods that may be employed in solving equations is beyond the intended scope of the present work, but the so-called fundamental theorem of algebra (Art. 120), accompanied by those propositions that are prerequisite to its demonstration, find a fitting place here.

117. The Remainder Theorem. If a polynomial of the nth degree in z be divided by zy, the remainder, after n successive divisions, is the result of substituting a for z in the polynomial.

Let f (z) denote the polynomial,

f(z) = a+a, ≈ + α2 ≈2 + . . . . + a, zn,

z

* Proved to be so by Abel: Journal für die reine und angewandte Mathematik (1826), Bd. I, pp. 65-84, and Œuvres completes de N. H. Abel, nouv. ed., Vol. I, pp. 66-94.

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