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. De III. Agenda. Problems in Orthomorphosis. scribe the graphical transformation of u+iv into each of the following functions : (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 in which r1/" is a tensor, (2k′′ +0)/n an amplitude, and k has one of the values o, I, 2, 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, 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. Hence A complex quantity has only n nth roots. 114. Agenda. Examples in the Determination of th Prove the following:* Roots. of (6). Find the fifth roots of +1 and the sixth roots (7). If w be one of the complex cube roots of + 1, 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 )}. 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. 2 116. Roots of Equations. The investigation of Art. 113 solves the problem of finding what are called the roots of the equation 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. |