Complex Numbers

Complex Numbers - 11 Chapter d t J r jlr 5 COMPLEX NUMBERS...

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11 .. Chapter d t 5 • The power series L::'=o anx n converges absolutely within the interval of convergence Ixl < R, where I an I R = lim -- . n-too a n +l Jr' COMPLEX NUMBERS • Relations between functions will be satisfied order by order when they are replaced by their power series. You must know how to expand functions of functions, out to some desired order within the common interval of conver- gence. jlr. • The power series representing a function may be integrated term by term and differentiated term by term within the interval of convergence to obtain the series for the integral or derivative of the function in question. ;jii. ili Pr! ,lh· 5.1. Introduction Let us consider the quadratic function f {x) = x 2 - 5x+6 and ask where it vanishes. If we plot it against x , we will find that it vanishes at x = 2 and x = 3. This is also clear if we write f in factorized form as f{x) = (x - 2){x - 3). We could equivalently use the well-known formula for the roots x ± of a quadratic equation: 89 A plot will show that this function is always positive and does not vanish for any point on the z -axis. We are then led to conclude that this quadratic equation has no roots. Let us pass from the graphical procedure which gives no solution to the algebraic one which does give some form of answer even now. It says The problemof course is that we do not know what to make of H since there is no real number whose square is ~3. Thus if we take the stand that a number is not a number unless it is a real number, we will have to conclude that some quadratic equations have roots and other do not. . "'~ ~ n ~ ~0 ;..J - ~ i! ~~ o ~ (]) ~~ ~ 00", ~ ~~ ~ 5~ ~~ ,...l >, ~ r--.Q:i §B ~ ,.... , . - ~ O ~ [~ <>.~ QQ UU >0 ='" ~ ~ 8"~ ~~ g$ o"ffi 'EQ ~ ~ ~ ~ ~8 -'25 .5 2 g~ 1l;; ~~ ~~ 'i~ ~~ ~g :/J = §~ u e." (5.1.4) (5.1.3) (5.1.2) (5.1.1) = - l ± H 2 x 2 + x + 1 = O. -b ± ,jb'1 - 4ac x ± = 2a to find the roots x ± = 2,3. Suppose instead we consider ax 2 + bx +c = 0 namely ~t. :>;'t f. :si)' ,} ~: ~. f)I" '. ~.;
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90 Chapter5 Complex Numbers 91 and refer to x and v as its real and imaginary parts and denote them by the symbols Re z and 1m z: A number withjust Y '# 0 is called a pure imaginary number. The solution to our quadratic equation has a real part x = -! and an imaginary part 'J :l.J 'J) -"~ h ~~ o~ §8 H ~0 - ~ ti (1) ~S ~ ~;; ~ ~ ~~ :5;: "8 "'- ~ >[email protected] f.... 4 . . ::i (1) §~ r"'I __ ~ O ~ [~ "'~ "" uu >-0 §C"'J 8'~ :"3 =Ol gf;i c-'- o ~ 'EQ gg a ~ .g8 ::2~ s~ ;s.Q 'g'5 ~~ (5.2.11) (5.2.8) (5.2.9) (5.2.10) (5.2.4) (5.2.5) rule (5.2.6) multiplication rule (5.2.7) Z2 = x 2 + iY2 implies X2 Y2 · = x - iy Xl + iYl X2 + iY2, we define. (Xl + X2) + i(Yl + Y2) addition (X1X2 - Y1Y2) + i(X1Y2 + X2Ytl Zl = + Yl Z2 Zl + Z2 ZlZ2 Suppose this were not true. This would imply Xl - X2 = i(Y2 - Yl). without both of them vanishing separately. Squaring both sides. we would find a positive definite left-hand side and a negative definite right-hand side. The only way to avoid a contradiction is for both sides to vanish, giving us 0 = -0, which is something we can live with. . Now, given any real number z, we can associate with it a unique number -x, called its negative. We can do that with a complex number Z = x + too, by negating x and y.
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Complex Numbers - 11 Chapter d t J r jlr 5 COMPLEX NUMBERS...

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