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Exponential Form of a Complex Number
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In this section we introduce the exponential ( expo NEN shal ) form of a complex number. We do this because certain operations can be greatly simpli ed if the complex numbers are written in this form. It is necessary, however, in order to give a proper explanation of the ORIGIN of the exponential form, that we make use of certain procedures from more advanced mathematics. In our explanations we ll state the facts as clearly as we can, so that you can understand, in a good general way, the origin of the exponential form. It is important that we do this, because it will give you added con dence in handling complex numbers in this form, which, as you ll be pleased to nd, is not at all hard to do. Let us proceed as follows. The ratio of the circumference of a circle to the diameter is certainly one of the best known, and most used, numbers in mathematics. It is a constant ratio, universally represented by the Greek letter  (pi), being an irrational number* having the approximate value  3:141 592 65: : : . Another number, equal in importance to , also exists. This number is denoted by the Greek letter  (epsilon) and, like , is an irrational number, having the approximate value  2:718 281 828 459: : : . The number represented by  arises in the study of the logarithmic function and, for a very speci c reason, is de ned to be equal to   1 n 2:718 28 : : : 146  lim 1 n!1 n Using your calculator, you can verify, for example, the following approximate values of  (rounded o to ve decimal places): n 10 n 100 n 1000 n 10 000  1:1 10  1:01 100  1:001 1000 2:593 74 2:704 81 2:716 92
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 1:0001 10 000 2:718 15
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Thus, as n becomes in nitely great  does not become in nitely great, but becomes an in nite string of non-repeating decimals, having a limiting value of something less than 2.72. As mentioned above, the number  arises in the study of the logarithmic function. In regard to logarithms, any system of logarithms has a base number, which let us denote by b. Now let y be any positive number; the logarithm of y is then de ned to be the POWER that the base number b must be raised to, to equal y. That is, the statement that 147 logb y x
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* See note 11 in Appendix.
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CHAPTER 6 Algebra of Complex Numbers
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y bx 148
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(where logb y is read as the logarithm of y to the base b ). The reason the foregoing is important is that the expression in eq. (146) appears at a critical point in the development of the logarithmic and exponential functions in the calculus. It is found that the formulas of the calculus, in regard to logarithmic and exponential work, are much simplifed if  is used as the base number. Hence, in advanced mathematics, it is always understood that  is the logarithmic base number; thus, for this case eq. (148) becomes y x 149
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which is the fundamental form of the exponential function as it appears in advanced mathematics. Next, eq. (149) can be written in the equivalent form of a power series in x;* thus x 1 x x2 x3 x4 x5 xn 2! 3! 4! 5! n! 150
in which the exclamation mark ! denotes the PRODUCT of all the positive integers from 1 to n inclusive; thus n! 1 2 3 4 n hence, 2! 1 2 2, 3! 1 2 3 6, 4! 1 2 3 4 24, and so on. The symbol ! can be read as factorial ; thus we have 2 factorial, 3 factorial, and so on. In the study of power series it is shown that the series form of eq. (150) is a valid representation of x for all positive and negative values of the variable x. Next, the sine and cosine functions can also be expressed in the power series form. Thus the following relationships are valid for all positive and negative values of x: sin x x cos x 1 x3 x5 x7 3! 5! 7! x2 x4 x6 2! 4! 6! 151 152
which, it should be noted, are valid in the above form only if angle x is measured in radians. (The series become awkward to write for x in degrees.) These are said to be alternating series, because the terms are alternately positive and negative in sign, as you can see. Now let us return to eq. (150) and boldly take the step of replacing x with jx. Assuming this is permissible (which it is), and upon taking careful note of the values of the powers of j from section 6.1, you should nd that eq. (150) becomes ! ! x2 x4 x6 x3 x5 x7 jx  1 j x 2! 4! 6! 3! 5! 7! Now compare the last equation with eqs. (151) and (152); doing this, we see that  jx cos x j sin x