ch3 - Chapter Three Elementary Functions 3.1 Introduction...

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Chapter Three Elementary Functions 3.1. Introduction. Complex functions are, of course, quite easy to come by—they are simply ordered pairs of real-valued functions of two variables. We have, however, already seen enough to realize that it is those complex functions that are differentiable that are the most interesting. It was important in our invention of the complex numbers that these new numbers in some sense included the old real numbers—in other words, we extended the reals. We shall find it most useful and profitable to do a similar thing with many of the familiar real functions. That is, we seek complex functions such that when restricted to the reals are familiar real functions. As we have seen, the extension of polynomials and rational functions to complex functions is easy; we simply change x ’s to z ’s. Thus, for instance, the function f defined by f z z 2 z 1 z 1 has a derivative at each point of its domain, and for z x 0 i , becomes a familiar real rational function f x x 2 x 1 x 1 . What happens with the trigonometric functions, exponentials, logarithms, etc ., is not so obvious. Let us begin. 3.2 . The exponential function . Let the so-called exponential function exp be defined by exp z e x cos y i sin y , where, as usual, z x iy . From the Cauchy-Riemann equations, we see at once that this function has a derivative every where—it is an entire function. Moreover, d dz exp z exp z . Note next that if z x iy and w u iv , then 3.1
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exp z w e x u cos y v i sin y v  e x e u cos y cos v sin y sin v i sin y cos v cos y sin v e x e u cos y i sin y  cos v i sin v exp z exp w . We thus use the quite reasonable notation e z exp z and observe that we have extended the real exponential e x to the complex numbers. Example Recall from elementary circuit analysis that the relation between the voltage drop V and the current flow I through a resistor is V RI ,where R is the resistance. For an inductor, the relation is V L dI dt L is the inductance; and for a capacitor, C dV dt I C is the capacitance. (The variable t is, of course, time.) Note that if V is sinusoidal with a frequency ,thensoalsois I . Suppose then that V A sin t . We can write this as V Im Ae i e i t Im Be i t B is complex. We know the current I will have this same form: I Im Ce i t . The relations between the voltage and the current are linear, and so we can consider complex voltages and currents and use the fact that e i t cos t i sin t . We thus assume a more or less fictional complex voltage
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This note was uploaded on 10/27/2009 for the course MBA 1918272 taught by Professor Peter during the Fall '09 term at Aberystwyth University.

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ch3 - Chapter Three Elementary Functions 3.1 Introduction...

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