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ch2 - Chapter Two Complex Functions 2.1 Functions of a real...

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Chapter Two Complex Functions 2.1. Functions of a real variable. A function : I C from a set I of reals into the complex numbers C is actually a familiar concept from elementary calculus. It is simply a function from a subset of the reals into the plane, what we sometimes call a vector-valued function. Assuming the function is nice, it provides a vector, or parametric, description of a curve. Thus, the set of all t : t e it cos t i sin t cos t ,sin t ,0 t 2 is the circle of radius one, centered at the origin. We also already know about the derivatives of such functions. If t x t iy t , then the derivative of is simply t x t iy t , interpreted as a vector in the plane, it is tangent to the curve described by at the point t . Example. Let t t it 2 , 1 t 1. One easily sees that this function describes that part of the curve y x 2 between x 1 and x 1: 0 1 -1 -0.5 0.5 1 x Another example. Suppose there is a body of mass M ”fixed” at the origin–perhaps the sun–and there is a body of mass m which is free to move–perhaps a planet. Let the location of this second body at time t be given by the complex-valued function z t . We assume the only force on this mass is the gravitational force of the fixed body. This force f is thus f GMm | z t | 2 z t | z t | where G is the universal gravitational constant. Sir Isaac Newton tells us that mz  t f GMm | z t | 2 z t | z t | 2.1
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Hence, z  GM | z | 3 z Next, let’s write this in polar form, z re i : d 2 dt 2 re i k r 2 e i where we have written GM k . Now, let’s see what we have. d dt re i r d dt e i dr dt e i Now, d dt e i d dt cos i sin sin i cos d dt i cos i sin d dt i d dt e i . (Additional evidence that our notation e i cos i sin is reasonable.) Thus, d dt re i r d dt e i dr dt e i r i d dt e i dr dt e i dr dt ir d dt e i . Now, 2.2
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d 2 dt 2 re i d 2 r dt 2 i dr dt d dt ir d 2 dt 2 e i dr dt ir d dt i d dt e i d 2 r dt 2 r d dt 2 i r d 2 dt 2 2 dr dt d dt e i Now, the equation d 2 dt 2 re i k r 2 e i becomes d 2 r dt 2 r d dt 2 i r d 2 dt 2 2 dr dt d dt k r 2 .
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