Assignment 5 Solutions

Page 66 #1 (6th edition: page 67 #1)

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Page 66 #3 (6th edition: page 68 #8)

First, some clarification: "ln" is the notation we use for the natural logarithm function of a positive real variable. It makes no sense to speak of "ln w" when w is a complex number (nor does it make sense when w is a negative real number, for that matter).

For complex variables, we let "log w" denote the set of all complex numbers z for which exp(z)=w. This is a multiple-valued function, unless we pick a particular branch.

These questions, however, come before we have even introduced the complex logarithm. You are simply asked to solve them using the definition of exp. It is wrong, for example, to try to tackle (a) by writing "z = ln(-2)"; the function ln is not defined at -2. And even if it were, it would only be giving you a single real-valued solution, not the set of all complex solutions as the question asked.

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