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Chapter 2: Q18P (page 76)
Show that tanh z never takes the values.
It has been proved thattanh z never takes the values.
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Express the following complex numbers in the form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others—try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.
17.
For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly findin your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.
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Use Problems 27 and 28 to find the following absolute values. If you understand Problems 27 and 28 and equation (5.1), you should be able to do these in your head.
Find the power series for and for from the series for in the following way: Write the series for ; put . Show that ; take real and imaginary parts of the equation, and put .
Findand as functions of for the example above, and verify for this case that are correctly given by the method of the example.
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