Example 5.10.  Find all the values of  z  for which  .

Solution.  Starting with Identity (5-35), we write

            .  

If we equate real and imaginary parts, then we get

               and   .  

The equation    implies either that  , where n is an integer, or that .  Using in the equation   leads to the impossible situation  .  Therefore  , where n is an integer.  Since    for all values of y, the term in the equation    must also be positive.  For this reason we eliminate the odd values of n and get  , where k is an integer.

    Finally, we solve the equation    and use the fact that is an even function to conclude that  .  Therefore the solutions to the equation    are  ,  where k is an integer.

Explore Solution 5.10.

First, use Mathematica's "Solve" procedure to find some of the solutions to  cos z = cosh 2.  

This is another way to solve the equation  cos(z) = cosh(2).

We can also list some of the solutions.

(c) 2006 John H. Mathews, Russell W. Howell