Example 5.12.  Suppose that we make specific choices in equation (5-47) by selecting  [Graphics:Images/ComplexFunTrigInverseMod_gr_137.gif]  as the value of the square root  [Graphics:Images/ComplexFunTrigInverseMod_gr_138.gif]  and using the principal value of the logarithm.  With  [Graphics:Images/ComplexFunTrigInverseMod_gr_139.gif],  the result is  

            [Graphics:Images/ComplexFunTrigInverseMod_gr_140.gif],  

and the corresponding value of the derivative is given by

            [Graphics:Images/ComplexFunTrigInverseMod_gr_141.gif][Graphics:Images/ComplexFunTrigInverseMod_gr_142.gif].  

Explore Solution 5.12.

Mathematica can find the solution.

[Graphics:../Images/ComplexFunTrigInverseMod_gr_149.gif]






[Graphics:../Images/ComplexFunTrigInverseMod_gr_150.gif]
[Graphics:../Images/ComplexFunTrigInverseMod_gr_151.gif]
[Graphics:../Images/ComplexFunTrigInverseMod_gr_152.gif]
[Graphics:../Images/ComplexFunTrigInverseMod_gr_153.gif]
[Graphics:../Images/ComplexFunTrigInverseMod_gr_154.gif]
[Graphics:../Images/ComplexFunTrigInverseMod_gr_155.gif]
[Graphics:../Images/ComplexFunTrigInverseMod_gr_156.gif]
[Graphics:../Images/ComplexFunTrigInverseMod_gr_157.gif]
[Graphics:../Images/ComplexFunTrigInverseMod_gr_158.gif]
[Graphics:../Images/ComplexFunTrigInverseMod_gr_159.gif]
[Graphics:../Images/ComplexFunTrigInverseMod_gr_160.gif]