Exercise 5.  Pick an appropriate branch of the logarithm (see Equation (5-20)), find  ,  and state where the formula is valid for  

5 (e).  .  

Solution 5 (e).

See text and/or instructor's solution manual.

Answer.     in the domain set   .  

The branch cuts for     are difficult to construct (portions of hyperbolas, or rays).  

We are done.   

Alternate solution.  For a specific example, we can choose  ,  then

the function    is defined for  ,  and

  analytic, and    for z in the domain  .

Remark.  The branch cuts for     are   ,

which is a ray starting at the branch point    and subtending the angle   with the horizontal line through  ,

together with a ray starting at the branch point    and subtending the angle   with the horizontal line through  .  

Remark.  The function maps the interval   onto the interval .

The function    maps the branch cuts    onto the ray

  which is the branch cut for  .  

We illustrate    by showing the image of the right half-plane    and left half-plane  ,  respectively.

                                                            The mappings    and    for the composition  .
                                                            The image of   is the Z-plane slit along the ray  .
                                                            The image of   is the w-plane slit along the ray  .

                                                            The mappings    and    for the composition  .
                                                            The image of   is the Z-plane slit along the ray  .
                                                            The image of   is the w-plane slit along the ray  .

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell