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

5 (a).  .

Solution 5 (a).

See text and/or instructor's solution manual.

Solution.  We require    because    is never defined.   

Observe that  .

Choose  ,  and use    with  .  With this restriction we have  .

Then we have      where     and  .  

Remark.  In the above description for      the domain set  is   .   

Notice that    is not analytic on the ray   .

which starts at the point    and subtending the angle   with the horizontal line through  .

          The mapping      and the image of the domain set  .

We are done.   

Remark.  You may feel comfortable with the usual polar coordinate form  ,  but if we translate by the amount    then we can extend the domain of definition by a "wee bit more", as shown below.

Alternate solution.  For example, we can choose  ,  then the function    is defined for  ,

where  ,   ,      and    .   

Notice that r is chosen in a different manner than in the above "Answer".

The function    analytic, and    where    and  .

Remark.  The branch cut for     is     

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

Notice that    is not analytic on it's branch cut   .

We are done.   

Aside.  We can let Mathematica double check our work.

                              The mapping      and it's branch cut   .

We are really done.   

And another solution.  For another example, we could choose  ,  then the function    is defined for  ,

where  ,   ,      and    .  

The function    analytic, and    where    and  .

Remark.  The branch cut for     is     

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

Notice that    is not analytic on it's branch cut   .

                              The mapping      and it's branch cut   .

We are really really done.   

Yet another solution.  For another example, we could choose  ,  then the function    is defined for  ,

where  ,   ,      and    .  

The function    analytic, and    where    and  .

Remark.  The branch cut for     is     

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

Notice that    is not analytic on it's branch cut   .

                                   The mapping      and it's branch cut   .

We are really really really done.   

And the standard solution.  For another example, we could choose  ,  then the function    

is defined for  ,  where  ,   ,      and    .  

The function    analytic, and    where    and  .

Remark.  The branch cut for     is     

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

Notice that    is not analytic on it's branch cut   .

                         The mapping      and it's branch cut   .

This solution is complements of the authors.

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