Exercise 11.  (Indirectly Conformal Mapping)  Let     be analytic at    and  .  
Show that the function    preserves the magnitude, but reverses the sense, of angles at  .  

Solution 11.

See text and/or instructor's solution manual.

Solution.  First show that the mapping    preserves the magnitude, but reverses the sense, of angles at  .

For a smooth curve that passes through the point  ,  we use the notation  

            ,   for  .   
    
A vector tangent to C at the point is given by  

            ,  
            
where the complex number    is expressed as a vector.

The angle of inclination of    with respect to the positive x axis is  

            .  

The image of C under the mapping    is the curve  K  in the w plane given by the formula

            .  

A vector    tangent to K at the point    is given by  

            .  

The angle of inclination of with respect to the positive u axis is

            .    

Therefore, the mapping    preserves the magnitude, but reverses the sense, of angles at  .

        Now consider the the mapping    as a composition  .  

Since    is analytic at  ,  the mapping    preserves angles at  .

The image of    under       is  .

The mapping    preserves the magnitude, but reverses the sense, of angles at  .

Therefore, the mapping    preserves the magnitude, but reverses the sense, of angles at  .

Remarks.  If    is analytic at    and    then     is said to be Indirectly Conformal at  .   

An Indirectly Conformal Mapping is sometimes referred to as an anti-conformal mapping.

We are done.

Aside.  For illustration purposes we can revisit Exercises 6 and 7 where we looked at     and  .

The mappings    and    will preserves the magnitude, but reverses the sense, of angles.

                                   At       preserves angles.

                                   At       preserves the magnitude, but reverses the sense of angles.  

                                   At       preserves angles.

                                   At       preserves the magnitude, but reverses the sense of angles.  

This solution is complements of the authors.

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