Exercise 14.  For the temperature function    in the upper half-plane  ,  

show that the isothermals    are portions of hyperbolas that have foci at the points    and  ,  as illustrated in Figure 11.34.

Solution 14.

See text and/or instructor's solution manual.

Solution.   In equations (10-26) and (10-27) in Section 10.4 we found the real and imaginary parts of  ,  respectively. Thus

                    .  
Hence, the temperature function is

                    .
        
The isothermal    can be written as  

                    .  

The quantity on the right side of the above equation is the sum of the distances from   to   and from   to .

Therefore, the equation      is a hyperbola that has foci at the points and .

We are done.   

                    A contour graph of the function   ,

                    where     for   .
                    
                    The isothermals    are portions of hyperbolas that have foci at the points    and  .

We are really done.   

                    A graph of the function   .

                    A graph of the function   .

This solution is complements of the authors.

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