Exercise 12.  Find the temperature function in the first quadrant  ,  

that satisfies the following boundary conditions (shown in Figure 11.32).  

                      

Solution 12.

See text and/or instructor's solution manual.

Answer.   .  

Solution.   Carefully look at Example 11.17 and discover that      has the boundary values

                          

Hence, if we use the rotation    we see that      has the boundary values

                      

Therefore the solution we seek is  

                         

Therefore,   

                    .

For computational purposes we can use the formulas for the real and imaginary parts of  ,  that were derived in Section 10.4.

                    .  

In particular,

(10-26)         .  

Now make the substitution      and  

                        and    .   

Thus,  

                       

Therefore,

                    .  

We are done.   

Aside.   We can graph the function   .   

                    A contour graph of the function   ,

                    where     for   .

                    A contour graph of the function   ,    

                      

We are really done.   

Aside.   We can check out some of the boundary values of   .   

This solution is complements of the authors.

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