Exercise 1.  Use Equations (6-1) and (6-2) to find  

1 (e).  .

Solution 1 (e).

See text and/or instructor's solution manual.

Answer  .  

Solution.   Given the real functions  u(t)  and  v(t) with    and  ,  we use the formula   

                    .  

First, expand the integrand  into it's real and imaginary parts

                    .

Here we have   

                    ,  and

                    .

Now integrate  u(t)  and  v(t)  and obtain  

                    ,  and

                    .

Compute values for  U(t)  and  V(t)  

                      

Compute the real definite integrals

                    


                    

Evaluate the complex definite integral

                      

Some points    in the interval of integration and their images ,  for  .

The right endpoint is and the last image point is .

We are done.   

Aside.  We can let Mathematica double check our work.

The details for this computation are:

Some points    in the interval of integration and their images ,  for  .

The right endpoint is and the last image point is .

We are done.   

Aside. After we have developed the topics of contour integrals (in Section 6.2), the independence of path for integration of an analytic function (in Section 6.3),

and established the Fundamental Theorem of Calculus (in Section 6.4) then we will be able to revisit this integral and use the more straightforward computation:

Remark.   It is important to understand whether you are permitted to use only real variables in computing an integral,

or whether you are permitted to use a complex variable () and complex functions in computing the integral.  

The three popular software packages Maple, Matlab and Mathematica use complex variable based computations.

This solution is complements of the authors.

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