Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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 Complex Integrals

We now turn our attention to the problem of integrating complex functions.  We will find that integrals of analytic functions are well behaved and that many properties from calculus carry over to the complex case.  

    We introduce the integral of a complex function by defining the integral of a complex-valued function of a real variable

 

Definition 

6.1 (Definite Integral of a Complex Integrand).  Let  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  where u(t) and v(t) are real-valued functions of the real variable t for  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev .  Then

   Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.        (6-1)

 

    We generally evaluate integrals of this type by finding the antiderivatives of u(t) and v(t) and evaluating the definite integrals on the right side of Equation (6-1).  That is, if    Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev,  we have  

  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.        (6-2)

 

Example 6.1.  Show that  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.  

Solution.  We write the integrand in terms of its real and imaginary parts, i.e.,   Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  and  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.  The integrals of u(t) and v(t) are  

Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.  

Hence, by Definition (6-1),  

   Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 

Example 6.2.  Show that  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.  

Solution :  We use the method suggested by Definitions (6-1) and (6-2).  

   Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  

We can evaluate each of the integrals via integration by parts.  For example,  

   Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev
Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev   

Adding  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  to both sides of this equation and then dividing by 2 gives Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.  Therefore,  

   Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.

Complex integrals have properties that are similar to those of real integrals.  We now trace through several commonalities.  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  and  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.

Using Definition (6-1), we can easily show that the integral of their sum is the sum of their integrals, that is

   .  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev         (6-3)

If we divide the interval  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  and integrate f(t) over these subintervals by using (6-1), then we get  

   Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.                           (6-4)

Similarly, if  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  denotes a complex constant, then

  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.                  (6-5)

If the limits of integration are reversed, then

    Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.                          (6-6)

The integral of the product f(t)g(t) becomes

   Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev                 (6-7) 

Example 6.3.  Let us verify property (6-5).  We start by writing

   Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  

Using Definition (6-1), we write the left side of Equation (6-5) as  

   Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  

which is equivalent to  

   Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev   

Therefore,  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.  


    It is worthwhile to point out the similarity between equation (6-2) and its counterpart in calculus.  Suppose that U and V are differentiable on  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  
and Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev,  equation (6-2) takes on the familiar form

 Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.               (6-8)   
   
where Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.  We can view Equation (6-8) as an extension of the fundamental theorem of calculus.  In Section 6.4 we show how to generalize this extension to analytic functions of a complex variable.  For now, we simply note an important case of Equation (6-8):

 Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.                 (6-9)    

Example 6.4.  Use Equation (6-8) to show that  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.  

Solution.  We seek a function F with the property that   Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.  We note that  Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev satisfies this requirement, so

   Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  

which is the same result we obtained in Example 6.2, but with a lot less work.

 

Remark 6.1  Example 6.4 illustrates the potential computational advantage we have when we lift our sights to the complex domain.  Using ordinary calculus techniques to evaluate Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev, for example, required a lengthy integration by parts procedure (Example 6.2).  When we recognize this expression as the real part of Evaluation of Integrals - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev, however, the solution comes quickly.  This is just one of the many reasons why good physicists and engineers, in addition to mathematicians, benefit from a thorough working knowledge of complex analysis.

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