6.5  Integral Representations for Analytic Functions

    We now present some major results in the theory of functions of a complex variable.  The first result is known as Cauchy's integral formula and shows that the value of an analytic function f(z) can be represented by a certain contour integral.  The    derivative,  ,  will have a similar representation.  In Section 7.2, we use the Cauchy integral formulas to prove Taylor's theorem and also establish the power series representation for analytic functions. The Cauchy integral formulas are a convenient tool for evaluating certain contour integrals.

Theorem 6.10 (Cauchy Integral Formula).  Let f(z) be analytic in the simply connected domain D,  and let C be a simple closed positively oriented contour that lies in D.  If    is a point that lies interior to C, then   

            .   

Proof.

Proof of Theorem 6.10 is in the book.
Complex Analysis for Mathematics and Engineering

Example 6.21.  Show that  ,  where C is the circle    with positive orientation.

Solution.  We have    and  .  The point    lies interior to the circle, so Cauchy's integral formula implies that  

            ,  

and multiplication by    establishes the desired result.

Explore Solution 6.21.

Example 6.22.  Show that  ,  where C is the circle    with positive orientation.

Solution.  Here we have  .  We manipulate the integral and use Cauchy's integral formula to obtain  

              

Explore Solution 6.22.

Example 6.23.  Show that  ,  where C is the circle    with positive orientation.

Solution.  We see that  .  The only zero of this expression that lies in the interior of C is .  
We set    and use Theorem 6.10 to conclude that  

               

Explore Solution 6.23.

Theorem 6.11 (Leibniz's Rule).  Let G be an open set,  and let    be an interval of real numbers.  Let and its partial derivative with respect to z be continuous functions for all z in G and all t in I.  Then  

               is analytic for z in G, and  

            .  

Proof.

Demonstration for Theorem 6.11.

    We now generalize Theorem 6.10 to give an integral representation for the derivative, . We use Leibniz's rule in the proof and note that this method of proof is a mnemonic device for remembering Theorem 6.12.

Theorem 6.12 (Cauchy's Integral Formulae for Derivatives).  Let be analytic in the simply connected domain D, and let C be a simple closed positively oriented contour that lies in D.  If z is a point that lies interior to C, then for any integer , we have

            .  

Proof.

 Proof of Theorem 6.12 is in the book.
Complex Analysis for Mathematics and Engineering

Example 6.24.  Let    denote a fixed complex value.  Show that, if C is a simple closed positively oriented contour such that    lies interior to C, then  

               ,   and
(6-50)
               ,   for any integer  . 

Solution.  We let .  Then for .  Theorem 6.10 implies that the value of the first integral in Equations (6-50) is

            ,  

and Theorem 6.12 further implies that  

            .  

This result is the same as that proven earlier in Corollary 6.1.  Obviously, though, the technique of using Theorems 6.10 and 6.12 is easier.

Explore Solution 6.24 (a).

Explore Solution 6.24 (b).

Example 6.25.  Show that,  where C is the circle    with positive orientation.

Solution.  If we set  ,  then a straightforward calculation shows that  .   Using Cauchy's integral formulas with , we conclude that  

              

Explore Solution 6.25.

    We now state two important corollaries of Theorem 6.12.

Corollary 6.2.  If is analytic in the domain D, then all derivatives     exists for    (and therefore are analytic in D).

Proof.

Proof of Corollary 6.2 is in the book.
Complex Analysis for Mathematics and Engineering

Remark 6.3.  This result is interesting, as it illustrates a big difference between real and complex functions.  A real function can have the property that exists everywhere in a domain D, but exists nowhere.  Corollary 6.2 states that if a complex function has the property that exists everywhere in a domain D, then, remarkably, all derivatives of exist in D.  

Corollary 6.3.  If is a harmonic function at each point in the domain D, then all partial derivatives  , , , ,   exists and are harmonic functions.  

Proof.

Proof of Corollary 6.3 is in the book.
Complex Analysis for Mathematics and Engineering

Extra Example 1.  Show that the partial derivatives of   are harmonic functions.  

Explore Extra Solution 1.

Exercises for Section 6.5.  Integral Representations for Analytic Functions