Module

for

Theorems of Morera and Liouville

The Maximum Modulus Principle

and Extensions

 

6.6 The Theorems of Morera and Liouville and Extensions

    In this section we investigate some of the qualitative properties of analytic and harmonic functions.  Our first result shows that the existence of an antiderivative for a continuous function is equivalent to the statement that the integral of f(z) is independent of the path of integration.  This result stated in a form that will serve as a converse to the Cauchy-Goursat theorem.

 

Theorem 6.13 (Morera's Theorem).  Let f(z) be a continuous function in a simply connected domain D.  If  [Graphics:Images/LiouvilleMoreraGaussMod_gr_1.gif]  for every closed contour in D, then f(z) is analytic in D.

Proof.

Proof of Theorem 6.13 is in the book.
Complex Analysis for Mathematics and Engineering

 

    Cauchy's integral formula show how the value [Graphics:Images/LiouvilleMoreraGaussMod_gr_2.gif] can be represented by a certain contour integral.  If we choose the contour of integration C to be a circle with center  [Graphics:Images/LiouvilleMoreraGaussMod_gr_3.gif],  then we can show that the value [Graphics:Images/LiouvilleMoreraGaussMod_gr_4.gif] is the integral average of the values of f(z) at points  z  on the circle C.

 

Theorem 6.14 (Gauss's Mean Value Theorem).  If f(z) is analytic in a simply connected domain  D  that contains the circle  [Graphics:Images/LiouvilleMoreraGaussMod_gr_5.gif],  then  

            [Graphics:Images/LiouvilleMoreraGaussMod_gr_6.gif].  

Proof.

Proof of Theorem 6.14 is in the book.
Complex Analysis for Mathematics and Engineering

 

    We now prove an important result concerning the modulus of an analytic function.

 

Theorem 6.15 (Maximum Modulus Principle).  Let f(z) be analytic and nonconstant in the bounded domain D.  Then  [Graphics:Images/LiouvilleMoreraGaussMod_gr_7.gif]  does not attain a maximum value at any point [Graphics:Images/LiouvilleMoreraGaussMod_gr_8.gif] in D.

Proof.

Proof of Theorem 6.15 is in the book.
Complex Analysis for Mathematics and Engineering

 

    We sometimes state the maximum modulus principle in the following form.

 

Theorem 6.16 (Maximum Modulus Principle).  Let f(z) be analytic and nonconstant in the bounded domain D.  If  f(z) is continuous on the closed region R that consists of D and all of its boundary points B, then [Graphics:Images/LiouvilleMoreraGaussMod_gr_9.gif] assumes its maximum value, and does so only at point(s) [Graphics:Images/LiouvilleMoreraGaussMod_gr_10.gif] on the boundary B.

Proof.

Proof of Theorem 6.16 is in the book.
Complex Analysis for Mathematics and Engineering

 

Example 6.26.  Let  [Graphics:Images/LiouvilleMoreraGaussMod_gr_11.gif].  If we set our domain D to be  [Graphics:Images/LiouvilleMoreraGaussMod_gr_12.gif],  then f(z) is continuous on the closed region  [Graphics:Images/LiouvilleMoreraGaussMod_gr_13.gif].  Prove that  

            [Graphics:Images/LiouvilleMoreraGaussMod_gr_14.gif],  

and this value is assumed by f(z) at a point  [Graphics:Images/LiouvilleMoreraGaussMod_gr_15.gif]  on the boundary of D.  

[Graphics:Images/LiouvilleMoreraGaussMod_gr_16.gif][Graphics:Images/LiouvilleMoreraGaussMod_gr_17.gif]

Solution.  From the triangle inequality and the fact that  [Graphics:Images/LiouvilleMoreraGaussMod_gr_18.gif] in D, it follows that  

(6-58)            [Graphics:Images/LiouvilleMoreraGaussMod_gr_19.gif].

If we choose  [Graphics:Images/LiouvilleMoreraGaussMod_gr_20.gif],  where  [Graphics:Images/LiouvilleMoreraGaussMod_gr_21.gif],  then  

            [Graphics:Images/LiouvilleMoreraGaussMod_gr_22.gif]  

so the vectors [Graphics:Images/LiouvilleMoreraGaussMod_gr_23.gif] and [Graphics:Images/LiouvilleMoreraGaussMod_gr_24.gif] lie on the same ray through the origin.  This is the requirement for the Inequality (6-58) to be an equality (see Exercise 19 in Section 1.3).  
Hence  [Graphics:Images/LiouvilleMoreraGaussMod_gr_25.gif],  and the result is established.

Explore Solution 6.26.

 

Extra Example 1.  Let  [Graphics:Images/LiouvilleMoreraGaussMod_gr_54.gif].  If we set our domain D to be  [Graphics:Images/LiouvilleMoreraGaussMod_gr_55.gif],  then f(z) is continuous on the closed region  [Graphics:Images/LiouvilleMoreraGaussMod_gr_56.gif].  Show that  [Graphics:Images/LiouvilleMoreraGaussMod_gr_57.gif],  and this value is assumed by [Graphics:Images/LiouvilleMoreraGaussMod_gr_58.gif] at a point  [Graphics:Images/LiouvilleMoreraGaussMod_gr_59.gif]  on the boundary of D.          

[Graphics:Images/LiouvilleMoreraGaussMod_gr_60.gif] [Graphics:Images/LiouvilleMoreraGaussMod_gr_61.gif]

Explore Solution for Extra Example 1.

 

Extra Example 2.  Let  [Graphics:Images/LiouvilleMoreraGaussMod_gr_71.gif].  If we set our domain D to be  [Graphics:Images/LiouvilleMoreraGaussMod_gr_72.gif],  then f(z) is continuous on the closed region  [Graphics:Images/LiouvilleMoreraGaussMod_gr_73.gif].  Show that  [Graphics:Images/LiouvilleMoreraGaussMod_gr_74.gif],  and this value is assumed by [Graphics:Images/LiouvilleMoreraGaussMod_gr_75.gif] at a point  [Graphics:Images/LiouvilleMoreraGaussMod_gr_76.gif]  on the boundary of D.  

Explore Solution for Extra Example 2.

 

Theorem 6.17 (Cauchy's Inequalities).  Let f(z) be analytic in the simply connected domain D that contains the circle  [Graphics:Images/LiouvilleMoreraGaussMod_gr_86.gif].  If  [Graphics:Images/LiouvilleMoreraGaussMod_gr_87.gif]  holds for all points [Graphics:Images/LiouvilleMoreraGaussMod_gr_88.gif],  then  

            [Graphics:Images/LiouvilleMoreraGaussMod_gr_89.gif]    for    [Graphics:Images/LiouvilleMoreraGaussMod_gr_90.gif].  

Proof.

Proof of Theorem 6.17 is in the book.
Complex Analysis for Mathematics and Engineering

 

    Theorem 6.18 shows that a nonconstant entire function cannot be a bounded function.  

 

Theorem 6.18 (Liouville's Theorem).  If f(z) is an entire function and is bounded for all values of z in the complex plane, then f(z) is constant.

Proof.

Proof of Theorem 6.18 is in the book.
Complex Analysis for Mathematics and Engineering

 

Example 6.27.  Show that the function   sin(z)   is not a bounded function.

Solution.  We established this characteristic with a somewhat tedious argument in Section 5.4.  All we need do now is observe that f(z) is not constant, and hence it is not bounded.         

[Graphics:Images/LiouvilleMoreraGaussMod_gr_91.gif] [Graphics:Images/LiouvilleMoreraGaussMod_gr_92.gif]

Explore Solution 6.27.

 

    We can use Liouville's theorem to establish an important theorem of algebra.  

 

Theorem 6.19 (Fundamental Theorem of Algebra).  If P(z) is a polynomial of degree [Graphics:Images/LiouvilleMoreraGaussMod_gr_100.gif], then P(z) has at least one zero.

Proof.

Proof of Theorem 6.19 is in the book.
Complex Analysis for Mathematics and Engineering

 

Corollary 6.4.  Let P(z) be a polynomial of degree [Graphics:Images/LiouvilleMoreraGaussMod_gr_101.gif].  Then P(z) can be expressed as the product of linear factors.  That is,  

            [Graphics:Images/LiouvilleMoreraGaussMod_gr_102.gif]  

where  [Graphics:Images/LiouvilleMoreraGaussMod_gr_103.gif]  are the zeros of P(z) counted according to multiplicity an A is a constant.  

Proof.

 

Extra Example 3.  Find the n zeros of the equation  [Graphics:Images/LiouvilleMoreraGaussMod_gr_104.gif].

Explore Solution for Extra Example 3.

 

Extra Example 4.  Find the n zeros of the equation  [Graphics:Images/LiouvilleMoreraGaussMod_gr_143.gif].

Explore Solution for Extra Example 4.

 

Extra Example 5.  Find the roots of the [Graphics:Images/LiouvilleMoreraGaussMod_gr_179.gif] Chebyshev polynomial.

Explore Solution for Extra Example 5.

 

Exercises for Section 6.6.  The Theorems of Morera and Liouville and Extensions

 

Library Research Experience for Undergraduates

Morera's Theorem  

Maximum Modulus Principle  

Liouville's Theorem  

Fundamental Theorem of Algebra

Schwarz Lemma  

 

 

 

 The Next Module is

The Fundamental Theorem of Algebra

 

 

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This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2012 John H. Mathews, Russell W. Howell