Section 3.3  Harmonic Functions and their Riemann sheets

        Let    be a continuous real-valued function of the two real variables    that is defined on a domain  .  

(Recall from Section 1.6 that a domain    is an connected and open set of points in the complex plane.)  

        The partial differential equation

(3-26)            ,  

is known as Laplace's equation and is sometimes referred to as the potential equation.  

If      are all continuous,

and if    satisfies Laplace's equation, then    is called a harmonic function.  

        In calculus we might have been asked to show that polynomial functions like  

                         and     ,  

and transcendental functions like  

                         and     ,  

and  

                         and     ,

are all harmonic functions.  These pairs of functions are not chosen at random, and there is an intimate relationship between them,

they are called the conjugate "harmonic functions."  It is our goal to understand how this concept is tied in with analytic functions.

        On the practical side, harmonic functions are important in the areas of applied mathematics, engineering, and mathematical physics.  

Harmonic functions are used to solve problems involving steady state temperatures, two-dimensional electrostatics, and ideal fluid flow.  

In Section 11.2 we will show how complex analysis techniques are used to solve these problems.   For example, the function  

                    ,  
                    
is harmonic in the upper half plane and takes on the boundary values   

                       and    .  

                                             The harmonic function   .  
Exploration
 

        We begin with an important theorem relating analytic and harmonic functions.

Theorem 3.8.  Let      be an analytic function on a domain  .   Then both  

  and    are harmonic functions on  .  In other words, the real and imaginary parts of an analytic function are harmonic.

Proof.  Since    is differentiable on  ,  the Cauchy-Riemann equations (Theorem 3.3 in Section 3.2) imply that

                         and     ,  

and that  

                    .  

In Corollary 6.2, (see Section 6.5), we will prove that if    is analytic on  ,  then     is also analytic on  .  

Since    is differentiable on  ,  the Cauchy-Riemann equations imply that the all the second partial derivatives:  

                         and     ,  

exist are and are continuous  on  .  

        Using these facts, we can start with the above mentioned Cauchy Riemann equations

and take the partial derivative with respect to of each side of these equations and obtain  

                         and     .  

Similarly, taking the partial derivative of each side with respect to yields  

                         and     .  

Since the partial derivatives    are all continuous,

we use a theorem from the calculus of real functions that states that the mixed partial derivatives are equal; that is,  

                         and     .  

Combining all these results finally gives

                    ,  
and
                    .  

Therefore both    and    are harmonic functions on  .  

Proof.

Definition (Harmonic Conjugate).  If we have a function    that is harmonic on the domain    and if we can find another

harmonic function    such that the partial derivatives for    and    satisfy the Cauchy-Riemann equations

throughout  ,  then we say that      is a harmonic conjugate of   .  

Furthermore, it then follows that the function      is analytic on  .  

        Theorem 3.8 unlocks the relationship among harmonic functions, conjugate harmonic functions and analytic functions.  

Specifically, it clearly states the special relationship between a harmonic function and it's conjugate harmonic function.  

Loosely speaking, the harmonic function is the real part of the given analytic function and the harmonic conjugate function

is the imaginary part of the given analytic function.  This concept is illustrated in Examples 3.11 and 3.12 and the Extra Examples.

Example 3.11.  Show that      is a harmonic function and find a conjugate harmonic function   ,  

and an analytic function  .    

Solution.   Given   ,   we have      and the second partial

derivatives are   .   It follows that  

                    ,   

hence      is a harmonic function for all  .  

If we choose   ,   we have      and the second partial

derivatives are   .   It follows that  

                    ,   

hence      is a harmonic function for all  .  

Therefore, the harmonic conjugate of   

                    ,  

                    is  

                    .  

        Furthermore,    satisfy the Cauchy-Riemann equations  

                    ,   and   

                    .  

Therefore,      is an analytic function.  

Alternative Solution.  

The function      is analytic for all values of  .

Hence, it follows from Theorem 3.8 that both   

                    ,     and  

                    ,  

are harmonic functions.  

Explore Solution 3.11.

Example 3.12.  Show that      is a harmonic conjugate of   .  

Solution.   Given   ,   we have      

and the second partial derivatives are   .   It follows that     

                    ,   

hence      is a harmonic function for all  .  

Similarly, for   ,   we have      

and the second partial derivatives are   .   It follows that     

                    ,   

hence      is a harmonic function for all  .  

        Furthermore,    satisfy the Cauchy-Riemann equations  

                    ,     and   

                    .

Using Theorem 3.4, we see that      is an analytic function.

Therefore, the harmonic conjugate of   

                    ,  

                    is  

                    .  

Alternative Solution.  

The function      is analytic for all values of  .

Hence, it follows from Theorem 3.8 that both   

                    ,     and  

                    ,  

are harmonic functions.  

Therefore, the harmonic conjugate of   

                    ,  

                    is  

                    .  

Explore Solution 3.12.

Aside.  Figures 3.2 and 3.3 show the graphs of      and   .  

The partial derivatives of    are  

                         and     ,   

and the partial derivatives of    are

                         and     .  

        They satisfy the Cauchy-Riemann equations because they are the real and imaginary parts of an analytic function.  
        
At the point   ,   we have      and   ,   and these partial derivatives appear along

the edges of the surfaces for      at the points      and   ,   respectively.

Similarly,  at the point   ,  we have       and      and these partial derivatives appear

along the edges of the surfaces for      at the points      and   ,   respectively.

                    Figure 3.2 a   .                                           Figure 3.3 a   .

                    Figure 3.2 b   ,                                           Figure 3.3 b   ,  

                    at we have .                                             at we have  .  

                    Figure 3.2 c   ,                                           Figure 3.3 c   ,  

                    at we have .                                           at we have  .  

                                                                                       Figures 3.2 and 3.3

                                        For the function      we see that

                                             and     .  

A question about the harmonic conjugate.

        If    is the harmonic conjugate of   ,   then is      is the harmonic conjugate of   ?

The following example shows that this is not the case, and      is not the harmonic conjugate of   .

Extra Example 3.12.1.  Given the harmonic functions        and    ,  

and the analytic function    .  

3.12.1 (a)  Show that      is not an analytic function.      

Explore Extra Solution 3.12.1 (a).

3.12.1 (b)  Show that      is an analytic function, for all  .  

Explore Extra Solution 3.12.1 (b).

        We can use complex analysis to show easily that certain combinations of harmonic functions are harmonic.  For example, if      is

a harmonic conjugate of   ,   then their product      is a harmonic function.  This can be verified directly

by computing the partial derivatives and showing that Laplace's equation (3-26) holds, but the details are tedious.  If we use complex variable

techniques instead, we can start with the fact that    is an analytic function.  Then we observe that

the square of    is also an analytic function, which is  

                    ,    which can be written as  

                    .  

        We then know immediately that the imaginary part,   ,   is a harmonic function by Theorem 3.8.   

Since a constant multiple of a harmonic function is harmonic, it follows that      is harmonic.  It is left as an exercise to

show that if      and      are two harmonic functions that are not related in the preceding fashion, then their product

need not be harmonic.

Method I. Construction of the Harmonic Conjugate of u(x,y) using Integration.

        We now introduce methods for the construction of a harmonic conjugate function.  

The first method uses familiar techniques of calculus.

Theorem 3.9 (Construction of a Conjugate).  Let      be harmonic in an -neighborhood of the point  .  

Then there exists a conjugate harmonic function      defined in this neighborhood such that  

                    ,  

is an analytic function.

Proof.  A conjugate harmonic function    will satisfy the Cauchy-Riemann equations

                         and     .  

Assuming that such a function exists, we determine what it would have to look like by using a two-step process.  

First, we integrate      (which should equal  ) with respect to    and get

(3-27)              

where is a function of    alone that is yet to be determined.  Second, we compute    by differentiating

both sides of this equation with respect to    and replacing    with    on the left side, which gives  

                      

It can be shown (we leave the details for the reader) that because u is harmonic, all terms except those involving    in the last equation

will cancel, revealing a formula for    involving   alone.  Elementary integration of the single-variable function    can

then be used to discover  .  We finally observe that the function       so created indeed has the properties we seek.

        The functions      and      are computed with the formulas:

                    ,   
                    
                    and     
            
                    .  

Remark.  If you prefer a more succinct formula, then the harmonic conjugate of       is given by

                    .  

Proof.

        Technically we should always specify the domain of a function when we define it.  When no such specification is given, it is often

assumed that the domain is the entire complex plane, or the largest set for which the expression defining the function which makes sense.

Example 3.13.  Show that      is a harmonic function and find the harmonic conjugate   .  

Solution.  We follow the construction process of Theorem 3.9.  The first partial derivatives are

(3-28)                   and     .  

To verify that    is harmonic, we compute the second partial derivatives and note that  

                    ,  

so    satisfies Laplace's Equation (3-26).  

To construct  ,  we start with Equation (3-27) and the first of Equations (3-28) and

the Cauchy-Riemann equation      and get    

                    

We now need to differentiate the left and right sides of this equation with respect to  ,  

                    .  

Use Equation (3-28) and the Cauchy-Riemann equation      to obtain   

                    

It follows easily that

                    ,  

then an easy integration yields   ,   where is a real constant.  

For convenience, we can choose .

Therefore,   

                    .  

Explore Solution 3.13.

The "ghost of the imaginary numbers" - the subtle connection between Harmonic and Analytic Functions.

        When you look at a family of level curves of a real function  ,  do you naturally think of complex numbers ?  

Certainly, it it not the first thing that pops into our minds.  However, it seems to be subtle fact when studying complex analysis.  

        We cannot fail to stress the importance of the harmonic function pair that is constructed with Theorem 3.8 and Theorem 3.9.  

The orthogonal grid formed by the families of a harmonic functions and how complex functions are used to find them is one goal

of this book and is discussed in detail in Chapter 11.  In reality, they are constructed with inverse functions  .  

It will take a while to feel comfortable with these concepts and that is why they are studied later in the book.

For the time being do not worry about them, they are merely ghosts of the imaginary numbers.  

        For practical purposes, it suffices to consider regions in the -plane and their image in the -plane.  However, the concept of

a Riemann surface as being an "two dimensional manifold" has been around for a long time.  So it is no surprise that things get sticky.  

The reader can do research and see that work being done regarding harmonic functions on Riemann surfaces (and also on foliations).  

Applications of Harmonic Functions

    In Section 11.4 we will introduce the complex potential  ,  which is an analytic function and  ,  

are harmonic functions.  It has many physical interpretations, some of which are listed below.

                                                     Interpretations for the level curves of and .

        We do not have time to explore all of these applications at this time.  So we will introduce the topic of ideal fluid flow.

Ideal Fluid Flow

        We assume that an incompressible and frictionless fluid flows over the complex plane and that all cross sections in planes parallel to the

complex plane are the same.  Situations such as this occur when fluid is flowing in a deep channel.  The velocity vector at the point    is

(3-29)              .  

        The assumption that the flow is irrotational and has no sources or sinks implies that both the curl and divergence vanish, that is,

(3-30)                   and     .  

Hence    obey the partial differential equations

(3-30)              ,    

(3-30)              and

                        .  

Equations (3-30) are similar to the Cauchy-Riemann equations and permit us to define a special complex function:

(3-31)              .  

Here we have  

(3-30)              ,   and  

(3-30)              


We can use Equations (3-30) to verify that the Cauchy-Riemann equations are satisfied for  :  

                    ,   

                    and    

                    .  

Assuming the functions    have continuous partials, Theorem 3.4 guarantees that function    

defined in Equation (3-31) is analytic, and that the fluid flow of Equation (3-29) is the conjugate of an analytic function, that is,

                    .  

        In Section 6.4 we will prove that every analytic function    has an analytic antiderivative  ;  

assuming this to be the case, we can write  

(3-32)              ,  

(3-30)              where   

(3-30)              .  

        Theorem 3.8 tells us that    is a harmonic function.  If we use the vector interpretation of a complex number we see that

the gradient of    can be written as

                    .

        The Cauchy-Riemann equations applied to      give   ;   

making this substitution in the last equation yields  

                    .  

Equation (3-14) says that   ,   which by the preceding equation and Equation (3-32) imply that  

                    .  

Finally, from Equation (3-29),   is the scalar potential function for the a fluid flow, so

                    .  

Definition.  Given the complex potential  .

          The curves      are called equipotentials,

          and

          the curves      are called streamlines.  

They are used to describe the path of fluid flow.  

In Section 11.4 we will see that the family of equipotentials is orthogonal to the family of streamlines.

Example 3.14.  Show that the harmonic function      is the scalar potential function for the fluid flow  

                    .  

Solution.  We can write the fluid flow expression as

                    .   

Then use the equation

                    .  

It is easy to see that an antiderivative of        is    .   

Therefore,      is the complex potential

The real part of    is the scalar potential function function:  

                    .  

Note that the hyperbolas   

                        are the equipotential curves,

and that the hyperbolas   

                        are the streamline curves,

these curves are orthogonal, as shown in Figure 3.6.  

                                             Figure 3.6  Red equipotential curves   ,  

                                                                 and blue streamline curves   ,  

                                                                 for the complex potential    .

Explore Solution 3.14.

Optional Material for the Internet

Method II. Construction of the Harmonic Conjugate of u(x,y) using Algebra.

        The usual method proposed for finding the harmonic conjugate uses integrals and derivatives and is shown above as Method I.  

A second method discovered by the British mathematician Louis Melville Milne-Thomson (1891-1974) uses novel algebraic construction.

His method appears in the article On the Relation of an Analytic Function of z to Its Real and Imaginary Parts, L. M. Milne-Thomson,

The Mathematical Gazette, Vol. 21, No. 244 (July 1937), pp. 228-229, Jstor.  A good reference to read is the recent article,

Recovering Holomorphic Functions from Their Real or Imaginary Parts without the Cauchy-Riemann Equations, William T. Shaw,

SIAM Review, Vol 46, No. 4, 2004, pp 717-718, Jstor.  

The Milne-Thomson Method for constructing a harmonic conjugate.  

(i)  Given the harmonic function    then construct   

                     .  

Under the proper conditions,    is a harmonic conjugate of  ,  and

                      

is an analytic function.  

Proof of (i).

(ii)  Given the harmonic function    then construct   

                     .  

Under the proper conditions,    is a harmonic conjugate of  ,  and

                       

is an analytic function.  

Proof of (ii).

Limitations of the Milne-Thomson Method.

Observe that in Milne-Thomson method, the term     will be transformed into  

                    

and that the term   will be transformed into

                      

Hence the method does not work if the given harmonic function contains a term that is the real or imaginary part of

                    

Hence it is applicable when the analytic function is a power series centered about the origin.  

The reader is encouraged to investigate the origins and limitations of the Milne-Thomson method.

Extra Example 1.  Use Methods I and II to construct the harmonic conjugate of   .  

Also, show that the underlying analytic function is   .  

          The orthogonal grid in the -plane and it's image under the analytic function   .  

Extra Solution 1.

Extra Example 2.  Use Methods I and II to construct the harmonic conjugate of   .  

Also, show that the underlying analytic function is   .  

          The orthogonal grid in the -plane and it's image under the analytic function   .  

Extra Solution 2.

Extra Example 3.  Use Methods I and II to construct the harmonic conjugate of   .  

Also, show that the underlying analytic function is   .  

          The orthogonal grid in the -plane and it's image under the analytic function   .  

Extra Solution 3.

Extra Example 4.  Use Methods I and II to construct the harmonic conjugate of   .  

Also, show that the underlying analytic function is   .  

          The orthogonal grid in the -plane and it's image under the analytic function   .  

Extra Solution 4.

Extra Example 5.  Use Methods I and II to construct the harmonic conjugate of   .  

Also, show that the underlying analytic function is   .  

                    The orthogonal grid in the -plane and it's image under the analytic function   .  

                    Remark.  There are infinitely many branches of the multi-valued inverse function   ,  
                                    and when the regions are combined, they will fill up the -plane.  

Extra Solution 5.

Extra Example 6.  Use Methods I and II to construct the harmonic conjugate of   .  

Also, show that the underlying analytic function is   .  

                    The orthogonal grid in the -plane and it's image under the analytic function   .  

                    Remark.  There are infinitely many branches of the multi-valued inverse function   ,  
                                    and when the regions are combined, they will fill up the -plane.  

Extra Solution 6.

Tutorial Exercises for Section 3.3.  Harmonic Functions

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

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