Section 3.3  Harmonic Functions

    Let    be a real-valued function of the two real variables x and y defined on a domain D.  (Recall that a domain is a connected open set.)  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.

    Harmonic functions are important in the areas of applied mathematics, engineering, and mathematical physics.  In calculus we might have been asked to show that the functions and   are harmonic (and also that the functions    and    are harmonic).  Harmonic functions are used to solve problems involving steady state temperatures, two-dimensional electrostatics, and ideal fluid flow.  In Section 11.2 and Section 12.2 we will see how complex analysis techniques can be used to solve the Dirichlet problem in the upper half-plane and unit disk, respectively.  For example, the function    is harmonic in the upper half plane and takes on the boundary values     and   ,  as shown in Figure 3.A.

        Figure 3.A  The harmonic function  .

Exploration

These pairs of functions are not chosen at random, and there is an intimate relationship between them.  We begin with an important theorem relating analytic and harmonic functions.

Theorem 3.8.  Let    be an analytic function on a domain D.  Then both    and    are harmonic functions on D.  In other words, the real and imaginary parts of an analytic function are harmonic.

Proof.  Corollary 6.3 (in Section 6.5) will show that, if f(z) is analytic, then all partial derivatives of u and v are continuous.  Using that result here, we see that, as f is analytic, u and v satisfy the Cauchy-Riemann equations

           and   .  

Taking the partial derivative with respect to x of each side of these equations gives  

           and   .  


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

           and   .  


The partial derivatives are all continuous, so 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 u and v are harmonic functions on D.

Proof.

Proof of Theorem 3.8 is also in the book
Complex Analysis for Mathematics and Engineering

    If we have a function that is harmonic on the domain D and if we can find another harmonic function such that the partial derivatives for u and v satisfy the Cauchy-Riemann equations throughout D, then we say that is a harmonic conjugate of .  It then follows that the function    is analytic on D.

Example 3.11.  If  ,  then  ;   hence u is a harmonic function for all z.  We find that is also a harmonic function and that

            ,   and   

            .

Therefore v is a harmonic conjugate of u, and the function f given by  

              

is an analytic function.

Explore Solution 3.11.

    Theorem 3.8 makes the construction of harmonic functions from known analytic functions an easy task.

Example 3.12. The function    is analytic for all values of z, hence it follows that  

              
is harmonic, and
              
        
is a harmonic conjugate of u(x,y).

Explore Solution 3.12.

    Figures 3.2 and 3.3 show the graphs of these two functions.  The partial derivatives 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 these partial derivatives appear along the edges of the surfaces for u and v where .  Similarly,    and    also appear along the edges of the surfaces for u and v where .

    Figure 3.2  .              Figure 3.3  .

    We can use complex analysis to show easily that certain combinations of harmonic functions are harmonic.  For example, if v is a harmonic conjugate of u, 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 f is also an analytic function, which is

        .  

    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.

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 v 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 y and get

(3-27)            

where is a function of x alone that is yet to be determined.  Second, we compute by differentiating both sides of this equation with respect to x and replacing    with    on the left side, which gives  

            

It can be shown (we omit the details) that because u is harmonic, all terms except those involving x in the last equation will cancel, revealing a formula for involving x alone.  Elementary integration of the single-variable function can then be used to discover .  We finally observe that the function v so created indeed has the properties we seek.

    The functions   and are computed with the formulas:

            ,   and     
            
            .  

Proof.
Proof of Theorem 3.9 is also in the book
Complex Analysis for Mathematics and Engineering

    Technically we should always specify the domain of function when defining it.  When no such specification is given, it is assumed that the domain is the entire complex plane, or the largest set for which the expression defining the function makes sense.

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

Execute this cell to activate the construction of the harmonic conjugate subroutine.

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 u is harmonic, we compute the second partial derivatives and note that  ,  so u satisfies Laplace's Equation (3-26).  To construct v, we start with Equation (3-27) and the first of Equations (3-28) to get  

            

Differentiating the left and right sides of this equation with respect to x and using    and Equations (3-28) on the left side yields

            

which implies that

            ,  

then an easy integration yields  ,  where C is a constant.  Now we substitute and obtain the desired solution

            .  

Explore Solution 3.13.

    Contour plots for the families of curves    and    are shown in Figures 3.B.  

    Figure 3.B  Contour plot for       and     .  

    In Section 11.4 we will show that if   is an analytic function.  Then the two families of level curves      and  ,  are orthogonal in the sense that if is a point in common to the two curves    and  ,  and  if   ,  then these two curves intersect orthogonally.  The orthogonal grid shown in Figure 3.C.  

        Figure 3.C  The orthogonal grid formed by and .

Extra Example 1.  Show that    is a harmonic function and find the harmonic conjugate  .

Explore Extra Solution 1.

Fluid Flow

    Harmonic functions are solutions to many physical problems.  Applications include two-dimensional models of heat flow, electrostatics, and fluid flow.  We now give an example of the latter.
    
    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, and .  Hence p and q obey the partial differential equations

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

            ,   and    

            .  

Assuming the functions p and q have continuous partials, Theorem 3.4 guarantees that function f 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 f has an analytic antiderivative F;  assuming this to be the case, we can write  

(3-32)            ,  

where .  

    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 preceeding equation and Equation (3-32) imply that  

            .  

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

            .

    The curves given by are called equipotentials.  The curves are called streamlines and 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

            .   

An antiderivative of    is  ,  and the real part of is the desired harmonic 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  Blue equipotential curves , and red streamline curves , for .

Explore Solution 3.14.

       There is an alternate method for constructing the harmonic conjugate function which involves only algebra,

it was discovered by L. M. Milne-Thomson, see On the Relation of an Analytic Function of z to Its Real and Imaginary Parts,

in The Mathematical Gazette, Vol. 21, No. 244 (July 1937), pp. 228-229.   

Theorem (Milne-Thomson Method for constructing a harmonic conjugate.)  

Formula (i)  Given the harmonic function    then construct   

                     .  

Show that under the proper conditions,    is a harmonic conjugate of  ,  and

                       is an analytic function.  

Formula (ii)  Given the harmonic function    then construct   

                     .  

Show that under the proper conditions,    is a harmonic conjugate of  ,  and

                       is an analytic function.  

Proof of (i).

Proof of (ii).

Extra Example 2.  Find an analytic function  ,  given the harmonic function  .

          The orthogonal grid formed by the image of three rectangular grids under the multivalued inverse function  , where  .  
Extra Solution 2.

Extra Example 3.  Find an analytic function  ,  given the harmonic function  .

          The orthogonal grid formed by the composite image of four rectangular grids under the multivalued inverse function  .  
Extra Solution 3.

    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.

     Table. Interpretations for the level curves of and .

Exercises for Section 3.3.  Harmonic Functions