Section 3.2  The Cauchy-Riemann Equations

    In Section 3.1 we showed that computing the derivative of complex functions written in a form such as    is a rather simple task. But life isn't always so easy.  
Many times we encounter complex functions written as  .   For example, suppose we had  

(3-13)            .  

Is there some criterion - perhaps involving the partial derivatives for  u(x,y),  and  v(x,y)  -  that we can use to determine whether f is differentiable, and if so, to find the value of ?

    The answer to this question is yes, thanks to the independent discovery of two important equations by the French mathematician Augustin Louis Cauchy (1789-1857) and the German mathematician Georg Friedrich Bernhard Riemann (1826-1866).

    First, let's reconsider the derivative of  .  As we have stated, the limit given in Equation (3-1) must not depend on how z approaches .  We investigate two such approaches: a horizontal approach and a vertical approach to .  Recall from our graphical analysis of    that the image of a square is a "curvilinear quadrilateral."  For convenience, we let the square have vertices  ,  ,  ,  and  .  Then the image points are  ,  ,  ,  and  , as shown in Figure 3.1.  

        Figure 3.1 The image of a small square with vertex  ,  under the mapping  .

Exploration

    We know that f is differentiable, so the limit of the difference quotient    exists no matter how we approach  .  

Thus we can approximate    by using a horizontal increment in z:  

               

And, we can approximate    by using a vertical increment in z:  

                  

Comparing these two results we see that

              

which leads us to speculate that .  

    These computations lead to the idea of taking limits along the horizontal and vertical directions. When we do so, we get  

               

and

    We now generalize this idea by taking limits of an arbitrary differentiable complex function and obtain an important result.

Explore Numerical Computations

Theorem 3.3 (Cauchy-Riemann Equations).  Suppose that  

              

is differentiable at the point .  Then the partial derivatives of  u  and  v  exist at the point , and  

(3-14)            ,   and also  

(3-15)            .  

Equating the real and imaginary parts of Equations (3-14) and (3-15) gives  

(3-16)         and .  

Proof.

Derivation of the Cauchy-Riemann Equations with Mathematica.

Exploration for the Cauchy-Riemann Equations.

    Note some of the important implications of this theorem.
    
(i).    If f is differentiable at , then the Cauchy-Riemann Equations (3-16) will be satisfied at , and we can use either either Equation (3-14) or (3-15) to evaluate .   
    
(ii).   Taking the contrapositive, if Equations (3-16) are not satisfied at , then we know automatically that f(z) is not differentiable at .

(iii).  Even if Equations (3-16) are satisfied at , we cannot necessarily conclude that f is differentiable at .

    We now illustrate each of these points.

Example 3.4.  We know that   is differentiable and that  .  We also have  

            .  

It is easy to verify that Equations (3-16) are indeed satisfied:

       and   .  

Using Equations (3-14) and (3-15), respectively, to compute   gives

        ,   and   

        ,

as expected.

Explore Solution 3.4.

Example 3.5.  Show that    is nowhere differentiable.

Solution.  We have  ,  where    and  .  Thus, for any point  ,    and  .  The Cauchy-Riemann equations are not satisfied at any point , so we conclude that    is nowhere differentiable.

Explore Solution 3.5.

Example 3.6.  Show that the function defined by

              

is not differentiable at the point even though the Cauchy-Riemann equations are satisfied at .

Solution.  We must use limits to calculate the partial derivatives at (0,0).

            ,

            ,   

            ,

            .

Thus we have shown that , , ,

Hence the Cauchy-Riemann equations hold at the point (0,0).

    We now show that f is not differentiable at .  Letting z approach 0 along the x axis gives

              

But if we let z approach 0 along the line    given by the parametric equations  ,  then  

              

The two limits are distinct, so f is not differentiable at the origin.

Explore Solution 3.6.

    Example 3.6 reiterates that the mere satisfaction of the Cauchy-Riemann equations is not sufficient to guarantee the differentiability of a function.  The following theorem, however, gives conditions that guarantee the differentiability of f at , so that which we can use Equation (3-14) or (3-15) to compute  .  They are referred to as the Cauchy-Riemann conditions for differentiability.

Theorem 3.4 (Cauchy-Riemann conditions for differentiability).  Let    be a continuous function that is defined in some neighborhood of the point  . If all the partial derivatives are continuous at the point and if the Cauchy-Riemann equations and hold at , then  f(z)  is differentiable at   and the derivative can be computed with either formula (3-14) or (3-15),  i.e.

            ,  or  

            .  

Proof.

Example 3.7.  At the beginning of this section (Equation (3-13)) we defined the function  

            .   

Show that this function is differentiable for all z, and find its derivative.

Solution.  We compute     and   ,  so the Cauchy-Riemann Equations (3-16), are satisfied.  Moreover,    are continuous everywhere.  By Theorem 3.4, f is differentiable everywhere, and, from Equation (3-14),  

              

Alternatively, from Equation (3-15),

          

This result isn't surprising because    and so the function f is really our old friend  .  

Explore Solution 3.7.

Extra Example 1.  Given  .   

Show that this function is differentiable for all z, and find its derivative.

Explore Extra Solution 1.

Example 3.8.  Show that the function    is differentiable for all and find its derivative.

Solution.  We first write    and then compute the partial derivatives.

            ,   and

            .  

We note that    are continuous functions and that the Cauchy-Riemann equations hold for all values of .  Hence, using Equation (3-14), we write  

            .

Aside.  Can you guess the "complex" form of    ?

Explore Solution 3.8.

Extra Example 2.  Show that the function   is differentiable for all    and find its derivative.

Explore Extra Solution 2.

    The Cauchy-Riemann conditions are particularly useful in determining the set of points for which a function f is differentiable.

Example 3.9.  Show that the function    is differentiable at points that lie on the x and y axes but analytic nowhere.

Solution.  Recall (Definition 3.1) that when we say a function is analytic at a point we mean that the function is differentiable not only at , but also at every point in some  neighborhood of .  With this in mind, we proceed to determine where the Cauchy-Riemann equations are satisfied.  We write    and    and compute the partial derivatives:

            ,   ,   and  

            ,   .  

Here    are continuous, and    holds for all  .  But    iff  ,  which is equivalent to  .  The Cauchy-Riemann equations hold only when  ,  and according to Theorem 3.4, f is differentiable only at points that lie on the coordinate axes.  But this means that f is nowhere analytic because any -neighborhood about a point on either axis contains points that are not on those axes.

Explore Solution 3.9.

    When polar coordinates are used to locate points in the plane, we use Expression (2-2) for a complex function for convenience;  that is,

              

where    are real functions of the real variables .  The polar form of the Cauchy-Riemann equations and a formula for finding f'(z) in terms of the partial derivatives of U(r,) and V(r,) are given in Theorem 3.5, which we ask you to prove in Exercise 10.  This theorem makes use of the validity of the Cauchy-Riemann equations for the functions u and v, so the relation between them and the functions   - namely,    and    - is important.

Theorem 3.5  (Polar Form of the Cauchy-Riemann equations).  Let    be a continuous function that is defined in some neighborhood of the point  .  If all the partial derivatives    are continuous at the point and if the polar form of the Cauchy-Riemann equations,

(3-22)               and   ,    

holds, then is differentiable at  , and we can compute the derivative by using either  

(3-23)            ,    or    

(3-24)            .  

Proof.

Example 3.10.  Show that, if f is is the principal square root function given by  

               

where the domain is restricted to be  ,  then the derivative is given by  

              

for every point in the domain  .  

Solution. We write  

            ,   and   
            
            .  
Thus,
            ,   and   

            .  

Since    are continuous at every point in the domain (note the strict inequality in ), we use Theorem 3.5 and Equation (3-23) to get  

            

Note that is discontinuous on the negative real axis and is undefined at the origin.  Using the terminology of Section 2.4, the negative real axis is a branch cut, and the origin is a branch point for this function.  

Explore Solution 3.10.

    Two important consequences of the Cauchy-Riemann equations close this section.

Theorem 3.6.  Let    be an analytic function on the domain D.  Suppose for all    that  ,  where K is a constant.  Then f is constant on D.  

Proof.

Theorem 3.7.  Let    be an analytic function on the domain D.   If    for all  ,  then f is constant on D.  

Proof.

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