5.2  The Complex Logarithm

    In Section 5.1, we showed that, if w is a nonzero complex number, then the equation    has infinitely many solutions.  Because the function    is a many-to-one function, its inverse (the logarithm) is multivalued.  

Definition 5.2.  (Multivalued Logarithm)  For , we define the function    as the inverse of the exponential function;  that is,  

(5-10)               if and only if   .  

    If we go through the same steps as we did in equations (5-8) and (5-9), we find that, for any complex number  ,  the solutions w to equation (5-10) take the form  

(5-11)            ,   for   ,  

where    and    denotes the natural logarithm of the positive number |z|.  Because    is the set  ,  we can express the set of values comprising as  

(5-12)        ,  
        or
(5-13)           for   ,   

where it is understood that identity (5-13) refers to the same set of numbers given in identity (5-12).

    Recall that Arg is defined so that for  ,  we have  .  We call any one of the values given in Identities (5-12) or (5-13) a logarithm of z.  Notice that the different values of    all have the same real part and that their imaginary parts differ by the amount  ,  where n is an integer.  When  ,  we have a special situation.

Definition 5.3.  (Principal Value of the Logarithm)  For  ,  we define the principal value of the logarithm as follows:

(5-14)         where   and .  

    The domain for the function    is the set of all nonzero complex numbers in the  z-plane, and its range is the horizontal strip    in the w-plane, and is shown in Figure 5.A.  We stress again that    is a single-valued function and corresponds to setting    in equation (5-12).  As we demonstrated in Section 2.4, the function    is discontinuous at each point along the negative x-axis, hence so is the function  .  In fact, because any branch of the multi-valued function    is discontinuous along some ray, a corresponding branch of the logarithm will have a discontinuity along that same ray.  

                Figure 5.A  The principal branch of the logarithm .

Caution.  A phenomenon inherent in constructing an logarithm function:  It must have a discontinuity!  This is the case because as we saw in Section 2.4, any branch we choose for    is necessarily a discontinuous function.  The principal branch,  ,  is discontinuous at each point along the negative x-axis.  

Extra Example 1.  Investigate the complex logarithm function.

Explore Solution for Extra Example 1.

Example 5.3.   Find the values of  .  

Solution.  By standard computations, we have  

              
            and
                
The principal values are

              

Explore Solution 5.3.

Extra Example 2.  The transformation  w = Log(z)  maps the z-plane punctured at the origin onto the horizontal strip in the w-plane.

Explore Solution for Extra Example 2.

    We now investigate some of the properties of log(z) and Log(z).  From Equations (5-10), (5-12), and (5-14), it follows that

(5-15)               for all z~=0   
            and
(5-16)            ,   provided   ,    

and that the mapping  w = Log(z)  is one-to-one from domain    in the z plane onto the horizontal strip    in the w plane.

     The following example illustrates that, even though Log(z) is not continuous along the negative real axis, it is still defined there.

Example 5.4.  Identity  reveals that  

    (a)  ,   and  

    (b)  .  

Explore Solution 5.4.

    When  ,  where x is a positive real number, the principal value of the complex logarithm of z is  

            ,  
        
where .  Hence Log is an extension of the real function ln(x) to the complex case.  Are there other similarities?  Let's use complex function theory to find the derivative of Log(z).   
When we use polar coordinates for  ,  equation (5-14) becomes  

where  .  Because is discontinuous only at points in its domain that lie on the negative real axis, U and V have continuous partials for any point in their domain, provided is not on the negative real axis, that is, provided    (Note the strict inequality for  here.).   In addition, the polar form of the Cauchy-Riemann equations holds in this region (see Equation (3-22) of Section 3.2), since    it follows that  

            
        and
            

Using Theorem 3.5 of Section 3.2 , we see that  

              

provided   .  Thus the principal branch of the complex logarithm has the derivative we would expect.  Other properties of the logarithm carry over, but only in specified regions of the complex plane.  

Example 5.5.  Show that the identity    is not always valid.  

Solution.  Let  .  Then  

              
        but
               

Since  ,  this is a counterexample for which   .  

Explore Solution 5.5.

    Our next result explains why the identity    did not hold for the particular numbers we chose.

Theorem 5.2.  The identity    holds true if and only if  .  

Proof of Theorem 5.2.

    As Example 5.5 and Theorem 5.2 illustrate, properties of the complex logarithm don't carry over when arguments of products combine in such a way that they drop down to or rise above . This is because of the restrictions placed on the domain of the function .  From the set of numbers associated with the multivalued logarithm, however, we can formulate properties that look exactly the same as those corresponding with the real logarithm.

Theorem 5.3.  Let be nonzero complex numbers. The multivalued function obeys the familiar properties of logarithms:

(5-17)            ,  

(5-18)            , and

(5-19)            .  

Proof of Theorem 5.3.

Property of  Log(z).  We know that , what about Log(z)?

Demonstration of Property of  Log(z).

    We can construct many different branches of the multivalued logarithm function that are continuous and differentiable except at points along any preassigned ray  .  If we let denote a real fixed number and choose the value of  ,  that lies in the range  ,  then the function    defined by  

            

where  ,  and  ,  is a single-valued branch of the logarithm function. The branch cut for    is the ray  ,  and each point along this ray is a point of discontinuity of .  Because  ,  we conclude that the mapping    is a one-to-one mapping of the domain onto the horizontal strip  .  If  ,  then the function    maps the set    one-to-one and onto the rectangle  .  Figure 5.4 shows the mapping , its branch cut , the set D, and its image R.

                    Figure 5.4  The branch of the logarithm.

    We can easily compute the derivative of any branch of the multivalued logarithm. For a particular branch for , and   (note the strict inequality for ),  we start with in Equations (5-10) and differentiate both sides to get  

              

Solving for gives  

            ,   for    for , and  .  

    The Riemann surface for the multivalued function    is similar to the one we presented for the square root function.  However, it requires infinitely many copies of the z plane cut along the negative x axis, which we label for .  Now, we stack these cut planes directly on each other so that the corresponding points have the same position.  We join the sheet as follows.  For each integer k, the edge of the sheet in the upper half-plane is joined to the edge of the sheet in the lower half-plane.  The Riemann surface for the domain of looks like a spiral staircase that extends upward on the sheets   and downward on the sheets ,  as shown in Figure 5-5.  We use polar coordinates for z on each sheet.  For , we use

              ,    where  
              
                and  .  

Again, for , the correct branch of on each sheet is  

            ,    where

                and  .  

                Figure 5.5  The Riemann surface for the mapping  .

Exercises for Section 5.2.  The Complex Logarithm