Chapter 2  Complex Functions

Overview

    In Chapter 1 we developed a basic theory of complex numbers. For the next few chapters we turn our attention to functions of complex numbers. They are defined in a similar way to functions of real numbers that you studied in calculus; the only difference is that they operate on complex numbers rather than real numbers. This chapter focuses primarily on very basic functions, their representations, and properties associated with functions such as limits and continuity. You will learn some interesting applications as well as some exciting new ideas.

2.1 Functions and Linear Mappings

    A complex-valued function f of the complex variable z is a rule that assigns to each complex number z in a set D one and only one complex number w.  We write    and call w the image of z under f.   A simple example of a complex-valued function is given by the formula  .  The set D is called the domain of f, and the set of all images    is called the range of f.  When the context is obvious, we omit the phrase complex-valued, and simply refer to a function f, or to a complex function f.

    We can define the domain to be any set that makes sense for a given rule, so for  ,  we could have the entire complex plane for the domain D, or we might artificially restrict the domain to some set such as  .  Determining the range for a function defined by a formula is not always easy, but we will see plenty of examples later on.  In some contexts functions are referred to as mappings or transformations.

    In Section 1.6, we used the term domain to indicate a connected open set.  When speaking about the domain of a function, however, we mean only the set of points on which the function is defined.  This distinction is worth noting, and context will make clear the use intended.

    Just as z can be expressed by its real and imaginary parts,  ,  we write  ,  where u and v are the real and imaginary parts of w, respectively.  Doing so gives us the representation  

            .

Because u and v depend on x and y, they can be considered to be real-valued functions of the real variables x and y; that is,

                and  .  

Combining these ideas, we often write a complex function f in the form

            .  

Figure 2.1 illustrates the notion of a function (mapping) using these symbols.

        Figure 2.1  The mapping  .

    There are two methods for defining a complex function in Mathematica.
Exploration.

We now give several examples that illustrate how to express a complex function.

Example 2.1.  Write    in the for  .

Solution.  Using the binomial formula, we obtain

            

so that  .

Explore Solution 2.1.

Example 2.2.  Express the function    in the form  .

Solution.  Using the elementary properties of complex numbers, it follows that  

            

so that  .

Explore Solution 2.2.

    Examples 2.1 and 2.2 show how to find u(x,y) and v(x,y) when a rule for computing f is given. Conversely, if u(x,y) and v(x,y) are two real-valued functions of the real
variables x and y, they determine a complex-valued function    ,  and we can use the formulas

              and  

to find a formula for f involving the variables z and .  

Example 2.3.  Express    by a formula involving the variables  .  

Solution.  Calculation reveals that

            

Explore Solution 2.3.

    Using    in the expression of a complex function f may be convenient.  It gives us the polar representation  

                ,

where U and V are real functions of the real variables r and .

Remark.  For a given function f, the functions u and v defined above are different from those used previously in    which used Cartesian coordinates instead of polar coordinates.

Example 2.4.  Express    in both Cartesian and polar form.

Solution.  For the Cartesian form, a simple calculation gives

            

so that  .

For the polar form, we get v

            

so that  .  

Explore Solution 2.4.

Remark.  Once we have defined u and v for a function f in Cartesian form, we must use different symbols if we want to express f in polar form. As is clear here, the functions u and U are quite different, as are v and V.  Of course, if we are working only in one context, we can use any symbols we choose.

For a given function f, the functions u and v defined here are different from those defined by equation (2-1), because equation (2-1) involves Cartesian coordinates and equation (2-2) involves polar coordinates.

Example 2.5.  Express    in polar form.

Solution. We obtain

              

so that  .  

Explore Solution 2.5.

    We now look at the geometric interpretation of a complex function.  If D is the domain of real-valued functions u(x,y) and v(x,y), the equations  

              and  

describe a transformation (or mapping) from D in the xy plane into the uv plane, also called the w plane. Therefore, we can also consider the function  

              

to be a transformation (or mapping) from the set D in the z plane onto the range R in the w plane.  This idea was illustrated in Figure 2.1. In the following paragraphs we present some additional key ideas. They are staples for any kind of function, and you should memorize all the terms in bold.  

    If A is a subset of the domain D of f, the set    is called the image of the set A, and f is said to map A onto B.  The image of a single point is a single point, and the image of the entire domain, D, is the range, R.  The mapping    is said to be from A into S if the image of A is contained in S.  Mathematicians use the notation   to indicate that a function maps A into S.  Figure 2.2 illustrates a function f whose domain is D and whose range is R.  The shaded areas depict that the function maps A onto B.  The function also maps A into R, and, of course, it maps D onto R.

            Figure 2.2   maps A onto B;   maps A into R.

    The inverse image of a point w is the set of all points z in D such that  .  The inverse image of a point may be one point, several points, or nothing at all.  If the last case occurs then the point w is not in the range of f.  For example, if  ,  the inverse image of the point is the single point , because  ,  and is the only point that maps to .  In the case of  ,  the inverse image of the point is the set .  You will learn in Section 5.1 that, if  ,  the inverse image of the point 0 is the empty set---there is no complex number z such that .

    The inverse image of a set of points, S, is the collection of all points in the domain that map into S.  If f maps D onto R it is possible for the inverse image of R to be function as well, but the original function must have a special property: a function f is said to be one-to-one if it maps distinct points    onto distinct points  .  Many times an easy way to prove that a function f is one-to-one is to suppose  ,  and from this assumption deduce that must equal .  Thus,    is one-to-one because if ,  then .  Dividing both sides of the last equation by gives  .  Figure 2.3 illustrates the idea of a one-to-one function: distinct points get mapped to distinct points.

            Figure 2.3  A function  w = f(z)  that is one-to-one.

    The function    is not one-to-one because ,  but  .  Figure 2.4 depicts this situation: at least two different points get mapped to the same point.

            Figure 2.4  A function that is not one-to-one.

    In the exercises we ask you to demonstrate that one-to-one functions give rise to inverses that are functions.  Loosely speaking, if    maps the set A one-to-one and onto the set B, then for each w in B there exists exactly one point z in A  A such that  .  For any such value of z we can take the equation    and "solve" for z as a function of w.  Doing so produces an inverse function    where the following equations hold:  

               

    Conversely, if and are functions that map A into B and B into A, respectively, and the above hold, then f maps the set A one-to-one and onto the set B.

    Further, if f is a one-to-one mapping from D onto T and if A is a subset of D, then f is a one-to-one mapping from A onto its image B.  We can also show that, if    is a one-to-one mapping from A onto B and    is a one-to-one mapping from B onto S, then the composite mapping    is a one-to-one mapping from A onto S.

    We usually indicate the inverse of by the symbol .  If the domains of and are A and B respectively, then we write  
    
               for all  ,  and  
        
               for all  .  
        
Also, for    and  .     

               iff   ,    and

               iff   .   

Example 2.6.  If    for any complex number z, find  .

Solution.  We can easily show f is one-to-one and onto the entire complex plane. We solve for z, given  ,  to get  .  This result implies that   for all complex numbers w.  

Remark.  Once we have specified    for all complex numbers w, we note that there is nothing magical about the symbol w.  We could just as easily write    for all complex numbers z.  

Explore Solution 2.6.

    We now show how to find the image B of a specified set A under a given mapping u+iv=w=f(z). The set A is usually described with an equation or inequality involving x and y. Using inverse functions, we can construct a chain of equivalent statements leading to a description of the set B in terms of an equation or an inequality involving u and v.

Example 2.7.  Show that the function    maps the line    in the xy plane onto the line    in the w plane.  

Solution.  Method 1:  With  ,  we want to describe  .  We let    and get  

              

where is the notation for "if and only if."  Note what this result says:  . The image of A under f, therefore, is the set  .  

Method 2:  We write    and note that the transformation can be given by the equations  .  Because A is described by  ,  we can substitute    into the equation    to obtain  ,  which we can rewrite as  .  If you use this method, be sure to pay careful attention to domains and ranges.

Explore Solution 2.7.

    We now look at some elementary mappings.  If we let    denote a fixed complex constant, the transformation  

              

is a one-to-one mapping of the z-plane onto the w-plane and is called a translation.  This transformation can be visualized as a rigid translation whereby the point  z  is displaced through the vector    to its new position  .  The inverse mapping is given by  

              

and shows that  T  is a one-to-one mapping from the z-plane onto the w-plane. The effect of a translation is depicted in Figure 2.5.

                Figure 2.5  The translation  .

    If we let    be a fixed real number, then for  ,  the transformation  

              

is a one-to-one mapping of the z-plane onto the w-plane and is called a rotation.  It can be visualized as a rigid rotation whereby the point  z  is rotated about the origin through an angle     to its new position  .  If we use polar coordinates and designate    in the w-plane, then the inverse mapping is

            .  

This analysis shows that  R  is a one-to-one mapping from the z-plane onto the w-plane.  The effect of rotation is depicted in Figure 2.6.

                    Figure 2.6  The rotation  .

Example 2.8.  The ellipse centered at the origin with a horizontal major axis of 4 units and vertical minor axis of 2 units can be represented by the parametric equation  

        ,   for  .  

Suppose we wanted to rotate the ellipse by an angle of radians and shift the center of the ellipse 2 units to the right and 1 unit up. Using complex arithmetic, we can easily generate a parametric equation r(t) that does so:

                
for  .  Figure 2.7 shows parametric plots of these ellipses.

             Figure 2.7  (a)  Plot of the original ellipse                 (b)  Plot of the rotated ellipse
                                                              

Explore Solution 2.8.

    If we let    be a fixed positive real number, then the transformation  

              

is a one-to-one mapping of the z-plane onto the w-plane and is called a magnification.  If   ,  it has the effect of stretching the distance between points by the factor  K.  If   ,  then it reduces the distance between points by the factor  K.  The inverse transformation is given by  

              

and shows that  S  is a one-to-one mapping from the z-plane onto the w-plane.  The effect of magnification is shown in Figure 2.8.

             Figure 2.8  The magnification  .

    Finally, if we let    and  , where    is a positive real number, then the transformation

            

is a one-to-one mapping of the z-plane onto the w-plane and is called a linear transformation.  It can be considered as the composition of a rotation, a magnification, and a translation.  It has the effect of rotating the plane though an angle given by  ,  followed by a magnification by the factor  , followed by a translation by the vector  .  The inverse mapping is given by    and shows that L is a one-to-one mapping from the z-plane onto the w-plane.

Example 2.9.  Show that the linear transformation    maps the right half plane    onto the upper half plane  .  

Solution.  Method 1:  Let  .  To describe  ,  we solve    for z to get  .  We have the following

              

Thus  ,  which is the same as saying  .  

Method 2: When we write    in Cartesian form as  

            ,  

we see that the transformation can be given by the equations   and .  Substituting    in the inequality    gives  ,  or  ,  which is the upper half-plane  .  

Method 3:  The effect of the transformation   is a rotation of the plane through the angle   (when z is multiplied by ) followed by a translation by the vector  .  The first operation yields the set  .  The second shifts this set up 1 unit, resulting in the set  .  We illustrate this result in Figure 2.9.

            Figure 2.9  The linear transformation  .

Explore Solution 2.9.

    Translations and rotations preserve angles.  First, magnifications rescale distance by a factor K, so it follows that triangles are mapped onto similar triangles, preserving angles.  Then, because a linear transformation can be considered to be a composition of a rotation, a magnification, and a translation, it follows that linear transformations preserve angles.  Consequently, any geometric object is mapped onto an object that is similar to the original object; hence linear transformations can be called similarity mappings.

Note. The usage of the phrase "linear transformation" in a "complex analysis course" is different than that the usage in "linear algebra courses".

Example 2.10.  Show that the image of the open disk    under the linear transformation    is the open disk  .  

Solution.  The inverse transformation is  ,  so if we designate the range of f as B, then

               
            
            

Hence the disk with center    and radius 1 is mapped one-to-one and onto the disk with center    and radius 5 as shown in Figure 2.10.

            Figure 2.10  The mapping  .  

Explore Solution 2.10.

Example 2.11.  Show that the image of the right half plane    under the linear transformation    is the half plane  .  

Solution.  The inverse transformation is given by  

            ,  

which we write as  

            .  

Substituting    into  e    gives  ,  which simplifies  .  Figure 2.11 illustrates the mapping.

            Figure 2.11  The the linear transformation  .  

Explore Solution 2.11.

Exercises for Section 2.1.  Functions and Linear Mappings