5.4  Trigonometric and Hyperbolic Functions

    Based on the success we had in using power series to define the complex exponential (see Section 5.1), we have reason to believe this approach will be fruitful for other elementary functions as well. The power series expansions for the real-valued sine and cosine functions are

            ,    and  
           
            

Thus, it is natural to make the following definitions.

Definition 5.5.   The series for  Sine and Cosine are

            ,    and  
           
            

    Clearly, these definitions agree with their real counterparts when z is real.  Additionally, it is easy to show that are entire functions.  (We leave the proof as an exercise.)

Exploration (i).  Investigate the series .  

Exploration (i).

Exploration (ii).  Investigate the series .  

Exploration (ii).

    With these definitions in place, it is now easy to create the other complex trigonometric functions, provided the denominators in the following expressions do not equal zero.

Definition 5.6.   ,  ,  ,  and  .  

Exploration for Definition 5.6.

    Since the series for the complex sine and cosine agree with the real sine and cosine when z is real, the remaining complex trigonometric functions likewise agree with their real counterparts. What additional properties are common? For starters, we have  

Theorem 5.4.   and are entire functions, with    and  .  

Proof of Theorem 5.4.

Exploration for Theorem 5.4.

    We now list several additional properties, providing proofs for some and leaving others as exercises.  For all complex numbers z,

              

    The verification that    and    comes from substituting -z for z in Definition 5.5.  We leave verification of the identity   as an exercise (with hints).

Exploration for identities.

A series exploration (i) The derivative of  sin(z)  is  cos(z).
A series exploration (i).

A series exploration (ii) The derivative of  cos(z)  is  -sin(z).
A series exploration (ii).

    For all complex numbers z for which the expressions are defined,  

              

Proof.

Exploration for the identities.

    To establish additional properties, it will be useful to express in the Cartesian form .  (Additionally, the applications in Chapters 10 and 11 will use these formulas.)  We begin by observing that the argument given to prove part (iii) in Theorem 5.1 easily generalizes to the complex case with the aid of Definition 5.5.  That is,  

(5-30)            ,

for all z, whether z is real or complex.  Hence,  

(5-31)              

    Adding the above two expressions and solving for gives

(5-32)             ,

and subtracting (5-32) from (5-31) and solving for gives

(5-33)            .  

                Figure 5.A  The mapping  .

These equations in turn are used to obtain the following important identities

(5-34)            ,

(5-35)            .  

Proof.

Exploration for the real and imaginary parts of Sin and Cos.

    Equipped with Identities (5-32) - (5-35), we can now establish many other properties of the trigonometric functions.  We begin with some periodic results.  For all complex numbers ,  

                 

Clearly, .  By Identity (5-34) this expression is  

               

Again, the proofs for the other periodic results are left as exercises.

Exploration for trigonometric identities.

    If    are any complex numbers, then

              

    We demonstrate that by making use of Identities (5-32) - (5-35):  

            ,   and  

            

Adding these expressions gives

            

which is what we wanted.

Demonstration.  Establish the identity  .  

Exploration Method (i).

Exploration Method (ii).

    A solution to the equation    is called a zero of the given function f.  As we now show, the zeros of the sine and cosine function are exactly where you might expect them to be.  We have    iff  ,  where n is any integer, and    iff  ,  where n is any integer.

    We show the result for and leave the result for as an exercise.  When we use Identity (5-35),    iff  

            .  

Equating the real and imaginary parts of this equation gives

               and   .  

The real-valued function cosh y is never zero, so the equation    implies that  ,  from which we obtain for any integer n .  

Using the values    in the equation    yields  

            .  

which implies that  ,  so the only zeros for are the values   for  n  an integer.

Exploration.

    What does the mapping look like?  We can get a graph of the mapping    by using parametric methods.  Let's consider the vertical line segments in the z plane obtained by successfully setting    for , and for each x value and letting y vary continuously, .  In the exercises we ask you to show that the images of these vertical segments are hyperbolas in the uv plane, as Figure 5.7 illustrates.  In Section 10.4, we give a more detailed analysis of the mapping  .

            Figure 5.7  Vertical segments mapped onto hyperbolas by  .

Exploration.

    Figure 5.7 suggests one big difference between the real and complex sine functions. The real sine has the property that    for all real x.  In Figure 5.7, however, the modulus of the complex sine appears to be unbounded, which is indeed the case.  Using Identity (5-34) gives  

               

The identities    and    then yield  

(5-36)            .  

A similar derivation produces  

(5-37)            .  

    If we set in Identity (5-36) and let  ,  we get  

            

As advertised, we have shown that is not a bounded function; it is also evident from Identity (5-37) that is unbounded.

Exploration.

    The periodic character of the trigonometric functions makes apparent that any point in their ranges is actually the image of infinitely many points.

Example 5.10.  Find all the values of  z  for which  .

Solution.  Starting with Identity (5-35), we write

            .  

If we equate real and imaginary parts, then we get

               and   .  

The equation    implies either that  , where n is an integer, or that .  Using in the equation   leads to the impossible situation  .  Therefore  , where n is an integer.  Since    for all values of y, the term in the equation    must also be positive.  For this reason we eliminate the odd values of n and get  , where k is an integer.

    Finally, we solve the equation    and use the fact that is an even function to conclude that  .  Therefore the solutions to the equation    are  ,  where k is an integer.

Explore Solution 5.10.

    The hyperbolic functions also have practical use in putting the tangent function into the Cartesian form .  
Using Definition 5.6, and Equations (5-34) and (5-35), we have  

(5-38)            .  

If we multiply each term on the right by the conjugate of the denominator, the simplified result is  

            

We leave it as an exercise to show that the identities    and    can be used in simplifying Equation (5-38) to get  

(5-39)            

Exploration.

    As with , we obtain a graph of the mapping parametrically.  Consider the vertical line segments in the z plane obtained by successively setting    for  ,  and for each z value letting y vary continuously, .  In the exercises we ask you to show that the images of these vertical segments are circular arcs in the uv plane, as Figure 5.8 shows.  In Section 10.4, we give a more detailed investigation of the mapping  .

    Figure 5.8  Vertical segments mapped onto circular arcs by  .

Exploration.

    How should we define the complex hyperbolic functions?  We begin with

Definition 5.7.   The hyperbolic cosine and hyperbolic sine functions are  

          

    With these definitions in place, we can now easily create the other complex hyperbolic trigonometric functions, provided the denominators in the following expressions are not zero.

Definition 5.8.   Identities for the hyperbolic trigonometric functions are  

                

    As the series for the complex hyperbolic sine and cosine agree with the real hyperbolic sine and cosine when z is real, the remaining complex hyperbolic trigonometric functions likewise agree with their real counterparts. Many other properties are also shared. We state several results without proof, as they follow from the definitions we gave using standard operations, such as the quotient rule for derivatives. We ask you to establish some of these identities in the exercises.

    The derivatives of the hyperbolic functions follow the same rules as in calculus:  

               

    The hyperbolic cosine and hyperbolic sine can be expressed as  

            

    Some of the important identities involving the hyperbolic functions are  

                

Exercises for Section 5.4.  Trigonometric and Hyperbolic Functions