Complex Trigonometric and Hyperbolic Functions
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
Thus, it is natural to make the following definitions.
5.5. The series for Sine
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 (ii). Investigate the series .
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,
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
for all z, whether z is real or complex. Hence,
Adding the above two expressions and
and subtracting (5-32) from (5-31) and solving for gives
Figure 5.A The mapping .
These equations in turn are used to obtain the following important
Exploration for the real and imaginary parts of Sin and Cos.
Equipped with Identities
(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
Again, the proofs for the other periodic results are left as exercises.
Exploration for trigonometric identities.
any complex numbers, then
We demonstrate that
by making use of Identities (5-32) -
Adding these expressions gives
which is what we wanted.
Demonstration. Establish the identity .
Exploration Method (i).
Exploration Method (ii).
A solution to the
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
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.
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 .
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
The identities and then yield
A similar derivation produces
If we set
in Identity (5-36) and
let , we
As advertised, we have shown that is not a bounded function; it is also evident from Identity (5-37) that is unbounded.
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
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
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
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 .
How should we define the complex hyperbolic functions? We begin with
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.
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
The Next Module is
Inverse Trigonometric and Hyperbolic Functions
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This material is coordinated with our book Complex Analysis for Mathematics and Engineering.
(c) 2012 John H. Mathews, Russell W. Howell