**for**

**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
(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 .

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).

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

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

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
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) .

**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 .

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.

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
(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.

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.

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)

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

**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**** **

**The Next Module
is**

**Inverse
Trigonometric and Hyperbolic Functions**

**Return to the Complex
Analysis Modules **

__Return
to the Complex Analysis Project__

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell