文档介绍：Section
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Hyperbolic Functions
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The final class of functions we will consider are the hyperbolic functions. In a sense these
functions are not new to us since they may all be expressed in terms of the exponential
function and its inverse, the natural logarithm function. However, we will see that they
have many interesting and useful properties.
Definition For any real number x, the hyperbolic sine of x, denoted sinh(x), is defined
by
1
sinh(x) = (ex − e−x) ()
2
and the hyperbolic cosine of x, denoted cosh(x), is defined by
1
cosh(x) = (ex + e−x). ()
2
Note that, for any real number t,
1 1
cosh2(t) − sinh2(t) = (et + e−t)2 −(et − e−t)2
4 4
1 1
= (e2t + 2ete−t + e−2t) −(e2t − 2ete−t + e−2t)
4 4
1
= (2 + 2)
4
= 1.
Thus we have the useful identity
cosh2(t) − sinh2(t) = 1 ()
for any real number t. Put another way, (cosh(t), sinh(t)) is a point on the hyperbola
x2 −y2 = 1. Hence we see an analogy between the hyperbolic cosine and sine functions and
the cosine and sine functions: Whereas (cos(t), sin(t)) is a point on the circle x2 + y2 = 1,
(cosh(t), sinh(t)) is a point on the hyperbola x2 − y2 = 1. In fact, the cosine and sine
functions are sometimes referred to as the circular cosine and sine functions. We shall see
many more similarities between the hyperbolic trigonometric functions and their circular
counterparts as we proceed with our discussion.
To understand the graphs of the hyperbolic sine and cosine functions, we first note
that, for any value of x,
1
sinh(−x) = (e−x − ex) = − sinh(x), ()
2
1
2 Hyperbolic Functions Section
10
5
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Figure Graph of y = sinh(x)
and
1
cosh(−x) = (e−x + ex) = cosh(x). ()
2
Now for large values of x, e−x ≈ 0, in which case
1 1
sinh(x) = (ex − e−x) ≈= ex
2 2
and
1
sinh(−x) = − sinh(x) ≈