2.4
The Hyperbolic Functions
These are constructed from the exponential function, but obey identities closely analogous
to those satisfied by the corresponding trigonometric functions (obtained by dropping the
final “h”).
2.4.1
Definitions
cosh x =
sinh x =
sinh x
cosh x
cosh x
1
=
coth x =
tanh x
sinh x
1
sech x =
cosh x
1
cosech x =
sinh x
tanh x =
2.4.2
=
=
=
=
1 x
(e + e−x ),
2
1 x
(e ¡ e−x ),
2
ex ¡ e−x
1 ¡ e−2x
e2x ¡ 1
,
=
=
ex + e−x
1 + e−2x
e2x + 1
ex + e−x
1 + e−2x
e2x + 1
=
=
ex ¡ e−x
1 ¡ e−2x
e2x ¡ 1
2
,
x
e + e−x
2
(x 6= 0).
x
e ¡ e−x
(2.31)
(2.32)
(2.33)
(x 6= 0),(2.34)
(2.35)
(2.36)
The Fundamental Identity
Corresponding to the well-known identity cos2 x + sin2 x = 1 for the trigonometric functions, we have for the hyperbolic functions the identity
cosh2 x ¡ sinh2 x = 1
(2.37)
(holding for all real x). NB the minus sign! To prove (2.37), just substitute the definitions
(2.31) and (2.32) into the LHS, to get
1 x −x 2 1 x −x 2 1 2x −2x
1
(e +e ) ¡ (e ¡e ) = (e +e +2ex e−x )¡ (e2x +e−2x ¡2ex e−x ) = ex e−x = ex−x = e0 = 1.
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4
4
4
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2.4.3
Further Identities
1 ¡ tanh2 x
coth2 x ¡ 1
sinh(x + y)
sinh(x ¡ y)
cosh(x + y)
cosh(x ¡ y)
sinh 2x
cosh 2x
=
=
=
=
=
=
=
=
cosh2 x =
sinh2 x =
sinh x cosh y =
cosh x cosh y =
sinh x sinh y =
sech2 x,
cosech2 x,
sinh x cosh y + cosh x sinh y,
sinh x cosh y ¡ cosh x sinh y,
cosh x cosh y + sinh x sinh y,
cosh x cosh y ¡ sinh x sinh y,
2 sinh x cosh x,
sinh2 x + cosh2 x,
1
(cosh 2x + 1),
2
1
(cosh 2x ¡ 1),
2
1
[sinh (x + y) + sinh (x ¡ y)] ,
2
1
[cosh (x + y) + cosh (x ¡ y)] ,
2
1
[cosh (x + y) ¡ cosh (x ¡ y)] .
2
(2.38)
(2.39)
(2.40)
(2.41)
(2.42)
(2.43)
(2.44)
(2.45)
(2.46)
(2.47)
(2.48)
(2.49)
(2.50)
Each of these has a trigonometric counterpart (see Appendix C), but there are some
important differences of sign, which should be carefully noted. The identities are all easy
to prove from the definitions above.
2.4.4
The Graphs of cosh, sinh and tanh
Using the definitions of the hyperbolic functions, and our knowledge of the exponential function, we can sketch the following graphs and deduce some properties for the
hyperbolic functions.
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cosh 0 = 1,
cosh x ! +1 as x ! §1,
cosh (¡x) = cosh (x) (i.e. it is an EVEN function).
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sinh 0
sinh x
sinh x
sinh (¡x)
=
!
!
=
0,
+1 as x ! +1,
¡1 as x ! ¡1,
¡ sinh (x) (i.e. it is an ODD function).
tanh 0
tanh x
tanh x
tanh (¡x)
=
!
!
=
0,
+1 as x ! +1,
¡1 as x ! ¡1,
¡ tanh (x) (i.e. it is an ODD function).
Like all even functions, cosh has a a graph which is symmetrical about Oy, i.e. invariant under reflection in Oy. It has a minimum value of 1 at x = 0, where its derivative
sinh x is zero. On the other hand, the graphs of the functions sinh and tanh, like those of
all odd functions, are invariant under rotation through 180◦ about O. Note that sinh and
tanh are both increasing functions and that the graph of tanh has horizontal asymptotes
y = §1 approached as x ! §1 respectively.
2.4.5
Symmetry and Asymptotic Properties of other hyperbolic
functions
Similarly, we can deduce the following properties of coth, sech and cosech:
coth(¡x) = ¡ coth x, sech (¡x) = sechx, cosech (¡x) = ¡cosechx.
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coth x ! 1 as x ! +1,
coth x ! +1 as x ! 0 from above ,
sech x ! 0
cosech x ! 0
sech 0 = 1
cosech x ! +1 as x ! 0 from above,
coth x ! ¡1 as x ! ¡1,
coth x ! ¡1 as x ! 0, from below,
as x ! §1,
as x ! §1,
cosech x ! ¡1 as x ! 0 from below.
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