1
Hyperbolic Functions
The hyperbolic functions are defined as
Hyperbolic
ex − e−x
sinh x =
2
ex + e−x
cosh x =
2
x
−x
e −e
sinh(x)
tanh x = x
=
−x
e +e
cosh(x)
Circular
eix − e−ix
sin x =
2
eix + e−ix
cos x =
2
ix
e − e−ix
sin(x)
tan x = ix
=
−ix
e +e
cos(x)
with cothx = 1/ tanh x, sechx = 1/ cosh x, and cosechx = 1/ sinh x also defined. These
are analagous to the definitions of the trignometric (circular) functions in terms of e,
as also given in the table above. For purely imaginary arguments you have cosh(iz) =
cos(z), sinh(iz) = i sin(z), which can be checked from the definitions above.
Recall the so-called circular functions: if x = cos θ, y = sin θ then as θ varies x2 + y 2 =
cos2 θ + sin2 θ = 1 sweeps out a circle.
Now, defining x = cosh θ, y = sinh θ then as θ varies a hyperbola, x2 − y 2 = 1 is swept
out. Actually the right branch of a hyperbola is swept out - for the other branch write,
x = − cosh θ.
1.1
Graphs of the Hyperbolic Functions
1
1.2
Properties
• sinh(x): an odd function ie sinh(−x) = − sinh(x) as for sin(x), and sinh(0) = 0
• cosh(x): an even function ie cosh(−x) = cosh(x) as for cos(x), and cosh(0) = 1. Also
cosh(x) ≥ 1∀x (unlike cos(x)).
• tanh(x): an odd function ie tanh(−x) = − tanh(x) as for tan(x), and tanh(0) = 0.
Also, −1 ≤ tanh(x) ≤ 1∀x (unlike tan(x)).
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1.3
Inverse Functions
The hyperbolic functions have associated inverse functions (as the trigonmetric functions
do) written sinh−1 x, cosh−1 x etc. Using the definitions in the first section it is easy to
show the following identities:
p
sinh−1 y = ln(y + y 2 + 1)
p
p
cosh−1 y = ± ln(y + y 2 − 1) = ln(y ± y 2 − 1)
1
1+y
−1
tanh y =
ln
2
1−y
1.4
Derivatives
From the definitions we have
d
d ex − e−x
ex + e−x
sinh(x) =
=
= cosh(x)
dx
dx
2
2
d
cosh(x) = sinh(x)
dx
d
tanh(x) = sech2 (x)
dx
For the inverses of the hyperbolic functions consider
d
sinh−1 (u) = 1/ cosh(y)
du
(1)
where y = sinh−1 u so u = sinh y and du/dy = cosh y so that using (du/dy)(dy/du) = 1
the result above follows. Using the identity cosh2 y − sinh2 y = 1 we get
d
1
sinh−1 u = √
du
1 + u2
(2)
and similarly
d
1
cosh−1 u = √
, cosh−1 u > 0
2
du
u −1
−1
= √
, cosh−1 u < 0
2
u −1
and so we get some useful identities for integration, namely
Z
Z
1
1
−1
√
√
du = cosh u + C ;
du = sinh−1 u + C
2
2
u −1
u +1
3
(3)
1.5
Uses
Many! Here are some ...
• in the solution of differential equations, eg cosh(x) describes a catenary curve ie a
hanging cable.
• in a formulation of the Schwarzchild metric in GR
• hyperbolic secant appears in the modelling function of the profile of a laminar jet.
• in complex analysis they are rational functions of exponentials → meromorphic.
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