Hyperbolic Functions
Dr. Farhana Shaheen
Yanbu University College
KSA
Hyperbolic Functions


Vincenzo Riccati
(1707 - 1775) is
given credit for
introducing the
hyperbolic functions.
Hyperbolic functions are very useful
in both mathematics and physics.
The hyperbolic functions are:
Hyperbolic sine:
Hyperbolic
cosine
Equilateral hyperbola


x = coshα , y = sinhα
x2 – y2= cosh2 α - sinh2 α = 1.
GRAPHS OF HYPERBOLIC
FUNCTIONS

y = sinh x

y = cosh x
Graphs of cosh and sinh functions
The St. Louis arch is in the shape of a
hyperbolic cosine.
Hyperbolic Curves
y = cosh x

The curve formed by a hanging
necklace is called a catenary. Its
shape follows the curve of
y = cosh x.
Catenary Curves

The curve described by a uniform,
flexible chain hanging under the
influence of gravity is called a
catenary curve. This is the familiar
curve of an electric wire hanging
between two telephone poles.
Catenary curves in
Architecture

In architecture, an inverted
catenary curve is often used to
create domed ceilings. This shape
provides an amazing amount of
structural stability as attested by
fact that many of ancient structures
like the pantheon of Rome which
employed the catenary in their
design are still standing.
Masjid in Kazkhistan
Fatima masjid in Kuwait
Kul sharif masjid in Russia
Masjid in Georgia
Great Masjid in China
Catenary Curve

The curve is described by a
COSH(theta) function
Example of catenary and non-catenary
curves
Sinh graphs
Graphs of tanh and coth functions

y = tanh x

y = coth x
Graphs of sinh, cosh, and tanh
Graphs of sech and csch functions

y = sech x

y = csch x

Useful relations
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
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
Hence:
1 - (tanh x)2 = (sech x)2.
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RELATIONSHIPS OF HYPERBOLIC
FUNCTIONS
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
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tanh x = sinh x/cosh x
coth x = 1/tanh x = cosh x/sinh x
sech x = 1/cosh x
csch x = 1/sinh x
cosh2x - sinh2x = 1
sech2x + tanh2x = 1
coth2x - csch2x = 1

The following list shows the
principal values of the inverse
hyperbolic functions expressed in
terms of logarithmic functions which
are taken as real valued.


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sinh-1 x = ln (x +
)
-∞ < x < ∞
cosh-1 x = ln (x +
)
x≥1
[cosh-1 x > 0 is principal value]
tanh-1x = ½ln((1 + x)/(1 - x))
-1 < x
<1
coth-1 x = ½ln((x + 1)/(x - 1))
x>1
or x < -1
sech-1 x = ln ( 1/x +
)
0 < x ≤ 1 [sech-1 a; > 0 is principal
value]
csch-1 x = ln(1/x +
)
x≠0
Hyperbolic Formulas for Integration

u
 sinh    C or ln (u  u 2  a 2 )
a
a2  u 2

u
 cosh    C or ln (u  u 2  a 2 )
a
u 2  a2
du
du
1
1
du
1
1
au
1  u 
 a 2  u 2  a tanh  a   C , u  a or 2a ln a  u  C, u  a
Hyperbolic Formulas for Integration
a  a2  u2
1
1
1 u
)  C, 0  u  a
 u a 2  u 2   a sec h a  C or  a ln ( u
du
RELATIONSHIPS OF HYPERBOLIC FUNCTIONS
a  a2  u2
1
1
1 u
)  C , u  0.
 u a 2  u 2   a csc h a  C or  a ln ( u
du

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The hyperbolic functions share many properties with
the corresponding circular functions. In fact, just as
the circle can be represented parametrically by
x = a cos t
y = a sin t,
a rectangular hyperbola (or, more specifically, its
right branch) can be analogously represented by
x = a cosh t
y = a sinh t
where cosh t is the hyperbolic cosine and sinh t is
the hyperbolic sine.

Just as the points (cos t, sin t) form
a circle with a unit radius, the
points (cosh t, sinh t) form the right
half of the equilateral hyperbola.
Animated plot of the trigonometric
(circular) and hyperbolic functions

In red, curve of equation
x² + y² = 1 (unit circle),
and in blue,
x² - y² = 1 (equilateral hyperbola),
with the points (cos(θ),sin(θ)) and
(1,tan(θ)) in red and
(cosh(θ),sinh(θ)) and (1,tanh(θ)) in
blue.
Animation of hyperbolic functions
Applications of Hyperbolic functions

Hyperbolic functions occur in the
solutions of some important linear
differential equations, for example
the equation defining a catenary,
and Laplace's equation in Cartesian
coordinates. The latter is important
in many areas of physics, including
electromagnetic theory, heat
transfer, fluid dynamics, and special
relativity.


The hyperbolic functions arise in many
problems of mathematics and
mathematical physics in which integrals
involving a 2  x 2 arise (whereas the
circular functions involve a 2  x 2 ).
For instance, the hyperbolic sine
arises in the gravitational potential of a
cylinder and the calculation of the Roche
limit. The hyperbolic cosine function is
the shape of a hanging cable (the socalled catenary).

The hyperbolic tangent arises in the
calculation and rapidity of special
relativity. All three appear in the
Schwarzschild metric using external
isotropic Kruskal coordinates in general
relativity. The hyperbolic secant arises
in the profile of a laminar jet. The
hyperbolic cotangent arises in the
Langevin function for magnetic
polarization.
Derivatives of Hyperbolic Functions

d/dx(sinh(x)) = cosh(x)

d/dx(cosh(x)) = sinh(x)

d/dx(tanh(x)) = sech2(x)
Integrals of Hyperbolic Functions

∫ sinh(x)dx = cosh(x) + c

∫ cosh(x)dx = sinh(x) + c.

∫ tanh(x)dx = ln(cosh x) + c.
Example :
Find d/dx (sinh2(3x))
Sol: Using the chain rule,
we have:

d/dx (sinh2(3x))
= 2 sinh(3x) d/dx (sinh(3x))
= 6 sinh(3x) cosh(3x)
Inverse hyperbolic functions


1
d (sinh−1 (x)) =
dx
1 x 2
1
d
(cosh−1 (x)) =
dx
x 2 1
d
(tanh−1 (x)) =
dx
1
1 x 2
Curves on Roller Coaster Bridge
Thank You
Animation of a Hypotrochoid
Complex Sinh.jpg
Sine/Cos Curves