What is Hyperbolic Function?

What is Hyperbolic Function?

Hyperbolic functions

We know that parametric co-ordinates of any point on the unit circle x2 + y2 = 1 is (cos θ, sin θ); so that these functions are called circular functions and co-ordinates of any point on unit hyperbola is
What is Hyperbolic Function 1
It means that the relation which exists amongst cos θ, sin θ and unit circle, that relation also exist amongst cosh θ, sinh θ and unit hyperbola. Because of this reason these functions are called as Hyperbolic functions.
For any (real or complex) variable quantity x,
What is Hyperbolic Function 2

Domain and range of hyperbolic functions

Let x is any real number

FunctionDomainRange
sinh xRR
cosh xR[1, ∞)
tanh xR(–1, 1)
coth xR0R – [–1, 1]
sech xR(0, 1]
cosech xR0R0

Graph of real hyperbolic functions

What is Hyperbolic Function 3

Formulae for hyperbolic functions

The following formulae can easily be established directly from above definitions
(1) Reciprocal formulae
What is Hyperbolic Function 4
(2) Square formulae
What is Hyperbolic Function 5
(3) Sum and difference formulae
What is Hyperbolic Function 6
(4) Formulae to transform the product into sum or difference
What is Hyperbolic Function 7
(5) Trigonometric ratio of multiple of an angle
What is Hyperbolic Function 8

Transformation of a hyperbolic functions

What is Hyperbolic Function 9
In a similar manner we can express cosh x, tanh x, coth x, ……………. in terms of other hyperbolic functions.

Expansion of hyperbolic functions

What is Hyperbolic Function 10
The expansion of coth x, cosech x does not exist because coth(0) = ∞ and cosech(0) = ∞.

Relation between hyperbolic, circular functions

What is Hyperbolic Function 11
Thus, we obtain the following relations between hyperbolic and trigonometrical functions.

(1)sin(ix) = i sinh x(2)cos(ix) = cosh x
sinh (ix) = i sin xcosh (ix) = cos x
sinh x = −i sin (ix)cosh x = cos (ix)
sin x = −i sinh (ix)cos x = cosh (ix)
(3)tan(ix) = i tanh x(4)cot(ix) = −i coth x
tanh (ix) = i tan xcoth (ix) = −i cot x
tanh x = −i tan(ix)coth x = i cot(ix)
tan x = −i tanh (ix)cot x = i coth (ix)
(5)sec(ix) = sech x(6)cosec(ix) = −i cosech x
sech (ix) = sec xcosech (ix) = −i cosec x
sech x = sec (ix)cosech x = i cosec (ix)
sec x = sech (ix)cosec x = i cosech (ix)

Period of hyperbolic functions

If for any function f(x), f(x +T) = f(x), then f(x) is called the Periodic function  and least positive value of T  is called the Period of the function.
∵ sinh x = sinh(2πi + x); cosh x = cosh(2πi + x) and tanh x = tanh(πi + x)
Therefore the period of these functions are respectively 2πi, 2πi and πi. Also period of cosech x, sech x and coth x are respectively 2πi, 2πi and πi.

  • Remember that if the period of f(x) is T, then period of f(nx) will be (T/n).
  • Hyperbolic function are neither periodic functions nor their curves are periodic but they show the algebraic properties of periodic functions and having imaginary period.

Inverse hyperbolic functions

If sinh y = x, then y is called the inverse hyperbolic sine of x and it is written as y = sinh1 x. Similarly cosech1 x, cosh1 x, tanh1 x etc. can be defined.
(1) Domain and range of Inverse hyperbolic function 

FunctionDomainRange
sinh1 xRR
cosh1 x[1, ∞)R
tanh1 x(–1, 1)R
coth1 xR – [–1, 1]R0
sech1 x(0, 1]R
cosech1 xR0R0

(2) Relation between inverse hyperbolic function and inverse circular function
What is Hyperbolic Function 12
(3) To express any one inverse hyperbolic function in terms of the other inverse hyperbolic functions
What is Hyperbolic Function 13
If x is real then all the above six inverse functions are single valued.
(4) Relation between inverse hyperbolic functions and logarithmic functions
What is Hyperbolic Function 14
By the above method we can obtain the following relations between inverse hyperbolic functions and principal values of logarithmic functions.
What is Hyperbolic Function 15
Formulae for values of cosech1 x, sech1 x, and coth1 x may be obtained by replacing x by 1/x in the values of sinh1 x, cosh1 x and tanh1 x respectively.

Separation of inverse trigonometric and inverse hyperbolic functions

If sin(α + iβ) = x + iy then (α + iβ), is called the inverse sine of (x + iy). We can write it as,  sin1(x + iy) = α + iβ.
Here the following results for inverse functions may be easily established.
What is Hyperbolic Function 16
Since each inverse hyperbolic function can be expressed in terms of logarithmic function, therefore for separation into real and imaginary parts of inverse hyperbolic function of complex quantities use the appropriate method.
Both inverse circular and inverse hyperbolic functions are many valued.