Math h Functions

Introduction:

The hyperbolic functions are defined as the analogs of common trigonometric functions. The fundamental hyperbolic functions include sin h called as the hyperbolic sine, cos h called the hyperbolic cosine and the tan h called the hyperbolic tangent function.

The hyperbolic functions are just the rational functions of the exponentials. The mathematical hyperbolic functions are also called as the math h functions. In this article we are going to see various math h functions.


Expressions of math h functions:


There are various mathematical algebraic expressions given to the math h functions.

The hyperbolic sine of x is given by sinh x = ½ (ex-e-x).

The hyperbolic cosine of x is given by cosh x = ½ (ex+e-x).

The hyperbolic tangent of x is given by tanh x = sinh x/cosh x

= (ex-e-x) / (ex+e-x)

= e2x-1 / e2x+1

The hyperbolic cosecant of x is given by csch x = (sinh x)-1

= 2 / ex-e-x

The hyperbolic secant of x is given by sech x = (cosh x)-1

= 2 / ex+e-x

The hyperbolic cotangent of x is given by coth x = cosh x / sinh x

= (ex+e-x) / (ex- e-x)

= e2x+1 / e2x-1 .


Math h functions with respect to circular functions:


The hyperbolic functions or math h function with respect to the circular functions is given by

x = a cos t and y = a sin t

Here the circle is given as a rectangular hyperbola.

The math h functions exist in many applications of mathematics which involve the integrals with v (1+x2) and the circular functions with v (1-x2)

The math h functions also include many identities similar to that of the trigonometric identites. The identites of the math h functions includes

Cosh2 x – sinh2 x = 1

Cosh x + sinh x = ex

Cosh x – sinh x = e-x

The identities for the complex arguments includes

Sinh (x+iy) = sinh x cos y + i cosh x sin y

Cosh (x+iy) = cosh x cosy + i sinh x sin y.