Hyperbolic Function

Hyperbolic functions in mathematics are linear of the average trigonometric functions but are determined using the hyperbola rather than the circle. Just as the constituents (cos t, sin t) form a circle with a unit radius, the points (cos h, sinh ) form the right half of the unit hyperbola. 

Hyperbolic functions are used in calculations of distances and angles in geometry hyperbolic. Laplace’s equations are important in many physics areas, including heat transfer, fluid dynamics, electromagnetic theory, and special relativity. In this article, we will discuss the integration formulas, integration by parts, and hyperbolic functions. 

Hyperbolic Functions

The functions that are very similar to trigonometric and circular functions are known as hyperbolic functions. The hyperbolic functions are mostly described with the help of algebraic expressions that indulge with inverse exponential functions(e-x) and exponential functions(ex), where e is known as the Euler’s constant. They also occur in the solutions of many linear differential equations, Laplace’s equation, cubic equation in Cartesian coordinates. The hyperbolic functions mostly include the real argument, also known as the hyperbolic angle. Generally, there are three basic hyperbolic functions, namely;

• Hyperbolic sine (sinh)

• Hyperbolic cosine (cosh)

• Hyperbolic tangent (tan h)

Furthermore, with the transformation of these hyperbolic functions, there can be many other functions such as cosecant(cosec h), hyperbolic secant(sech), and hyperbolic cotangent(cot h) function can be found or derived.

Hyperbolic Integration formulas 

Below-mentioned are the basic hyperbolic functions integration formulas.

The basic formulas of hyperbolic integral formulas function are described as:

• Sinh x = ex – e-x/2

• Cosh x = ex + e-x/2

• Tan h x = sinh/cosh =  ex – e-x/ ex + e-x

• Coth x = cosh x/ sinh x = ex + e-x/ ex – e-x

• Sec h x = 1/coth x =  2/ ex + e-x

• Cosech x = 1/sin h x = 2/ ex – e-x

Hyperbolic Functions Properties

The belongings of the hyperbolic integration formula functions correspond to the trigonometric functions. Some of them are:

1. Sinh (-x) = -sinh x

2. Cosh (-x) = cosh x

3. Cosh 2x = cosh2x + sinh2x

4. Sinh 2x =  2 sinh x cosh x

The deduced function of the hyperbolic function is:

1. d/dx sinh (x) = cosh x

2. d/dx cosh (x) = sinh x

Some of the basic relations of hyperbolic integral formulas into the trigonometric functions are as follows:

1. Sinh x = – i sin(ix)

2. Cosh x = cos (ix)

3. Tan h x = – i tan(ix)

What are Identities of Hyperbolic Function

The identities of trigonometric functions are very similar to identities of hyperbolic functions, such as;

Identities of Pythagorean Trigonometric

• Cos h2 (x) – sin2(x) = 1 

• Tan h2 (x) + sec h2 (x) = 1

• Cot h2 (x) – cosec h2 (x) = 1

Conversion of Sum into Product

• Sinh x + sinh y = 2 sinh((x+y/2) cosh((x-y)/2)

• Sinh x – sinh y = 2 cos((x+y)/2) cosh((x-y)/2)

• Cosh x + cosh y = 2 cosh((x+y)/2) sin h((x-y)/2)

Conversion of Product into Sum

• 2 sinh x cosh y = sinh(x+y) + sinh(x-y)

• 2 cosh x sinh y = sinh(x+y) – sinh(x-y) 

• 2 sinh x sinh y = cosh(x+y) – cosh(x-y)

• 2 cosh x cosh y = cosh(x+y) + cosh(x-y)

Identities Difference and Sum

• Sinh(x ± y) = sinh x cosh x ± cosh x sinh y

• Cosh(x ± y) = cosh x cosh y ± sinh x sinh y

• Tanh(x ± y) = (tanh x ± tanh y) / (1± tanh x tan h y)

• Coth(x ± y) = (coth x coth y ± 1) / (coth y ± coth x)

The Inverse of Hyperbolic Functions 

The inverse of hyperbolic functions is also called inverse hyperbolic function. The inverse of hyperbolic functions is also called the area hyperbolic function. The inverse of hyperbolic functions or inverse hyperbolic function provides the angles of hyperbolic, which correspond to the value of hyperbola given in the hyperbolic function. The function of inverse hyperbolic integration formula function is denoted as;

Sin h-1, tan h-1, cos h-1, sec h-1, cosec h-1, and cot-1. 

The functions of inverse hyperbolic function in a complex plane can be represented as;

• Sinh-1 x = ln(x + √[1+x2])

• Cosh-1 x = ln(x + √[x2-1])

• Tanh-1 x = (½)[ln(1+x) – ln(1-x)

Let’s talk another look at a hyperbolic function with the help of an example:

1. Find the solution of cosh2 x – sinh2 x

Solution: 

Cosh x = ex + e-x/2

Sinh x = ex – e-x/2

Cosh2 x – sinh2 x =  [ [ex + e-x]/2 ]2 – [ [ex – e-x]/2 ]2

Cosh2 x – sinh2 x = 4(1)/4 = 1

Therefore, cosh2 x – sinh2 x = 1

Conclusion 

There are six hyperbolic functions used: sinh x, tanh x, cosh x, cot h x, sec h x, and cosec h x. The properties of trigonometric functions correspond to the properties of the hyperbolic function.  The hyperbolic functions are only described or defined for a given hyperbola. A hyperbolic function can be a combination of functions that can be exponential. A function of a hyperbola can contain inverse exponential function(e-x) and exponential function (ex). The hyperbolic functions can also be differentiated into trigonometric functions resulting in complex arguments.