# Origin of hyperbolic functions

• SpartanG345
In summary, hyperbolic functions are mathematical functions that are the solutions to various elementary forms of differential equations and are the basis for some orthogonal function expansions. They play a similar role with hyperbolas as circular trig functions do with circles and ellipses. They have various applications, including in special relativity where they are used to make changes of coordinates that preserve distances between points. The analogy between hyperbolic and trig functions was first discovered in 1757 by Riccati.
SpartanG345
I read the wiki on hyperbolic functions but i don't really understand. I also plotted cosht,sinht on wolfram it made sideways v lying on the x axis. Can anyone explain why people made hyperbolic functions.

I still don't really understand what it means apart from cosht being the x coordinate of the intercept of a ray passing through the origin intercepts a hyperbola, where the rays angle is 2A

I can think off the top of the head they are solutions to various elementary forms of differential equations and the basis for some orthogonal function expansions.

The hyperbolic trig functions play the same role with hyperbolas that the ordinary (circular) trig functions play with circles and ellipses.

I found this page useful when I first met the hyperbolic functions in the context of special relativity. This is just one example of an application, rather than a comment how they originated, but you might find it interesting as it shows another parallel between circular and hyperbolic functions.

If you rotate an orthonormal coordinate system for Euclidean space, the way the coordinates of any point change can be expressed with a matrix of circular functions. For example, a rotation about the z axis:

$$\begin{bmatrix}\cos{\theta} & \sin{\theta} & 0\\ -\sin{\theta} & \cos{\theta} & 0\\ 0 & 0 & 1\\\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}$$

In special relativity, besides rotating an orthonormal coordinate system for spacetime in this way, you can make another kind of change of coordinates that also preserves (spacetime) distances between points. This switches to a coordinate system moving at some velocity relative to the one you started with, for example moving along the x axis:

$$\begin{bmatrix}\cosh{\phi} & -\sinh{\phi} & 0 & 0\\ -\sinh{\phi} & \cosh{\phi} & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}t\\x\\ y\\ z\end{bmatrix}$$

where $\phi = \text{tanh}^{-1}\left({\frac{v}{c}}\right)$, the inverse hyperbolic tangent, "artanh", of the speed of the new coordinate system as a fraction of the speed of light.

Based on a recent book I read on the history of the number e, y = 1/2(ex+e-x), which is cosh(x), is the solution to the 'hanging chain' problem posed by Jakob Bernoulli in 1690. One year later, three correct solutions were published (Leibniz, Johann Bernoulli, and Huygens). The analogy between the hyperbolic and trig functions appears to have been discovered in 1757 by Riccati.

## 1. What are hyperbolic functions?

Hyperbolic functions are a set of mathematical functions that are closely related to the trigonometric functions. They are defined in terms of the exponential function and are used to describe the behavior of hyperbolic curves.

## 2. When and how were hyperbolic functions discovered?

Hyperbolic functions were first introduced in the 18th century by Swiss mathematician Leonhard Euler. He discovered them while studying the properties of logarithms and exponential functions.

## 3. What is the origin of the term "hyperbolic" in hyperbolic functions?

The term "hyperbolic" comes from the Greek word "hyperbolē" meaning "excess" or "exaggeration". This refers to the fact that hyperbolic functions have a similar shape to hyperbolas, which are a type of conic section.

## 4. What are the main applications of hyperbolic functions?

Hyperbolic functions have many practical applications in mathematics, physics, engineering, and other fields. They are used to model natural phenomena such as heat flow, population growth, and electrical circuits.

## 5. How are hyperbolic functions related to trigonometric functions?

Hyperbolic functions and trigonometric functions share many similar properties and identities. In fact, they can be expressed in terms of each other using complex numbers. This relationship is known as the Euler's formula for complex numbers.

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