# Why are hyperbolic functions defined in terms of exponentials?

• neginf
In summary, the definitions of hyperbolic functions in terms of exponentials may have originated from someone trying to substitute i*x for x in the formulas for e^ix and e^-ix, resulting in functions that behaved similarly to regular trigonometric functions. However, the hyperbolic trigonometric functions were originally defined geometrically based on the rectangular hyperbola. It is possible to derive the definitions in terms of exponentials from the geometric definition, but they are more commonly defined using power series of complex variables. These functions have many useful applications in advanced mathematics and physics, despite their origins in trigonometry.
neginf
Where do the definitions of hyperbolic functions in terms of exponentials come from ?

Maybe somebody tried e^i*x=cos x+i*sin x and e^-i*x=cos x-i*sin x with i*x instead of x and got functions that acted a lot like regular trig functions so they named them a lot like trig functions, like cos(i*x)=cosh x.
Could that be where those definitions come from ?
If so,do they come from somewhere else also ?

AFAIK, the hyperbolic trigonometric functions were originally defined geometrically, analogously to the ordinary (circular) trigonometric functions, just based on the rectangular hyperbola rather than the circle.

Thank you for that.
I wonder if the definitions in terms of exponentials can be gotten from the geometric definition. That would seem more natural than adding and subtracting trig functions with imaginary arguments.

neginf said:
Thank you for that.
I wonder if the definitions in terms of exponentials can be gotten from the geometric definition. That would seem more natural than adding and subtracting trig functions with imaginary arguments.

The most "obvious" formula from the geometry of a hyperbola would be cosh^2 x = sinh^2 x + 1.

Another "obvious" starting point from a rectangular hyperbola is to define log(x) as the integral of 1/x, and exp(x) as the inverse of log(x).

From the point of view of "advanced" math, probably the simplest way to define the trig hyperbolic and exponential functions is using power series of complex variables. Then the relations between them are obvious (and they are all so-called analytic functions defined for all complex arguments, which means they have lots of nice properties), but you then have to prove they have something to do with angles, circles, and hyperbolas. But many of the their uses in "advanced" math and physics don't have much to do with angles and hyperbolas anyway...

## 1. Why are hyperbolic functions important in mathematics?

Hyperbolic functions are important in mathematics because they have a wide range of applications in various fields such as physics, engineering, and statistics. They are particularly useful in solving problems involving curves and surfaces.

## 2. How are hyperbolic functions related to exponential functions?

Hyperbolic functions are defined in terms of exponential functions. Specifically, the hyperbolic sine (sinh) and cosine (cosh) functions can be expressed as combinations of the exponential function e^x and its inverse, ln(x).

## 3. What is the purpose of defining hyperbolic functions in terms of exponentials?

The purpose of defining hyperbolic functions in terms of exponentials is to make calculations and equations involving hyperbolic functions easier. This is because exponential functions have well-established properties and can be manipulated algebraically, making it simpler to work with hyperbolic functions.

## 4. Are there any real-life applications of hyperbolic functions?

Yes, hyperbolic functions have many real-life applications. For example, they are used in modeling the shape of a hanging chain or cable, calculating the trajectory of a satellite orbit, and analyzing heat and diffusion processes.

## 5. Can hyperbolic functions be graphed?

Yes, just like any other mathematical function, hyperbolic functions can also be graphed. They have distinct curves that resemble the graphs of sine and cosine functions, but with different properties and behaviors.

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