# Power Series expansion of hyperbolic functions

• thanksie037
In summary, the conversation discusses finding the power series expansion of a given function, specifically ((cosh x)/(sinh x)) - (1/x). The equations for cosh x and sinh x are also mentioned, and the conversation includes a discussion about simplifying the function and finding its Taylor series.
thanksie037

## Homework Statement

power series expansion of:

((cosh x)/(sinh x)) - (1/x)

## Homework Equations

cosh x = (1/2)(ex + e-x)
sinh x = (1/2)(ex - e-x)

## The Attempt at a Solution

what i have so far:

I simplified the first part of the eq to read :
e2x-1
e2x-1

You simplified for it to be $$\frac{e^{2x}-1}{e^{2x}-1}$$? Isn't that just 1?

error in simplification:

$$\frac{e^{2x}+1}{e^{2x}-1}$$

I'm sorry that was a typo. Should I just expand both was like you would ex? how about the 1/x part?

I would and then hopefully things will cancel, for example what's the expansion for $$e^{2x} - 1$$?

How do one usually find the taylor series of a given function?
now you have a function, what do you do?

## 1. What are hyperbolic functions?

Hyperbolic functions, also known as hyperbolic trigonometric functions, are mathematical functions that are analogous to the trigonometric functions used in traditional geometry. They are defined in terms of the exponential function and are used to describe hyperbolic geometry, as well as many other phenomena in mathematics and physics.

## 2. What is a power series expansion?

A power series expansion is a representation of a function as an infinite sum of powers of a variable. In other words, it is a way of expressing a function as a polynomial with an infinite number of terms. This is useful for approximating and analyzing functions, especially those that are difficult to evaluate directly.

## 3. How are power series expansions related to hyperbolic functions?

Power series expansions can be used to approximate and evaluate hyperbolic functions. By expressing a hyperbolic function as a power series, we can break it down into simpler terms and calculate its value at any point. This is especially useful for complex functions, such as hyperbolic functions, that cannot be evaluated directly.

## 4. What is the general form of a power series expansion of a hyperbolic function?

The general form of a power series expansion of a hyperbolic function is:
f(x) = a0 + a1x + a2x^2 + a3x^3 + ... + anx^n
where a0, a1, a2, ... , an are constants and x is the variable. The value of x can be any real number, and the more terms included in the expansion, the more accurate the approximation will be.

## 5. How are hyperbolic functions and their power series expansions used in real-world applications?

Hyperbolic functions and their power series expansions have numerous applications in mathematics, physics, and engineering. They are used to model and solve problems related to waves, vibrations, and oscillations, as well as in the study of surfaces and volumes in hyperbolic geometry. They are also used in financial mathematics, signal processing, and statistics.

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