Maclaurin Series for Hyperbolic Functions: An In-Depth Guide

The Maclaurin series is a fundamental concept in the study of calculus and series expansion. It allows us to express complex functions in a simpler form, making it easier to analyze and understand their behavior. One such interesting application is in the expansion of hyperbolic functions, specifically sineh (sinhx) and coshx. This guide will explore how the Maclaurin series is used to derive the expansions of these functions from the exponential function.

Definition and Simplification

The hyperbolic sine (sinhx) and hyperbolic cosine (coshx) functions are defined as follows:

1. Hyperbolic Sine (sinhx)

$$sinhx frac{e^x - e^{-x}}{2}$$

2. Hyperbolic Cosine (coshx)

$$coshx frac{e^x e^{-x}}{2}$$

Maclaurin Series for the Exponential Function

The Maclaurin series for the exponential function (e^x) is a powerful tool that allows for the expansion of the function into an infinite series. This series is given by:

$$e^x 1 x frac{x^2}{2!} frac{x^3}{3!} frac{x^4}{4!} ldots frac{x^n}{n!} ldots$$

Similarly, the Maclaurin series for (e^{-x}) is:

$$e^{-x} 1 - x frac{x^2}{2!} - frac{x^3}{3!} frac{x^4}{4!} - ldots (-1)^n frac{x^n}{n!} ldots$$

Derivation of Maclaurin Series for sinh(x) and cosh(x)

To derive the Maclaurin series for sinh(x) and coshx, we start by subtracting the series for (e^{-x}) from (e^x).

1. Maclaurin Series for sinh(x)

Subtracting (e^{-x}) from (e^x), we get:

$$e^x - e^{-x} (1 x frac{x^2}{2!} frac{x^3}{3!} frac{x^4}{4!} ldots) - (1 - x frac{x^2}{2!} - frac{x^3}{3!} frac{x^4}{4!} - ldots)$$

Simplifying, we obtain:

$$e^x - e^{-x} 2x 2frac{x^3}{3!} 2frac{x^5}{5!} ldots$$

Dividing by 2, we get the Maclaurin series for sinh(x):

$$sinhx x frac{x^3}{3!} frac{x^5}{5!} ldots frac{x^{2n-1}}{(2n-1)!} ldots$$

2. Maclaurin Series for cosh(x)

Adding (e^{-x}) to (e^x), we get:

$$e^x e^{-x} (1 x frac{x^2}{2!} frac{x^3}{3!} frac{x^4}{4!} ldots) (1 - x frac{x^2}{2!} - frac{x^3}{3!} frac{x^4}{4!} - ldots)$$

Simplifying, we obtain:

$$e^x e^{-x} 2 2frac{x^2}{2!} 2frac{x^4}{4!} ldots$$

Dividing by 2, we get the Maclaurin series for coshx:

$$coshx 1 frac{x^2}{2!} frac{x^4}{4!} ldots frac{x^{2n}}{(2n)!} ldots$$

Understanding the Properties

It is important to understand the properties and applications of these series. Since sinh(x) is an odd function, its Maclaurin series contains only odd powers of (x). On the other hand, coshx is an even function, so its series contains only even powers of (x).

Conclusion

The Maclaurin series for hyperbolic functions are crucial in various mathematical and scientific applications. By breaking down the exponential functions and using subtraction and addition, we can derive these series. This approach not only simplifies complex functions but also deepens our understanding of the underlying mathematical principles.

Keywords: Maclaurin series, hyperbolic functions, exponential functions