# Fourier Series Expansion of coshx

• rayman123
In summary, the task is to expand the function f(x) = coshx, |x|\leq \pi into a Fourier series. The equation for the Fourier series is C_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}(\frac{e^{x}+e^{-x}}{2}})e^{-inx}}dx, which can be calculated by solving two separate integrals. The final result for the Fourier series is C_{n}=\frac{(-1)^n}{2(1-in)}[2].
rayman123

## Homework Statement

Expand the function into Fourier series
$$f(x) = coshx, |x|\leq \pi$$

## Homework Equations

Fourier series will be
$$C_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}(\frac{e^{x}+e^{-x}}{2}})e^{-inx}}dx$$

$$\frac{1}{4\pi}\int_{-\pi}^{\pi}({e^{x}e^{-inx})dx+ \frac{1}{4\pi}\int_{-\pi}^{\pi}(e^{-x}e^{-inx})dx$$

## The Attempt at a Solution

I calculate both integrals separately

$$\frac{1}{4\pi}\int_{-\pi}^{\pi}({e^{x}e^{-inx})dx=\frac{1}{4\pi}\int_{-\pi}^{\pi}({e^{x-inx})dx=\frac{1}{4\pi}\frac{e^{x-inx}}{1-in}$$

I substitute $$x=\pm\pi \Rightarrow \frac{(-1)^n}{1-in}[e^{\pi}-e^{-\pi}]$$

## Homework Statement

and the other one gives me

$$\frac{(-1)^n}{-1-in}[e^{-\pi}-e^{\pi}]$$

is this correct so far?

Finally I add them up and get the Fourier series C_{n}=\frac{(-1)^n}{2(1-in)}[e^{-\pi}+e^{\pi}]=\frac{(-1)^n}{2(1-in)}[2]

## 1. What is the Fourier series expansion of coshx?

The Fourier series expansion of coshx is an infinite sum of cosine functions with different frequencies and amplitudes. It is represented by the equation:
coshx = 1 + x^2/2! + x^4/4! + x^6/6! + ... = ∑ (x^(2n)/(2n)!), where n = 0, 1, 2, ...

## 2. How is the Fourier series expansion of coshx derived?

The Fourier series expansion of coshx is derived using the Fourier series representation of a function, which states that any periodic function can be represented as a sum of trigonometric functions. By applying this representation to coshx, we can obtain the infinite series mentioned above.

## 3. What is the significance of the Fourier series expansion of coshx?

The Fourier series expansion of coshx is significant in mathematics and physics as it allows us to approximate a periodic function with a finite number of terms. This can be useful in solving differential equations and analyzing the behavior of various physical systems.

## 4. What is the domain and range of the Fourier series expansion of coshx?

The domain of the Fourier series expansion of coshx is all real numbers, while the range is from 1 to positive infinity. This means that the series is valid for any value of x, but the resulting sum will always be positive and never equal to 0 or a negative number.

## 5. Can the Fourier series expansion of coshx be used for non-periodic functions?

No, the Fourier series expansion of coshx can only be used for periodic functions. This is because the series relies on the periodicity of the function to accurately represent it as a sum of trigonometric functions. For non-periodic functions, other methods such as the Fourier transform are used for approximation.

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