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Problem 400

# Let $$\mathrm{f}(\mathrm{x})$$ be a real valued function of one variable, $$\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$$, with a period $$2 c$$ (i.e. $$\mathrm{f}(\mathrm{x}+2 \mathrm{c})=\mathrm{f}(\mathrm{x})$$ for all $$\mathrm{x} \in \mathrm{R}$$ ). Define the Fourier series of $$\mathrm{f}$$.

Expert verified
The Fourier series representation of a real-valued periodic function $$f(x)$$ with period $$2c$$ is given by: $f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos{(\frac{n\pi x}{c})} + b_n \sin{(\frac{n\pi x}{c})})$ The coefficients can be found using the following formulas: 1. $$a_0 = \frac{1}{c} \int_{-c}^{c} f(x)dx$$ 2. $$a_n = \frac{1}{c} \int_{-c}^{c} f(x)\cos{(\frac{n\pi x}{c})}dx$$ 3. $$b_n = \frac{1}{c} \int_{-c}^{c} f(x)\sin{(\frac{n\pi x}{c})}dx$$ For n = 1, 2, 3, ... To compute the Fourier coefficients, substitute the given function $$f(x)$$ (if provided) into the formulas and integrate over the interval $$[-c, c]$$.
See the step by step solution

## Step 1: Finding the Fourier Series Representation

The general form of a Fourier series for a function with period 2c can be represented as: $f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos{(\frac{n\pi x}{c})} + b_n \sin{(\frac{n\pi x}{c})})$ To find the Fourier coefficients $$a_0, a_n$$, and $$b_n$$, we will use the following formulas: 1. $$a_0 = \frac{1}{c} \int_{-c}^{c} f(x)dx$$ 2. $$a_n = \frac{1}{c} \int_{-c}^{c} f(x)\cos{(\frac{n\pi x}{c})}dx$$ 3. $$b_n = \frac{1}{c} \int_{-c}^{c} f(x)\sin{(\frac{n\pi x}{c})}dx$$ For n = 1, 2, 3, ...

## Step 1: Determine the Period and Coefficients Formulas

First, identify the period of the function, which is given as 2c. Using the period, we can write down the general form of the Fourier series and the formulas to find the coefficients: Function: $$f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos{(\frac{n\pi x}{c})} + b_n \sin{(\frac{n\pi x}{c})})$$ Coefficients: $$a_0 = \frac{1}{c} \int_{-c}^{c} f(x)dx$$ $$a_n = \frac{1}{c} \int_{-c}^{c} f(x)\cos{(\frac{n\pi x}{c})}dx$$ $$b_n = \frac{1}{c} \int_{-c}^{c} f(x)\sin{(\frac{n\pi x}{c})}dx$$

## Step 2: Compute the Coefficients

The problem did not specify the form of the function f(x). To find the unknown coefficients for the general Fourier series representation, we would need to know the function f(x). Once the function f(x) is given, you can substitute it into the coefficients formulas above and integrate over the interval [-c, c] to find the Fourier coefficients.

## Step 4: Conclusion

The Fourier series representation of any periodic function allows us to represent it as an infinite series of trigonometric functions. For a function with period 2c, the general Fourier series form and the coefficients can be computed. However, without the specific function f(x), we cannot compute the Fourier coefficients in this problem. Upon obtaining f(x), the formulas provided can be utilized to find the Fourier series representation of the given periodic function.

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