You are given f(x) on an interval, say 0< x< b. Sketch...

You are given $f(x)$ on an interval, say $0< x< b$. Sketch several periods of the even function $f_{c}$ of period $2 b,$ the odd function $f_{s}$ of period $2 b,$ and the function $f_{p}$ of period $b$, each of which equals $f(x)$ on $0< x< b$. Expand each of the three functions in an appropriate Fourier series. $$f(x)=|\cos x|, \quad 0< x< \pi$$

Key Concepts

Fourier Series

A method for expressing a periodic function as an infinite sum of sine and cosine functions. This representation is based on the idea that even complicated periodic functions can be decomposed into simple, harmonic oscillations, making it a fundamental tool in analyzing functions in fields such as signal processing, heat transfer, and differential equations.

Even Extension and Fourier Cosine Series

When a function is defined on a half-interval, it can be extended to an even function on a symmetric interval by reflecting it about the origin. This even extension leads to a Fourier cosine series, which consists solely of cosine terms. The symmetry ensures that the sine coefficients vanish, simplifying the analysis and computation of the series representation.

Odd Extension and Fourier Sine Series

Similarly, extending a function defined on a half-interval as an odd function involves reflecting it anti-symmetrically about the origin. This results in a Fourier sine series where only sine terms appear. The odd symmetry causes the cosine coefficients to be zero, which is particularly useful in solving boundary value problems with homogeneous Dirichlet conditions.

Periodic Extension

Periodic extension is the process of extending a function defined on a finite interval to the entire real line by repeating its behavior in a regular pattern, thereby creating a periodic function. Depending on the type of extension (even, odd, or periodic with a different fundamental period), the Fourier expansion adapts, and the appropriate series (either full, sine-only, or cosine-only) is derived.

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