Half-Range Expansions

Fourier Analysis · Axiom Academy

Extending functions to create pure sine or cosine series Consider a function defined only on the interval [0, L]. For example, let's take f(x) = x on [0, π] . We want to represent this function as a Fourier series, but standard Fourier series are defined on symmetric intervals like [-L, L]. How do we proceed? The even extension reflects the function across the y-axis, creating a symmetric function where f(-x) = f(x) . For our example f(x) = x on [0, π], the even extension creates the function |x| on [-π, π]. Because the even extension creates an even function, the Fourier series contains only cosine terms (and possibly a constant term). All sine coefficients b_n = 0 . The half-range cosine expansion is particularly elegant for our example. The odd extension creates an antisymmetric function where f(-x) = -f(x) . We reflect across the y-axis and then flip the sign. For f(x) = x on [0, π], the odd extension gives us the same function: f(-x) = -x on [-π, 0], maintaining f(-x) = -f(x). Because the odd extension creates an odd function, the Fourier series contains only sine terms. All cosine coefficients a_n = 0 (including a_0). For our example f(x) = x on [0, π], this produces a particularly clean sine series. 6. When to Use Which Extension The choice between even and odd extensions depends on the boundary conditions of your problem: Use Cosine Series (Even Extension) when: Derivative boundary conditions (Neumann): f'(0) = 0, f'(L) = 0

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