Computing Fourier Coefficients

Fourier Analysis · Axiom Academy

LESSON Computing Fourier Coefficients A step-by-step guide with the concrete example f(x) = x Every Fourier series computation follows the same systematic approach. For our example function f(x) = x on [-π, π], we have period 2L = 2π, so L = π. The Three Coefficient Formulas: 2. Computing a 0 (The Average Value) The coefficient a 0 represents twice the average value of the function over one period. For f(x) = x, this is the integral of x from -π to π. 3. Computing a n (Cosine Coefficients) The coefficients a n capture the even (symmetric) components. We integrate f(x)cos(nx) over one period. For our function, this requires integration by parts. Integration by parts: Let u = x, dv = cos(nx) dx 4. Computing b n (Sine Coefficients) The coefficients b n capture the odd (antisymmetric) components. For f(x) = x, an odd function, all the information is in these coefficients. We use integration by parts again. Integration by parts: Let u = x, dv = sin(nx) dx 5. The Complete Fourier Series Now we assemble all the pieces! With a 0 = 0, a n = 0, and b n = (-1) n+1 · 2/n, we can write the complete Fourier series for f(x) = x on [-π, π]. Identify the period and determine L Compute a 0 using the average value integral Compute a n using the cosine integral (often needs integration by parts) Compute b n using the sine integral (often needs integration by parts) Assemble the complete series and simplify

This is the written version of the interactive lesson above. See the full Fourier Analysis course.