5.6 Symmetry and Half-Range Expansions
Before computing any Fourier series it is worth asking whether half the coefficients are zero, because very often they are and the symmetry that makes them so is visible at a glance.
Theorem 5.6.1 (Symmetry). Let \(f\) be Riemann integrable on \([-\pi ,\pi ]\).
- (i)
- If \(f\) is even, then \(b_n = 0\) for all \(n\) and \(a_n = \dfrac {2}{\pi }\displaystyle \int ^{\pi }_{0}f(x)\cos nx\,dx\): the series is a cosine series.
- (ii)
- If \(f\) is odd, then \(a_n = 0\) for all \(n\) and \(b_n = \dfrac {2}{\pi }\displaystyle \int ^{\pi }_{0}f(x)\sin nx\,dx\): the series is a sine series.
Proof. If \(f\) is even then \(f(x)\sin nx\) is odd, and the integral of an odd function over a symmetric interval vanishes, giving \(b_n=0\); while \(f(x)\cos nx\) is even, so its integral over \([-\pi ,\pi ]\) is twice that over \([0,\pi ]\). The odd case is identical with the roles exchanged. □
Note 5.6.2. The saving is real and worth taking. Half the integrals disappear, and the remaining ones are over half the interval. It is also a useful check on a completed calculation: a cosine term appearing in the expansion of an odd function is an arithmetic slip, not a discovery.
A function given only on \([0,\pi ]\) may be extended to \([-\pi ,\pi ]\) in either way, and the choice is ours.
Definition 5.6.3 (Half-range expansions). For \(f\) defined on \([0,\pi ]\), the half-range cosine series is the Fourier series of its even extension, and the half-range sine series that of its odd extension. Both represent \(f\) on \([0,\pi ]\); they differ outside it.
Note 5.6.4. This is not a technicality. In a boundary-value problem the boundary conditions choose the extension for you: a condition \(y(0)=y(L)=0\) requires eigenfunctions vanishing at the ends, hence the sine series, while an insulated boundary \(y'(0)=y'(L)=0\) requires the cosine series. Read against Table 4.4.1, both are the same Sturm–Liouville problem \(y''+\lambda y=0\) under different boundary conditions, and it is the boundary condition — not the function being expanded — that selects the system.
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