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Eigenvalues

Many of the appearances of π in the formulae of mathematics and the sciences have to do with its close relationship with geometry. However, π also appears in many natural situations having apparently nothing to do with geometry.

In many applications, it plays a distinguished role as an eigenvalue. For example, an idealized vibrating string can be modelled as the graph of a function f on the unit interval, with fixed ends f (0) = f (1) = 0. The modes of vibration of the string are solutions of the differential equation f ″ (x) + λ f (x) = 0 {\displaystyle f''(x)+\lambda f(x)=0}, or f ″ (t) = − λ f (x) {\displaystyle f''(t)=-\lambda f(x)}. Thus λ is an eigenvalue of the second derivative operator f ↦ f ″ {\displaystyle f\mapsto f''}, and is constrained by Sturm–Liouville theory to take on only certain specific values. It must be positive, since the operator is negative definite, so it is convenient to write λ = ν, where ν > 0 is called the wavenumber. Then f (x) = sin(π x) satisfies the boundary conditions and the differential equation with ν = π.

The value π is, in fact, the least such value of the wavenumber, and is associated with the fundamental mode of vibration of the string. One way to show this is by estimating the energy, which satisfies Wirtinger's inequality: for a function f: → C {\displaystyle f:\to \mathbb {C} } with f (0) = f (1) = 0 and f, f ′ both square integrable, we have: π 2 ∫ 0 1 | f (x) | 2 d x ≤ ∫ 0 1 | f ′ (x) | 2 d x, {\displaystyle \pi ^{2}\int _{0}^{1}|f(x)|^{2}\,dx\leq \int _{0}^{1}|f'(x)|^{2}\,dx,} with equality precisely when f is a multiple of sin(π x). Here π appears as an optimal constant in Wirtinger's inequality, and it follows that it is the smallest wavenumber, using the variational characterization of the eigenvalue. As a consequence, π is the smallest singular value of the derivative operator on the space of functions on vanishing at both endpoints (the Sobolev space H 0 1 {\displaystyle H_{0}^{1}}).