Editing Pi (section)

=== Eigenvalues ===
[[File:Harmonic partials on strings.svg|thumb|right|The [[overtone]]s of a vibrating string are [[eigenfunction]]s of the second derivative, and form a [[harmonic series (music)|harmonic progression]]. The associated eigenvalues form the [[arithmetic progression]] of integer multiples of {{pi}}.]]
Many of the appearances of {{pi}} in the formulae of mathematics and the sciences have to do with its close relationship with geometry. However, {{pi}} 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 {{math|''f''}} on the unit interval {{closed-closed|0, 1}}, with [[boundary conditions|fixed ends]] {{math|1=''f''(0) = ''f''(1) = 0}}. The modes of vibration of the string are solutions of the [[differential equation]] <math>f''(x) + \lambda f(x) = 0</math>, or <math>f''(t) = -\lambda f(x)</math>. Thus {{math|λ}} is an eigenvalue of the second derivative [[differential operator|operator]] <math>f \mapsto f''</math>, 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 {{math|1=''λ'' = ''ν''<sup>2</sup>}}, where {{math|''ν'' > 0}} is called the [[wavenumber]]. Then {{math|1=''f''(''x'') = sin(''π'' ''x'')}} satisfies the boundary conditions and the differential equation with {{math|1=''ν'' = ''π''}}.<ref>{{cite book |last1=Hilbert |first1=David |author1-link=David Hilbert |last2=Courant |first2=Richard |author2-link=Richard Courant |title=Methods of mathematical physics |volume=1 |pages=286–290 |year=1966 |publisher=Wiley}}</ref>

The value {{pi}} 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 functions|Wirtinger's inequality]]:{{sfn|Dym|McKean|1972|page=47}} for a function <math>f : [0, 1] \to \Complex</math> with {{math|1=''f''(0) = ''f''(1) = 0}} and {{math|''f''}}, {{math|''f''{{′}}}} both [[square integrable]], we have:

<math display=block>\pi^2\int_0^1|f(x)|^2\,dx\le \int_0^1|f'(x)|^2\,dx,</math>

with equality precisely when {{math|''f''}} is a multiple of {{math|sin(π ''x'')}}. Here {{pi}} appears as an optimal constant in Wirtinger's inequality, and it follows that it is the smallest wavenumber, using the [[variational theorem|variational characterization]] of the eigenvalue. As a consequence, {{pi}} is the smallest [[singular value]] of the derivative operator on the space of functions on {{closed-closed|0, 1}} vanishing at both endpoints (the [[Sobolev space]] <math>H^1_0[0,1]</math>).