Editing Pi (section)

== Role and characterizations in mathematics ==
Because {{pi}} is closely related to the circle, it is found in [[List of formulae involving π|many formulae]] from the fields of geometry and trigonometry, particularly those concerning circles, spheres, or ellipses. Other branches of science, such as statistics, physics, [[Fourier analysis]], and number theory, also include {{pi}} in some of their important formulae.

=== Geometry and trigonometry ===
[[File:Circle Area.svg|thumb|alt=A diagram of a circle with a square coving the circle's upper right quadrant.|right|The area of the circle equals {{pi}} times the shaded area. The area of the [[unit circle]] is {{pi}}.]]

{{pi}} appears in formulae for areas and volumes of geometrical shapes based on circles, such as [[ellipse]]s, [[sphere]]s, [[cone (geometry)|cones]], and [[torus|tori]].<ref>{{mathworld| |title=Circle |id=Circle|ref=none}}</ref>  Below are some of the more common formulae that involve {{pi}}.<ref>{{harvnb|Bronshteĭn|Semendiaev|1971|pp=200, 209}}.</ref>
* The circumference of a circle with radius {{math|''r''}} is {{math|2π''r''}}.<ref>{{mathworld |title=Circumference |id=Circumference|ref=none}}</ref>
* The [[area of a disk|area of a circle]] with radius {{math|''r''}} is {{math|π''r''<sup>2</sup>}}.
* The area of an ellipse with semi-major axis {{math|''a''}} and semi-minor axis {{math|''b''}} is {{math|π''ab''}}.<ref>{{mathworld |title=Ellipse |id=Ellipse|ref=none}}</ref>
* The volume of a sphere with radius {{math|''r''}} is {{math|{{sfrac|4|3}}π''r''<sup>3</sup>}}.
* The surface area of a sphere with radius {{math|''r''}} is {{math|4π''r''<sup>2</sup>}}.
Some of the formulae above are special cases of the volume of the [[N-ball|''n''-dimensional ball]] and the surface area of its boundary, the [[n-sphere|(''n''−1)-dimensional sphere]], given [[#The gamma function and Stirling's approximation|below]].

Apart from circles, there are other [[Curve of constant width|curves of constant width]]. By [[Barbier's theorem]], every curve of constant width has perimeter {{pi}} times its width. The [[Reuleaux triangle]] (formed by the intersection of three circles with the sides of an [[equilateral triangle]] as their radii) has the smallest possible area for its width and the circle the largest. There also exist non-circular [[Smoothness|smooth]] and even [[algebraic curve]]s of constant width.<ref>{{cite book |last1=Martini |first1=Horst |last2=Montejano |first2=Luis |last3=Oliveros |first3=Déborah |author3-link=Déborah Oliveros |doi=10.1007/978-3-030-03868-7 |isbn=978-3-030-03866-3 |mr=3930585 |publisher=Birkhäuser |s2cid=127264210 |title=Bodies of Constant Width: An Introduction to Convex Geometry with Applications |year=2019}}{{pb}}
See Barbier's theorem, Corollary 5.1.1, p. 98; Reuleaux triangles, pp. 3, 10; smooth curves such as an analytic curve due to Rabinowitz, §&nbsp;5.3.3, pp. 111–112.</ref>

[[Integral|Definite integrals]] that describe circumference, area, or volume of shapes generated by circles typically have values that involve {{pi}}. For example, an integral that specifies half the area of a circle of radius one is given by:<ref>{{cite book |last1=Herman |first1=Edwin |last2=Strang |first2=Gilbert |author2-link=Gilbert Strang |contribution=Section 5.5, Exercise 316 |contribution-url=https://openstax.org/books/calculus-volume-1/pages/5-5-substitution |page=594 |publisher=[[OpenStax]] |title=Calculus |volume=1 |year=2016}}</ref>

<math display=block>\int_{-1}^1 \sqrt{1-x^2}\,dx = \frac{\pi}{2}.</math>

In that integral, the function <math>\sqrt{1-x^2}</math> represents the height over the <math>x</math>-axis of a [[semicircle]] (the [[square root]] is a consequence of the [[Pythagorean theorem]]), and the integral computes the area below the semicircle.  The existence of such integrals makes {{pi}} an [[Period (algebraic geometry)|algebraic period]].<ref>{{cite book |last1=Kontsevich |first1=Maxim |author-link1=Maxim Kontsevich |contribution=Periods |date=2001 |title=Mathematics Unlimited — 2001 and Beyond |pages=771–808 |editor-last=Engquist |editor-first=Björn |doi=10.1007/978-3-642-56478-9_39 |location=Berlin, Heidelberg |publisher=Springer |language=en |isbn=978-3-642-56478-9 |last2=Zagier |first2=Don |editor2-last=Schmid |editor2-first=Wilfried}}</ref>

=== Units of angle ===
{{Main|Units of angle measure}}
[[File:Sine cosine one period.svg|thumb|upright=1.5|alt=Diagram showing graphs of functions|[[Sine]] and [[cosine]] functions repeat with period 2{{pi}}.|left]]The [[trigonometric function]]s rely on angles, and mathematicians generally use [[radian]]s as units of measurement. {{pi}} plays an important role in angles measured in radians, which are defined so that a complete circle spans an angle of 2{{pi}}&nbsp;radians. The angle measure of 180° is equal to {{pi}} radians, and {{nowrap|1=1° = {{pi}}/180 radians}}.{{sfn|Abramson|2014|loc=[https://openstax.org/books/precalculus/pages/5-1-angles Section 5.1: Angles]}}

Common trigonometric functions have periods that are multiples of {{pi}}; for example, sine and cosine have period 2{{pi}}, so for any angle {{math|''θ''}} and any integer {{math|''k''}},{{sfn|Bronshteĭn|Semendiaev|1971|pp=210–211}}
<math display=block> \sin\theta = \sin\left(\theta + 2\pi k \right) \text{ and } \cos\theta = \cos\left(\theta + 2\pi k \right).</math>

=== 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>).

=== Inequalities ===
[[File:Sir William Thompson illustration of Carthage.png|thumb|The [[ancient Carthage|ancient city of Carthage]] was the solution to an isoperimetric problem, according to a legend recounted by [[Lord Kelvin]]:<ref>{{cite journal |first=William |last=Thompson |author-link=Lord Kelvin |title=Isoperimetrical problems |year=1894 |journal=Nature Series: Popular Lectures and Addresses |volume=II |pages=571–592}}</ref> those lands bordering the sea that [[Dido|Queen Dido]] could enclose on all other sides within a single given oxhide, cut into strips.|left]]

The number {{pi}} serves appears in similar eigenvalue problems in higher-dimensional analysis.  As mentioned [[#Definition|above]], it can be characterized via its role as the best constant in the [[isoperimetric inequality]]: the area {{mvar|A}} enclosed by a plane [[Jordan curve]] of perimeter {{mvar|P}} satisfies the inequality

<math display=block>4\pi A\le P^2,</math>

and equality is clearly achieved for the circle, since in that case {{math|1=''A'' = π''r''{{sup|2}}}} and {{math|1=''P'' = 2π''r''}}.<ref>{{cite book |first=Isaac |last=Chavel |title=Isoperimetric inequalities |publisher=Cambridge University Press |year=2001}}</ref>

Ultimately, as a consequence of the isoperimetric inequality, {{pi}} appears in the optimal constant for the critical [[Sobolev inequality]] in ''n'' dimensions, which thus characterizes the role of {{pi}} in many physical phenomena as well, for example those of classical [[potential theory]].<ref>{{cite journal |last=Talenti |first=Giorgio |title=Best constant in Sobolev inequality |journal=Annali di Matematica Pura ed Applicata |volume=110 |number=1 |pages=353–372 |issn=1618-1891 |doi=10.1007/BF02418013 |citeseerx=10.1.1.615.4193 |year=1976 |s2cid=16923822}}</ref><ref>{{cite arXiv |title=Best constants in Poincaré inequalities for convex domains |eprint=1110.2960 |author1=L. Esposito |author2=C. Nitsch |author3=C. Trombetti |year=2011 |class=math.AP}}</ref><ref>{{cite journal |title=Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions |first1=M. |last1=Del Pino |first2=J. |last2=Dolbeault |journal=Journal de Mathématiques Pures et Appliquées |year=2002 |volume=81 |issue=9 |pages=847–875 |doi=10.1016/s0021-7824(02)01266-7 |citeseerx=10.1.1.57.7077 |s2cid=8409465}}</ref>  In two dimensions, the critical Sobolev inequality is

<math display=block>2\pi\|f\|_2 \le \|\nabla f\|_1</math>

for ''f'' a smooth function with compact support in {{math|'''R'''<sup>2</sup>}}, <math>\nabla f</math> is the [[gradient]] of ''f'', and <math>\|f\|_2</math> and <math>\|\nabla f\|_1</math> refer respectively to the [[Lp space|{{math|L<sup>2</sup>}} and {{math|L<sup>1</sup>}}-norm]].  The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants.

Wirtinger's inequality also generalizes to higher-dimensional [[Poincaré inequality|Poincaré inequalities]] that provide best constants for the [[Dirichlet energy]] of an ''n''-dimensional membrane.  Specifically, {{pi}} is the greatest constant such that

<math display=block>
\pi \le \frac{\left (\int_G |\nabla u|^2\right)^{1/2}}{\left (\int_G|u|^2\right)^{1/2}}
</math>

for all [[convex set|convex]] subsets {{math|''G''}} of {{math|'''R'''<sup>''n''</sup>}} of diameter 1, and square-integrable functions ''u'' on {{math|''G''}} of mean zero.<ref>{{cite journal |last1=Payne |first1=L. E. |last2=Weinberger |first2=H. F. |title=An optimal Poincaré inequality for convex domains |year=1960 |journal=Archive for Rational Mechanics and Analysis |volume=5 |issue=1 |issn=0003-9527 |pages=286–292 |doi=10.1007/BF00252910 |bibcode=1960ArRMA...5..286P |s2cid=121881343}}</ref>  Just as Wirtinger's inequality is the [[calculus of variations|variational]] form of the [[Dirichlet eigenvalue]] problem in one dimension, the Poincaré inequality is the variational form of the [[Neumann problem|Neumann]] eigenvalue problem, in any dimension.

=== Fourier transform and Heisenberg uncertainty principle ===
[[File:Animation of Heisenberg geodesic.gif|thumb|right|An animation of a [[Heisenberg group#As a sub-Riemannian manifold|geodesic in the Heisenberg group]]]]
The constant {{pi}} also appears as a critical spectral parameter in the [[Fourier transform]].  This is the [[integral transform]], that takes a complex-valued integrable function {{math|''f''}} on the real line to the function defined as:

<math display=block>\hat{f}(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi i x\xi}\,dx.</math>

Although there are several different conventions for the Fourier transform and its inverse, any such convention must involve {{pi}} ''somewhere''. The above is the most canonical definition, however, giving the unique unitary operator on {{math|''L''{{sup|2}}}} that is also an algebra homomorphism of {{math|''L''{{sup|1}}}} to {{math|''L''{{sup|∞}}}}.<ref>{{cite book |title=Harmonic analysis in phase space |first=Gerald |last=Folland |publisher=Princeton University Press |year=1989 |page=5 |author-link=Gerald Folland}}</ref>

The [[Heisenberg uncertainty principle]] also contains the number {{pi}}.  The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: with our conventions for the Fourier transform,

<math display=block>
\left(\int_{-\infty}^\infty x^2|f(x)|^2\,dx\right)
\left(\int_{-\infty}^\infty \xi^2|\hat{f}(\xi)|^2\,d\xi\right)
 \ge
\left(\frac{1}{4\pi}\int_{-\infty}^\infty |f(x)|^2\,dx\right)^2.
</math>

The physical consequence, about the uncertainty in simultaneous position and momentum observations of a [[quantum mechanical]] system, is [[#Describing physical phenomena|discussed below]].  The appearance of {{pi}} in the formulae of Fourier analysis is ultimately a consequence of the [[Stone–von Neumann theorem]], asserting the uniqueness of the [[Schrödinger representation]] of the [[Heisenberg group]].<ref name=howe>{{cite journal |first=Roger |last=Howe |title=On the role of the Heisenberg group in harmonic analysis |journal=[[Bulletin of the American Mathematical Society]] |volume=3 |pages=821–844 |number=2 |year=1980 |doi=10.1090/S0273-0979-1980-14825-9 |mr=578375 |doi-access=free}}</ref>

=== Gaussian integrals ===
[[File:E^(-x^2).svg|thumb|A graph of the [[Gaussian function]] {{math|1=''ƒ''(''x'') = ''e''{{sup|−''x''{{sup|2}}}}}}. The coloured region between the function and the ''x''-axis has area {{math|{{sqrt|π}}}}.|left]]

The fields of [[probability]] and [[statistics]] frequently use the [[normal distribution]] as a simple model for complex phenomena; for example, scientists generally assume that the observational error in most experiments follows a normal distribution.<ref>Feller, W. ''An Introduction to Probability Theory and Its Applications, Vol. 1'', Wiley, 1968, pp. 174–190.</ref> The [[Gaussian function]], which is the [[probability density function]] of the normal distribution with [[mean]] {{math|μ}} and [[standard deviation]] {{math|σ}}, naturally contains {{pi}}:<ref name="GaussProb">{{harvnb|Bronshteĭn|Semendiaev|1971|pp=106–107, 744, 748}}.</ref>

<math display=block>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu )^2/(2\sigma^2)}.</math>

The factor of <math>\tfrac{1}{\sqrt{2\pi}}</math> makes the area under the graph of {{math|''f''}}  equal to one, as is required for a probability distribution. This follows from a [[integration by substitution|change of variables]] in the [[Gaussian integral]]:{{r|GaussProb}}

<math display=block>\int_{-\infty}^\infty e^{-u^2} \, du=\sqrt{\pi}</math>

which says that the area under the basic [[bell curve]] in the figure is equal to the square root of {{pi}}.

The [[central limit theorem]] explains the central role of normal distributions, and thus of {{pi}}, in probability and statistics. This theorem is ultimately connected with the [[#Fourier transform and Heisenberg uncertainty principle|spectral characterization]] of {{pi}} as the eigenvalue associated with the Heisenberg uncertainty principle, and the fact that equality holds in the uncertainty principle only for the Gaussian function.{{sfn|Dym|McKean|1972|loc=Section 2.7}} Equivalently, {{pi}} is the unique constant making the Gaussian normal distribution {{math|''e''{{sup|−π''x''{{sup|2}}}}}} equal to its own Fourier transform.<ref>{{cite book |first1=Elias |last1=Stein |first2=Guido |last2=Weiss |title=Fourier analysis on Euclidean spaces |year=1971 |publisher=Princeton University Press |page=6 |author1-link=Elias Stein}}; Theorem 1.13.</ref> Indeed,  according to {{harvtxt|Howe|1980}}, the "whole business" of establishing the fundamental theorems of Fourier analysis reduces to the Gaussian integral.{{r|howe}}

=== Topology ===
[[File:Order-7 triangular tiling.svg|thumb|right|[[Uniformization theorem|Uniformization]] of the [[Klein quartic]], a surface of [[genus (mathematics)|genus]] three and Euler characteristic −4, as a quotient of the [[Poincaré disk model|hyperbolic plane]] by the [[symmetry group]] [[PSL(2,7)]] of the [[Fano plane]].  The hyperbolic area of a fundamental domain is {{math|8π}}, by Gauss–Bonnet.]]
The constant {{pi}} appears in the [[Gauss–Bonnet formula]] which relates the [[differential geometry of surfaces]] to their [[topology]].  Specifically, if a [[compact space|compact]] surface {{math|Σ}} has [[Gauss curvature]] ''K'', then

<math display=block>\int_\Sigma K\,dA = 2\pi \chi(\Sigma)</math>

where {{math|''χ''(Σ)}} is the [[Euler characteristic]], which is an integer.<ref>{{cite book |title=A Comprehensive Introduction to Differential Geometry |volume=3 |first=Michael |last=Spivak |year=1999 |publisher=Publish or Perish Press |author-link=Michael Spivak}}; Chapter 6.</ref>  An example is the surface area of a sphere ''S'' of curvature 1 (so that its [[radius of curvature]], which coincides with its radius, is also 1.)  The Euler characteristic of a sphere can be computed from its [[homology group]]s and is found to be equal to two.  Thus we have

<math display=block>A(S) = \int_S 1\,dA = 2\pi\cdot 2 = 4\pi</math>

reproducing the formula for the surface area of a sphere of radius 1.

The constant appears in many other integral formulae in topology, in particular, those involving [[characteristic class]]es via the [[Chern–Weil homomorphism]].<ref>{{cite book |last1=Kobayashi |first1=Shoshichi |last2=Nomizu |first2=Katsumi |title=Foundations of Differential Geometry |volume=2 |publisher=[[Wiley Interscience]] |year=1996 |edition=New |page=293 |title-link=Foundations of Differential Geometry}}; Chapter XII ''Characteristic classes''</ref>

=== Cauchy's integral formula ===
[[File:Factorial05.jpg|thumb|right|Complex analytic functions can be visualized as a collection of streamlines and equipotentials, systems of curves intersecting at right angles.  Here illustrated is the complex logarithm of the Gamma function.]]

One of the key tools in [[complex analysis]] is [[contour integration]] of a function over a positively oriented ([[rectifiable curve|rectifiable]]) [[Jordan curve]] {{math|''γ''}}.  A form of [[Cauchy's integral formula]] states that if a point {{math|''z''<sub>0</sub>}} is interior to {{math|''γ''}}, then<ref>{{cite book |first=Lars |last=Ahlfors |title=Complex analysis |publisher=McGraw-Hill |year=1966 |page=115 |author-link=Lars Ahlfors}}</ref>

<math display=block>\oint_\gamma \frac{dz}{z-z_0} = 2\pi i.</math>

Although the curve {{math|''γ''}} is not a circle, and hence does not have any obvious connection to the constant {{pi}}, a standard proof of this result uses [[Morera's theorem]], which implies that the integral is invariant under [[homotopy]] of the curve, so that it can be deformed to a circle and then integrated explicitly in polar coordinates.  More generally, it is true that if a rectifiable closed curve {{math|γ}} does not contain {{math|''z''<sub>0</sub>}}, then the above integral is {{math|2π''i''}} times the [[winding number]] of the curve.

The general form of Cauchy's integral formula establishes the relationship between the values of a [[complex analytic function]] {{math|''f''(''z'')}} on the Jordan curve {{math|''γ''}} and the value of {{math|''f''(''z'')}} at any interior point {{math|''z''<sub>0</sub>}} of {{math|γ}}:<ref>{{cite book |last=Joglekar |first=S. D. |title=Mathematical Physics |publisher=Universities Press |year=2005 |page=166 |isbn=978-81-7371-422-1}}</ref>

<math display=block>\oint_\gamma { f(z) \over z-z_0 }\,dz = 2\pi i f (z_{0})</math>

provided {{math|''f''(''z'')}} is analytic in the region enclosed by {{math|''γ''}} and extends continuously to {{math|''γ''}}.  Cauchy's integral formula is a special case of the [[residue theorem]], that if {{math|''g''(''z'')}} is a [[meromorphic function]] the region enclosed by {{math|''γ''}} and is continuous in a neighbourhood of {{math|''γ''}}, then

<math display=block>\oint_\gamma g(z)\, dz =2\pi i \sum \operatorname{Res}( g, a_k ) </math>

where the sum is of the [[residue (mathematics)|residues]] at the [[pole (complex analysis)|poles]] of {{math|''g''(''z'')}}.

=== Vector calculus and physics ===
The constant {{pi}} is ubiquitous in [[vector calculus]] and [[potential theory]], for example in [[Coulomb's law]],<ref>{{cite book |first=H. M. |last=Schey |year=1996 |title=Div, Grad, Curl, and All That: An Informal Text on Vector Calculus |publisher=W. W. Norton |isbn=0-393-96997-5}}</ref> [[Gauss's law]], [[Maxwell's equations]], and even the [[Einstein field equations]].<ref>{{cite book |last=Yeo |first=Adrian |title=The pleasures of pi, e and other interesting numbers |publisher=World Scientific Pub. |year=2006 |page=21 |isbn=978-981-270-078-0}}</ref><ref>{{cite book |last=Ehlers |first=Jürgen |author-link=Jürgen Ehlers |title=Einstein's Field Equations and Their Physical Implications |publisher=Springer |year=2000 |page=7 |isbn=978-3-540-67073-5}}</ref>  Perhaps the simplest example of this is the two-dimensional [[Newtonian potential]], representing the potential of a point source at the origin, whose associated field has unit outward [[flux]] through any smooth and oriented closed surface enclosing the source:

<math display="block">\Phi(\mathbf x) = \frac{1}{2\pi}\log|\mathbf x|.</math>

The factor of <math>1/2\pi</math> is necessary to ensure that <math>\Phi</math> is the [[fundamental solution]] of the [[Poisson equation]] in <math>\mathbb R^2</math>:<ref name="Elliptic PDE2">{{cite book |first1=D. |last1=Gilbarg |first2=Neil |author-link1=David Gilbarg |last2=Trudinger |authorlink2=Neil Trudinger |title=Elliptic Partial Differential Equations of Second Order |publisher=Springer |publication-place=New York |year=1983 |isbn=3-540-41160-7}}</ref>

<math display="block">\Delta\Phi = \delta</math>

where <math>\delta</math> is the [[Dirac delta function]].

In higher dimensions, factors of {{pi}} are present because of a normalization by the n-dimensional volume of the unit [[n sphere]].  For example, in three dimensions, the Newtonian potential is:{{r|Elliptic PDE2}}

<math display="block">\Phi(\mathbf x) = -\frac{1}{4\pi|\mathbf x|},</math>

which has the 2-dimensional volume (i.e., the area) of the unit 2-sphere in the denominator.

=== Total curvature ===
[[File:Winding Number Around Point.svg|thumb|This curve has total curvature {{math|6''π''}} and turning number {{math|3}}; it has [[winding number]] {{math|2}} about {{mvar|p}} and an additional loop which does not contain {{mvar|p}}.]]
In the [[differential geometry of curves]], the ''[[total curvature]]'' of a smooth plane curve is the amount it turns anticlockwise, in radians, from start to finish, computed as the integral of signed [[curvature]] with respect to arc length:

<math display=block>\int_a^b k(s)\,ds</math>

For a closed curve, this quantity is equal to {{math|2''πN''}} for an integer {{mvar|N}} called the ''[[turning number]]'' or ''index'' of the curve. {{mvar|N}} is the [[winding number]] about the origin of the [[hodograph]] of the curve parametrized by arclength, a new curve lying on the unit circle, described by the normalized [[tangent vector]] at each point on the original curve. Equivalently, {{mvar|N}} is the [[Degree of a continuous mapping|degree of the map]] taking each point on the curve to the corresponding point on the hodograph, analogous to the [[Gauss map]] for surfaces.

=== The gamma function and Stirling's approximation ===
[[File:Gamma plot points marked.svg|thumb|Plot of the gamma function on the real axis]]
The [[factorial]] function <math>n!</math> is the product of all of the positive integers through {{math|''n''}}. The [[gamma function]] extends the concept of [[factorial]] (normally defined only for non-negative integers) to all complex numbers, except the negative real integers, with the identity <math>\Gamma(n)=(n-1)!</math>. When the gamma function is evaluated at half-integers, the result contains {{pi}}. For example, <math> \Gamma\bigl(\tfrac12\bigr) = \sqrt{\pi} </math> and <math display="inline">\Gamma\bigl(\tfrac52\bigr) = \tfrac 34 \sqrt{\pi} </math>.<ref>{{harvnb|Bronshteĭn|Semendiaev|1971|pp=191–192}}.</ref>

The gamma function is defined by its [[Weierstrass product]] development:<ref>{{cite book |title=The Gamma Function |first=Emil |last=Artin |publisher=Holt, Rinehart and Winston |year=1964 |series=Athena series; selected topics in mathematics |edition=1st |author-link=Emil Artin}}</ref>

<math display=block>\Gamma(z) = \frac{e^{-\gamma z}}{z}\prod_{n=1}^\infty \frac{e^{z/n}}{1+z/n}</math>

where {{math|γ}} is the [[Euler–Mascheroni constant]].  Evaluated at {{tmath|1= z = \tfrac12 }} and squared, the equation {{tmath|1=\textstyle \gamma\bigl(\tfrac12\bigr)\vphantom)^2 = \pi}} reduces to the Wallis product formula.  The gamma function is also connected to the [[Riemann zeta function]] and identities for the [[functional determinant]], in which the constant {{pi}} [[#Number theory and Riemann zeta function|plays an important role]].

The gamma function is used to calculate the volume {{math|''V''<sub>''n''</sub>(''r'')}} of the [[n-ball|''n''-dimensional ball]] of radius ''r'' in Euclidean ''n''-dimensional space, and the surface area {{math|''S''<sub>''n''−1</sub>(''r'')}} of its boundary, the [[n-sphere|(''n''−1)-dimensional sphere]]:<ref>{{cite book |first=Lawrence |last=Evans |author-link=Lawrence C. Evans |title=Partial Differential Equations |publisher=AMS |year=1997 |page=615}}</ref>

<math display=block>V_n(r) = \frac{\pi^{n/2}}{\Gamma\bigl(\frac{n}{2}+1\bigr)}r^n,</math>

<math display=block>S_{n-1}(r) = \frac{n\pi^{n/2}}{\Gamma\bigl(\tfrac{n}{2}+1\bigr)}r^{n-1}.</math>

Further, it follows from the [[functional equation]] that

<math display=block>2\pi r = \frac{S_{n+1}(r)}{V_n(r)}.</math>

The gamma function can be used to create a simple approximation to the factorial function {{math|''n''!}} for large {{math|''n''}}: <math display="inline"> n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n</math> which is known as [[Stirling's approximation]].<ref>{{harvnb|Bronshteĭn|Semendiaev|1971|p=190}}.</ref>  Equivalently,

<math display=block>\pi = \lim_{n\to\infty} \frac{e^{2n}n!^2}{2 n^{2n+1}}.</math>

As a geometrical application of Stirling's approximation, let {{math|Δ<sub>''n''</sub>}} denote the [[simplex|standard simplex]] in ''n''-dimensional Euclidean space, and {{math|(''n''&nbsp;+&nbsp;1)Δ<sub>''n''</sub>}} denote the simplex having all of its sides scaled up by a factor of {{math|''n''&nbsp;+&nbsp;1}}. Then

<math display=block>\operatorname{Vol}((n+1)\Delta_n) = \frac{(n+1)^n}{n!} \sim \frac{e^{n+1}}{\sqrt{2\pi n}}.</math>

[[Ehrhart's volume conjecture]] is that this is the (optimal) upper bound on the volume of a [[convex body]] containing only one [[lattice point]].<ref>{{cite journal |author1=Benjamin Nill |author2=Andreas Paffenholz |title=On the equality case in Erhart's volume conjecture |year=2014 |arxiv=1205.1270 |journal=Advances in Geometry |volume=14 |issue=4 |pages=579–586 |issn=1615-7168 |doi=10.1515/advgeom-2014-0001 |s2cid=119125713}}</ref>

=== Number theory and Riemann zeta function ===
[[File:Prüfer.png|thumb|right|Each prime has an associated [[Prüfer group]], which are arithmetic localizations of the circle.  The [[L-function]]s of analytic number theory are also localized in each prime ''p''.]]
[[File:ModularGroup-FundamentalDomain.svg|thumb|right|Solution of the Basel problem using the [[Weil conjecture on Tamagawa numbers|Weil conjecture]]: the value of {{math|''ζ''(2)}} is the [[Poincaré half-plane model|hyperbolic]] area of a fundamental domain of the [[modular group]], times {{math|{{pi}}/2}}.]]
The [[Riemann zeta function]] {{math|''ζ''(''s'')}} is used in many areas of mathematics. When evaluated at {{math|1=''s'' = 2}} it can be written as

<math display=block>\zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots</math>

Finding a [[closed-form expression|simple solution]] for this infinite series was a famous problem in mathematics called the [[Basel problem]]. [[Leonhard Euler]] solved it in 1735 when he showed it was equal to {{math|π<sup>2</sup>/6}}.{{sfn|Posamentier|Lehmann|2004|p=284}} Euler's result leads to the [[number theory]] result that the probability of two random numbers being [[relatively prime]] (that is, having no shared factors) is equal to {{math|6/π<sup>2</sup>}}.<ref>{{harvnb|Arndt|Haenel|2006|pp=41–43}}. {{pb}} This theorem was proved by [[Ernesto Cesàro]] in 1881. For a more rigorous proof than the intuitive and informal one given here, see {{cite book |last=Hardy |first=G. H. |author-link=G. H. Hardy |title=An Introduction to the Theory of Numbers |publisher=Oxford University Press |year=2008 |isbn=978-0-19-921986-5 |at=Theorem 332}}</ref> This probability is based on the observation that the probability that any number is [[divisible]] by a prime {{math|''p''}} is {{math|1/''p''}} (for example, every 7th integer is divisible by 7.) Hence the probability that two numbers are both divisible by this prime is {{math|1/''p''<sup>2</sup>}}, and the probability that at least one of them is not is {{math|1&nbsp;−&nbsp;1/''p''<sup>2</sup>}}. For distinct primes, these divisibility events are mutually independent; so the probability that two numbers are relatively prime is given by a product over all primes:<ref>{{cite book |author1-link=C. Stanley Ogilvy |last1=Ogilvy |first1=C. S. |last2=Anderson |first2=J. T. |title=Excursions in Number Theory |publisher=Dover |year=1988 |pages=29–35 |isbn=0-486-25778-9}}</ref>

<math display=block>\begin{align}
\prod_p^\infty \left(1-\frac{1}{p^2}\right) &= \left( \prod_p^\infty \frac{1}{1-p^{-2}} \right)^{-1}\\[4pt]
&= \frac{1}{1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots }\\[4pt]
&= \frac{1}{\zeta(2)} = \frac{6}{\pi^2} \approx 61\%.
\end{align}</math>

This probability can be used in conjunction with a [[random number generator]] to approximate {{pi}} using a Monte Carlo approach.<ref>{{harvnb|Arndt|Haenel|2006|p=43}}.</ref>

The solution to the Basel problem implies that the geometrically derived quantity {{pi}} is connected in a deep way to the distribution of prime numbers.  This is a special case of [[Weil's conjecture on Tamagawa numbers]], which asserts the equality of similar such infinite products of ''arithmetic'' quantities, localized at each prime ''p'', and a ''geometrical'' quantity: the reciprocal of the volume of a certain [[locally symmetric space]].  In the case of the Basel problem, it is the [[hyperbolic 3-manifold]] {{math|[[SL2(R)|SL<sub>2</sub>('''R''')]]/[[modular group|SL<sub>2</sub>('''Z''')]]}}.<ref>{{cite book |title=Algebraic Groups and Number Theory |first1=Vladimir |last1=Platonov |author-link1=Vladimir Platonov |first2=Andrei |last2=Rapinchuk |publisher=Academic Press |year=1994 |pages=262–265}}</ref>

The zeta function also satisfies Riemann's functional equation, which involves {{pi}} as well as the gamma function:

<math display=block>\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s).</math>

Furthermore, the derivative of the zeta function satisfies

<math display=block>\exp(-\zeta'(0)) = \sqrt{2\pi}.</math>

A consequence is that {{pi}} can be obtained from the [[functional determinant]] of the [[harmonic oscillator]].  This functional determinant can be computed via a product expansion, and is equivalent to the Wallis product formula.<ref>{{cite journal |last=Sondow |first=J. |title=Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series |journal=[[Proceedings of the American Mathematical Society]] |volume=120 |issue=2 |pages=421–424 |year=1994 |doi=10.1090/s0002-9939-1994-1172954-7 |citeseerx=10.1.1.352.5774 |s2cid=122276856}}</ref>  The calculation can be recast in [[quantum mechanics]], specifically the [[Calculus of variations|variational approach]] to the [[Bohr model|spectrum of the hydrogen atom]].<ref>{{cite journal |doi=10.1063/1.4930800 |first=T. |last=Friedmann |first2=C. R. |last2=Hagen |title=Quantum mechanical derivation of the Wallis formula for pi |journal=Journal of Mathematical Physics |volume=56 |issue=11 |pages=112101 |year=2015 |arxiv=1510.07813 |bibcode=2015JMP....56k2101F |s2cid=119315853}}</ref>

=== Fourier series ===
[[File:2-adic integers with dual colorings.svg|thumb|{{pi}} appears in characters of [[p-adic numbers]] (shown), which are elements of a [[Prüfer group]]. [[Tate's thesis]] makes heavy use of this machinery.<ref>{{cite conference |last1=Tate |first1=John T. |author-link=John Tate (mathematician) |editor1-first=J. W. S. |editor1-last=Cassels |editor2-first=A. |editor2-last=Fröhlich |title=Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) |publisher=Thompson, Washington, DC |isbn=978-0-9502734-2-6 |mr=0217026 |year=1950 |contribution=Fourier analysis in number fields, and Hecke's zeta-functions |pages=305–347}}</ref>|left]]
The constant {{pi}} also appears naturally in [[Fourier series]] of [[periodic function]]s.  Periodic functions are functions on the group {{math|'''T''' {{=}}'''R'''/'''Z'''}} of fractional parts of real numbers.  The Fourier decomposition shows that a complex-valued function {{math|''f''}} on {{math|'''T'''}} can be written as an infinite linear superposition of [[unitary character]]s of {{math|'''T'''}}.   That is, continuous [[group homomorphism]]s from {{math|'''T'''}} to the [[circle group]] {{math|''U''(1)}} of unit modulus complex numbers.  It is a theorem that every character of {{math|'''T'''}} is one of the complex exponentials <math>e_n(x)= e^{2\pi i n x}</math>.

There is a unique character on {{math|'''T'''}}, up to complex conjugation, that is a group isomorphism.  Using the [[Haar measure]] on the circle group, the constant {{pi}} is half the magnitude of the [[Radon–Nikodym derivative]] of this character.  The other characters have derivatives whose magnitudes are positive integral multiples of 2{{pi}}.{{r|Nicolas Bourbaki}} As a result, the constant {{pi}} is the unique number such that the group '''T''', equipped with its Haar measure, is [[Pontrjagin dual]] to the [[lattice (group)|lattice]] of integral multiples of 2{{pi}}.{{sfn|Dym|McKean|1972|loc=Chapter 4}}  This is a version of the one-dimensional [[Poisson summation formula]].

=== Modular forms and theta functions ===
[[File:Lattice with tau.svg|thumb|right|Theta functions transform under the [[lattice (group)|lattice]] of periods of an elliptic curve.]]

The constant {{pi}} is connected in a deep way with the theory of [[modular form]]s and [[theta function]]s.  For example, the [[Chudnovsky algorithm]] involves in an essential way the [[j-invariant]] of an [[elliptic curve]].

[[Modular form]]s are [[holomorphic function]]s in the [[upper half plane]] characterized by their transformation properties under the [[modular group]] <math>\mathrm{SL}_2(\mathbb Z)</math> (or its various subgroups), a lattice in the group <math>\mathrm{SL}_2(\mathbb R)</math>.  An example is the [[Jacobi theta function]]

<math display=block>\theta(z,\tau) = \sum_{n=-\infty}^\infty e^{2\pi i nz \ +\  \pi i n^2\tau}</math>

which is a kind of modular form called a [[Jacobi form]].<ref name="Mumford 1983 1–117">{{cite book |first=David |last=Mumford |author-link=David Mumford |title=Tata Lectures on Theta I |year=1983 |publisher=Birkhauser |location=Boston |isbn=978-3-7643-3109-2 |pages=1–117}}</ref>  This is sometimes written in terms of the [[nome (mathematics)|nome]] <math>q=e^{\pi i \tau}</math>.

The constant {{pi}} is the unique constant making the Jacobi theta function an [[automorphic form]], which means that it transforms in a specific way.  Certain identities hold for all automorphic forms.  An example is

<math display=block>\theta(z+\tau,\tau) = e^{-\pi i\tau -2\pi i z}\theta(z,\tau),</math>

which implies that {{math|θ}} transforms as a representation under the discrete [[Heisenberg group]].  General modular forms and other [[theta function]]s also involve {{pi}}, once again because of the [[Stone–von Neumann theorem]].{{r|Mumford 1983 1–117}}

=== Cauchy distribution and potential theory ===
[[File:Witch of Agnesi, construction.svg|thumb|The [[Witch of Agnesi]], named for [[Maria Gaetana Agnesi|Maria Agnesi]] (1718–1799), is a geometrical construction of the graph of the Cauchy distribution.|left]]
[[File:2d random walk ag adatom ag111.gif|left|thumb|The Cauchy distribution governs the passage of [[Brownian motion|Brownian particles]] through a membrane.]]
The [[Cauchy distribution]]

<math display=block>g(x)=\frac{1}{\pi}\cdot\frac{1}{x^2+1}</math>

is a [[probability density function]].  The total probability is equal to one, owing to the integral:

<math display=block>\int_{-\infty }^{\infty } \frac{1}{x^2+1} \, dx = \pi.</math>

The [[Shannon entropy]] of the Cauchy distribution is equal to {{math|ln(4π)}}, which also involves {{pi}}.

The Cauchy distribution plays an important role in [[potential theory]] because it is the simplest [[Furstenberg boundary|Furstenberg measure]], the classical [[Poisson kernel]] associated with a [[Brownian motion]] in a half-plane.<ref>{{cite book |first1=Sidney |last1=Port |first2=Charles |last2=Stone |title=Brownian motion and classical potential theory |publisher=Academic Press |year=1978 |page=29}}</ref> [[Conjugate harmonic function]]s and so also the [[Hilbert transform]] are associated with the asymptotics of the Poisson kernel. The Hilbert transform ''H'' is the integral transform given by the [[Cauchy principal value]] of the [[singular integral]]

<math display=block>Hf(t) = \frac{1}{\pi}\int_{-\infty}^\infty \frac{f(x)\,dx}{x-t}.</math>

The constant {{pi}} is the unique (positive) normalizing factor such that ''H'' defines a [[linear complex structure]] on the Hilbert space of square-integrable real-valued functions on the real line.<ref>{{cite book |last=Titchmarsh |first=E. |author-link=Edward Charles Titchmarsh |title=Introduction to the Theory of Fourier Integrals |isbn=978-0-8284-0324-5 |year=1948 |edition=2nd |publication-date=1986 |publisher=Clarendon Press |location=Oxford University}}</ref>  The Hilbert transform, like the Fourier transform, can be characterized purely in terms of its transformation properties on the Hilbert space {{math|L<sup>2</sup>('''R''')}}: up to a normalization factor, it is the unique bounded linear operator that commutes with positive dilations and anti-commutes with all reflections of the real line.<ref>{{cite book |first=Elias |last=Stein |author-link=Elias Stein |title=Singular Integrals and Differentiability Properties of Functions |publisher=Princeton University Press |year=1970}}; Chapter II.</ref>  The constant {{pi}} is the unique normalizing factor that makes this transformation unitary.

=== In the Mandelbrot set ===
[[File:Mandel zoom 00 mandelbrot set.jpg|alt=An complex black shape on a blue background.|thumb|The [[Mandelbrot set]] can be used to approximate {{pi}}.]]

An occurrence of {{pi}} in the [[fractal]] called the [[Mandelbrot set]] was discovered by David Boll in 1991. He examined the behaviour of the Mandelbrot set near the "neck" at {{math|(−0.75, 0)}}. When the number of iterations until divergence for the point {{math|(−0.75, ''ε'')}} is multiplied by {{mvar|ε}}, the result approaches {{pi}} as {{mvar|ε}} approaches zero. The point {{math|(0.25 + ''ε'', 0)}} at the cusp of the large "valley" on the right side of the Mandelbrot set behaves similarly: the number of iterations until divergence multiplied by the square root of {{mvar|ε}} tends to {{pi}}.<ref>{{cite journal |last1=Klebanoff |first1=Aaron |year=2001 |title=Pi in the Mandelbrot set |journal=Fractals |volume=9 |issue=4 |pages=393–402 |url=http://home.comcast.net/~davejanelle/mandel.pdf |archive-url=https://web.archive.org/web/20111027155739/http://home.comcast.net/~davejanelle/mandel.pdf |archive-date=27 October 2011 |access-date=14 April 2012 |doi=10.1142/S0218348X01000828 |url-status=dead}} {{pb}} {{cite book |last=Peitgen |first=Heinz-Otto |author-link=Heinz-Otto Peitgen |title=Chaos and fractals: new frontiers of science |publisher=Springer |year=2004 |pages=801–803 |isbn=978-0-387-20229-7}}</ref>