Because is closely related to the circle, it is found in
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 in some of their important formulae.
Geometry and trigonometry is . appears in formulae for areas and volumes of geometrical shapes based on circles, such as
ellipses,
spheres,
cones, and
tori. Below are some of the more common formulae that involve . • The circumference of a circle with radius is . • The
area of a circle with radius is . • The area of an ellipse with semi-major axis and semi-minor axis is . • The volume of a sphere with radius is . • The surface area of a sphere with radius is . Some of the formulae above are special cases of the volume of the
n-dimensional ball and the surface area of its boundary, the
(n−1)-dimensional sphere, given
below. Apart from circles, there are other
curves of constant width. By
Barbier's theorem, every curve of constant width has perimeter 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
smooth and even
algebraic curves of constant width.
Definite integrals that describe circumference, area, or volume of shapes generated by circles typically have values that involve . For example, an integral that specifies half the area of a circle of radius one is given by: \int_{-1}^1 \sqrt{1-x^2}\,dx = \frac{\pi}{2}. In that integral, the function \sqrt{1-x^2} represents the height over the x-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 an
algebraic period.
Unit of angle and
cosine functions repeat with period 2.|leftThe
trigonometric functions rely on
angles, and mathematicians generally use the
radian as a unit of measurement. plays an important role in angles measured in radians, which are defined so that a complete circle spans an angle of 2radians. The angle measure of 180° is equal to radians, and . Common trigonometric functions have periods that are multiples of ; for example, sine and cosine have period 2, so for any angle and any integer , \sin\theta = \sin\left(\theta + 2\pi k \right) \text{ and } \cos\theta = \cos\left(\theta + 2\pi k \right).
Eigenvalues s of a vibrating string are
eigenfunctions of the second derivative, and form a
harmonic progression. The associated eigenvalues form the
arithmetic progression of integer multiples of . Many of the appearances of in the formulae of mathematics and the sciences have to do with its close relationship with geometry. However, 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 on the unit interval , with
fixed ends . The modes of vibration of the string are solutions of the
differential equation , or . Thus is an eigenvalue of the second derivative
operator , 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 , where is called the
wavenumber. Then satisfies the boundary conditions and the differential equation with . The value 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 : [0, 1] \to \Complex with and , both
square integrable, we have: \pi^2\int_0^1|f(x)|^2\,dx\le \int_0^1|f'(x)|^2\,dx, with equality precisely when is a multiple of . Here 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, is the smallest
singular value of the derivative operator on the space of functions on vanishing at both endpoints (the
Sobolev space H^1_0[0,1]).
Inequalities was the solution to an isoperimetric problem, according to a legend recounted by
Lord Kelvin: those lands bordering the sea that
Queen Dido could enclose on all other sides within a single given oxhide, cut into strips.|left The number serves appears in similar eigenvalue problems in higher-dimensional analysis. As mentioned
above, it can be characterized via its role as the best constant in the
isoperimetric inequality: the area enclosed by a plane
Jordan curve of perimeter satisfies the inequality 4\pi A\le P^2, and equality is clearly achieved for the circle, since in that case and . Ultimately, as a consequence of the isoperimetric inequality, appears in the optimal constant for the critical
Sobolev inequality in
n dimensions, which thus characterizes the role of in many physical phenomena as well, for example those of classical
potential theory. In two dimensions, the critical Sobolev inequality is 2\pi\|f\|_2 \le \|\nabla f\|_1 for
f a smooth function with compact support in , \nabla f is the
gradient of
f, and \|f\|_2 and \|\nabla f\|_1 refer respectively to the
and -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 inequalities that provide best constants for the
Dirichlet energy of an
n-dimensional membrane. Specifically, is the greatest constant such that \pi \le \frac{\left (\int_G |\nabla u|^2\right)^{1/2}}{\left (\int_G|u|^2\right)^{1/2}} for all
convex subsets of of diameter 1, and square-integrable functions
u on of mean zero. Just as Wirtinger's inequality is the
variational form of the
Dirichlet eigenvalue problem in one dimension, the Poincaré inequality is the variational form of the
Neumann eigenvalue problem, in any dimension.
Fourier transform and Heisenberg uncertainty principle The constant also appears as a critical spectral parameter in the
Fourier transform. This is the
integral transform, that takes a complex-valued integrable function on the real line to the function defined as: \hat{f}(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi i x\xi}\,dx. Although there are several different conventions for the Fourier transform and its inverse, any such convention must involve
somewhere. The above is the most canonical definition, however, giving the unique unitary operator on that is also an algebra homomorphism of to . The
Heisenberg uncertainty principle also contains the number . 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, \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. The physical consequence, about the uncertainty in simultaneous position and momentum observations of a
quantum mechanical system, is
discussed below. The appearance of 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.
Gaussian integrals . The coloured region between the function and the
x-axis has area .|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 experiments follows a normal distribution. The
Gaussian function, which is the
probability density function of the normal distribution with
mean and
standard deviation , naturally contains : f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu )^2/(2\sigma^2)}. The factor of \tfrac{1}{\sqrt{2\pi}} makes the area under the graph of equal to one, as is required for a probability distribution. This follows from a
change of variables in the
Gaussian integral: \int_{-\infty}^\infty e^{-u^2} \, du=\sqrt{\pi} , which says that the area under the basic
bell curve in the figure is equal to the square root of . The
central limit theorem explains the central role of normal distributions, and thus of , in probability and statistics. This theorem is ultimately connected with the
spectral characterization of as the eigenvalue associated with the Heisenberg uncertainty principle, and the fact that equality holds in the uncertainty principle only for the Gaussian function. Equivalently, is the unique constant making the Gaussian normal distribution equal to its own Fourier transform. Indeed, according to , the "whole business" of establishing the fundamental theorems of Fourier analysis reduces to the Gaussian integral.
Topology of the
Klein quartic, a surface of
genus three and Euler characteristic −4, as a quotient of the
hyperbolic plane by the
symmetry group PSL(2,7) of the
Fano plane. The hyperbolic area of a fundamental domain is , by Gauss–Bonnet. The constant appears in the
Gauss–Bonnet formula which relates the
differential geometry of surfaces to their
topology. Specifically, if a
compact surface has
Gauss curvature K, then \int_\Sigma K\,dA = 2\pi \chi(\Sigma) where is the
Euler characteristic, which is an integer. 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 groups and is found to be equal to two. Thus we have A(S) = \int_S 1\,dA = 2\pi\cdot 2 = 4\pi 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 classes via the
Chern–Weil homomorphism.
Cauchy's integral formula One of the key tools in
complex analysis is
contour integration of a function over a positively oriented (
rectifiable)
Jordan curve . A form of
Cauchy's integral formula states that if a point is interior to , then \oint_\gamma \frac{dz}{z-z_0} = 2\pi i. Although the curve is not a circle, and hence does not have any obvious connection to the constant , 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 does not contain , then the above integral is 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 on the Jordan curve and the value of at any interior point of : \oint_\gamma { f(z) \over z-z_0 }\,dz = 2\pi i f (z_{0}) provided is analytic in the region enclosed by and extends continuously to . Cauchy's integral formula is a special case of the
residue theorem, that if is a
meromorphic function the region enclosed by and is continuous in a neighbourhood of , then \oint_\gamma g(z)\, dz =2\pi i \sum \operatorname{Res}( g, a_k ) where the sum is of the
residues at the
poles of .
Vector calculus and physics The constant is ubiquitous in
vector calculus and
potential theory, for example in
Coulomb's law,
Gauss's law,
Maxwell's equations, and even the
Einstein field equations. 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: \Phi(\mathbf x) = \frac{1}{2\pi}\log|\mathbf x|. The factor of 1/2\pi is necessary to ensure that \Phi is the
fundamental solution of the
Poisson equation in \mathbb R^2: \Delta\Phi = \delta where \delta is the
Dirac delta function. In higher dimensions, factors of 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: \Phi(\mathbf x) = -\frac{1}{4\pi|\mathbf x|}, which has the 2-dimensional volume (i.e., the area) of the unit 2-sphere in the denominator.
Total curvature about and an additional loop which does not contain . 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: \int_a^b k(s)\,ds For a closed curve, this quantity is equal to for an integer called the
turning number or
index of the curve. 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, is the
degree of the map taking each point on the curve to the corresponding point on the hodograph, analogous to the
Gauss map for surfaces.
Gamma function and Stirling's approximation The
factorial function n! is the product of all of the positive integers through . 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 \Gamma(n)=(n-1)!. When the gamma function is evaluated at half-integers, the result contains . For example, \Gamma\bigl(\tfrac12\bigr) = \sqrt{\pi} and \Gamma\bigl(\tfrac52\bigr) = \tfrac 34 \sqrt{\pi} . The gamma function is defined by its
Weierstrass product development: \Gamma(z) = \frac{e^{-\gamma z}}{z}\prod_{n=1}^\infty \frac{e^{z/n}}{1+z/n} where is the
Euler–Mascheroni constant. Evaluated at and squared, the equation 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
plays an important role. The gamma function is used to calculate the volume of the
n-dimensional ball of radius
r in Euclidean
n-dimensional space, and the surface area of its boundary, the
(n−1)-dimensional sphere: V_n(r) = \frac{\pi^{n/2}}{\Gamma\bigl(\frac{n}{2}+1\bigr)}r^n, S_{n-1}(r) = \frac{n\pi^{n/2}}{\Gamma\bigl(\tfrac{n}{2}+1\bigr)}r^{n-1}. Further, it follows from the
functional equation that 2\pi r = \frac{S_{n+1}(r)}{V_n(r)}. The gamma function can be used to create a simple approximation to the factorial function for large : n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n which is known as
Stirling's approximation. Equivalently, \pi = \lim_{n\to\infty} \frac{e^{2n}n!^2}{2 n^{2n+1}}. As a geometrical application of Stirling's approximation, let denote the
standard simplex in
n-dimensional Euclidean space, and denote the simplex having all of its sides scaled up by a factor of . Then \operatorname{Vol}((n+1)\Delta_n) = \frac{(n+1)^n}{n!} \sim \frac{e^{n+1}}{\sqrt{2\pi n}}.
Ehrhart's volume conjecture is that this is the (optimal) upper bound on the volume of a
convex body containing only one
integer lattice point.
Number theory and Riemann zeta function , which are arithmetic localizations of the circle. The
L-functions of analytic number theory are also localized in each prime
p. : the value of is the
hyperbolic area of a fundamental domain of the
modular group, times . The
Riemann zeta function is used in many areas of mathematics. When evaluated at it can be written as \zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots Finding a
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 . 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 . This probability is based on the observation that the probability that any number is
divisible by a prime is (for example, every 7th integer is divisible by 7). Hence the probability that two numbers are both divisible by this prime is , and the probability that at least one of them is not is . 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: \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} This probability can be used in conjunction with a
random number generator to approximate using a Monte Carlo approach. The solution to the Basel problem implies that the geometrically derived quantity 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 . The zeta function also satisfies Riemann's functional equation, which involves as well as the gamma function: \zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s). Furthermore, the derivative of the zeta function satisfies \exp(-\zeta'(0)) = \sqrt{2\pi}. A consequence is that 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. The calculation can be recast in
quantum mechanics, specifically the
variational approach to the
spectrum of the hydrogen atom.
Fourier series (shown), which are elements of a
Prüfer group.
Tate's thesis makes heavy use of this machinery.|left The constant also appears naturally in
Fourier series of
periodic functions. Periodic functions are functions on the group of fractional parts of real numbers. The Fourier decomposition shows that a complex-valued function on can be written as an infinite linear superposition of
unitary characters of . That is, continuous
group homomorphisms from to the
circle group of unit modulus complex numbers. It is a theorem that every character of is one of the complex exponentials e_n(x)= e^{2\pi i n x}. It is a theorem that every character of is one of the complex exponentials e_n(x)= e^{2\pi i n x}. Among these, there is a unique character, up to complex conjugation, that is a group isomorphism from onto the multiplicative group of complex numbers of absolute value one. A definition of the constant, due to
Nicolas Bourbaki, takes to be half the magnitude of the derivative of this isomorphism. Equivalently, using the
Haar measure on the circle group, the other characters have derivatives whose magnitudes are positive integral multiples of 2. Thus is the unique positive number such that the group , equipped with its Haar measure, is
Pontrjagin dual to the
lattice of integral multiples of 2. This is a version of the one-dimensional
Poisson summation formula. In Fourier analysis, the appearance of rather than 2 often signals the passage from ordinary characters to
projective representations. The basic exponential e^{\pi i x} is not a character of the group , since it changes sign after one turn of the circle; instead, it defines a representation of the
double cover of . This is the simplest example of a projective representation, and the same phenomenon recurs more broadly in harmonic analysis.
Spinors, for example, represent rotations only after passing to a double cover, and
metaplectic representations similarly arise from a double cover of SL(2,R)|. The same ideas also connect Fourier analysis with arithmetic, since theta functions and modular forms of half-integral weight are naturally governed by these metaplectic and Heisenberg-theoretic constructions.
Modular forms and theta functions of periods of an elliptic curve. The constant is connected in a deep way with the theory of
modular forms and
theta functions. For example, the
Chudnovsky algorithm involves in an essential way the
j-invariant of an
elliptic curve.
Modular forms are
holomorphic functions in the
upper half plane characterized by their transformation properties under the
modular group \mathrm{SL}_2(\mathbb Z) (or its various subgroups), a lattice in the group {{tmath| \mathrm{SL}_2(\R) }}. An example is the
Jacobi theta function \theta(z,\tau) = \sum_{n=-\infty}^\infty e^{2\pi i nz \ +\ \pi i n^2\tau} which is a kind of modular form called a
Jacobi form. This is sometimes written in terms of the
nome q=e^{\pi i \tau}. The constant 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 \theta(z+\tau,\tau) = e^{-\pi i\tau -2\pi i z}\theta(z,\tau), which implies that transforms as a representation under the discrete
Heisenberg group. General modular forms and other
theta functions also involve , once again because of the
Stone–von Neumann theorem.
Cauchy distribution and potential theory , named for
Maria Agnesi (1718–1799), is a geometrical construction of the graph of the Cauchy distribution.|left through a membrane. The
Cauchy distribution g(x)=\frac{1}{\pi}\cdot\frac{1}{x^2+1} is a
probability density function. The total probability is equal to one, owing to the integral: \int_{-\infty }^{\infty } \frac{1}{x^2+1} \, dx = \pi. The
Shannon entropy of the Cauchy distribution is equal to , which also involves . The Cauchy distribution plays an important role in
potential theory because it is the simplest
Furstenberg measure, the classical
Poisson kernel associated with a
Brownian motion in a half-plane.
Conjugate harmonic functions 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 Hf(t) = \frac{1}{\pi}\int_{-\infty}^\infty \frac{f(x)\,dx}{x-t}. The constant 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. The Hilbert transform, like the Fourier transform, can be characterized purely in terms of its transformation properties on the Hilbert space : 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. The constant is the unique normalizing factor that makes this transformation unitary.
In the Mandelbrot set can be used to approximate . An occurrence of 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 . When the number of iterations until divergence for the point is multiplied by , the result approaches as approaches zero. The point 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 tends to . == Outside mathematics ==