Pi

The number ( ; spelled out as pi) is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining , to avoid relying on the definition of the length of a curve. The number is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as are commonly used to approximate it. Consequently, its decimal representation never ends, nor does it enter a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an algebraic equation involving only finite sums, products, powers, and integers. The transcendence of implies that it is impossible to solve the ancient problem of squaring the circle with a compass and straightedge. The decimal digits of appear to be evenly distributed, but no proof of this conjecture has been found. Mathematicians have long attempted to extend their understanding of , sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate with arbitrary accuracy.…

The number ( ; spelled out as pi) is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining , to avoid relying on the definition of the length of a curve. The number is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as are commonly used to approximate it. Consequently, its decimal representation never ends, nor does it enter a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an algebraic equation involving only finite sums, products, powers, and integers. The transcendence of implies that it is impossible to solve the ancient problem of squaring the circle with a compass and straightedge. The decimal digits of appear to be evenly distributed, but no proof of this conjecture has been found. Mathematicians have long attempted to extend their understanding of , sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for , based on infinite series, was discovered a millennium later. The development of calculus in the 17th century soon led to the calculation of hundreds of digits of , enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have extended the decimal representation of to hundreds of trillions of digits. Because it relates to a circle, is found in formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is found as well in formulae from cosmology, fractals, thermodynamics, mechanics, and electromagnetism. It also appears in areas such as number theory, statistics, and in modern mathematical analysis is ubiquitous. \pi = \frac{C}{d}. In Euclidean geometry, the ratio \frac{C}{d} is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle, it will also have twice the circumference, preserving the ratio \frac{C}{d} . Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined using limits a concept in calculus. For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation x^2+y^2=1 , as the integral: \pi = \int_{-1}^1 \frac{dx}{\sqrt{1-x^2}}. An integral such as this was proposed as a definition of by Karl Weierstrass, who defined it directly as an integral in 1841. Integration is no longer commonly used in a first analytical definition because, as explains, differential calculus typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of that does not rely on the latter. One such definition, due to Richard Baltzer and popularized by Edmund Landau, is the following: is twice the smallest positive number at which the cosine function equals 0. is also the smallest positive number at which the sine function equals zero, and the difference between consecutive zeroes of the sine function. The cosine and sine can be defined independently of geometry as a power series, or as the solution of a differential equation. In a similar spirit, can be defined using properties of the complex exponential, , of a complex variable . Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which is equal to one is then an (imaginary) arithmetic progression of the form: \{\dots,-2\pi i, 0, 2\pi i, 4\pi i,\dots\} = \{2\pi ki\mid k\in\mathbb Z\} and there is a unique positive real number with this property. \pi = 3+\textstyle \cfrac{1}{7+\textstyle \cfrac{1}{15+\textstyle \cfrac{1}{1+\textstyle \cfrac{1}{292+\textstyle \cfrac{1}{1+\textstyle \cfrac{1}{1+\textstyle \cfrac{1}{1+\ddots}}}}}}} Truncating the continued fraction at any point yields a rational approximation for ; the first four of these are , , , and . These numbers are among the best-known and most widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to than any other fraction with the same or a smaller denominator. Because is transcendental, it is by definition not algebraic and so cannot be a quadratic irrational. Therefore, cannot have a periodic continued fraction. Although the simple continued fraction for (with numerators all 1, shown above) also does not exhibit any other obvious pattern, several non-simple continued fractions do, such as: {{overflow| \begin{align} \pi &= 3+ \cfrac {1^2}{6+ \cfrac {3^2}{6+ \cfrac {5^2}{6+ \cfrac {7^2}{6+ \ddots}}}} = \cfrac {4}{1+ \cfrac {1^2}{2+ \cfrac {3^2}{2+ \cfrac {5^2}{2+ \ddots}}}} = \cfrac {4}{1+ \cfrac {1^2}{3+ \cfrac {2^2}{5+ \cfrac {3^2}{7+ \ddots}}}} \end{align} }} z = r\cdot(\cos\varphi + i\sin\varphi), where is the imaginary unit satisfying i^2=-1 . The frequent appearance of in complex analysis can be related to the behaviour of the exponential function of a complex variable, described by Euler's formula: e^{i\varphi} = \cos \varphi + i\sin \varphi, where the constant is the base of the natural logarithm. This formula establishes a correspondence between imaginary powers of and points on the unit circle centred at the origin of the complex plane. Setting \varphi=\pi in Euler's formula results in Euler's identity, celebrated in mathematics due to it containing five important mathematical constants: e^{i \pi} + 1 = 0. There are different complex numbers satisfying , and these are called the " th roots of unity" and are given by the formula: e^{2 \pi i k/n} \qquad (k = 0, 1, 2, \dots, n - 1). Surviving approximations of prior to the 2nd century CE are accurate to one or two decimal places at best. The earliest written approximations are found in Babylon and Egypt, both within one percent of the true value. In Babylon, a clay tablet dated 1900–1600 BCE has a geometrical statement that, by implication, treats as . In Egypt, the Rhind Papyrus, dated around 1650 BCE but copied from a document dated to 1850 BCE, has a formula for the area of a circle that treats as \bigl(\frac{16}{9}\bigr)^2\approx3.16 . Although some pyramidologists have theorized that the Great Pyramid of Giza was built with proportions related to , this theory is not widely accepted by scholars. In the Shulba Sutras of Indian mathematics, dating to an oral tradition from the 1st or 2nd millennium BCE, approximations are given which have been variously interpreted as approximately 3.08831, 3.08833, 3.004, 3, or 3.125. In ancient China, values for included 3.1547 (around 1 AD), \sqrt{10} (100 AD, approximately 3.1623), and (3rd century, approximately 3.1556). Around 265 AD, the Cao Wei mathematician Liu Hui created a polygon-based iterative algorithm, with which he constructed a 3,072-sided polygon to approximate as 3.1416. Liu later invented a faster method of calculating and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4. Around 480 AD, Zu Chongzhi calculated that 3.1415926 and suggested the approximations \pi \approx \frac{355}{113} = 3.14159292035\ldots and \pi \approx \frac{22}{7} = 3.142857142857\ldots , which he termed the milü ('close ratio') and yuelü ('approximate ratio') respectively, iterating with Liu Hui's algorithm up to a 12,288-sided polygon. With a correct value for its seven first decimal digits, Zu's result remained the most accurate approximation of for the next 800 years. The Indian astronomer Aryabhata used a value of 3.1416 in his Āryabhaṭīya (499 AD). Around 1220, Fibonacci computed 3.1418 using a polygonal method devised independently of Archimedes. Italian author Dante apparently employed the value 3+\frac{\sqrt{2}}{10} \approx 3.14142 . The Persian astronomer Jamshīd al-Kāshī produced nine sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424, using a polygon with 3\times 2^{28} sides, which stood as the world record for about 180 years. French mathematician François Viète in 1579 achieved nine digits with a polygon of 3\times 2^{17} sides. Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593. In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, was called the "Ludolphian number" in Germany until the early 20th century). Dutch scientist Willebrord Snellius reached 34 digits in 1621, and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630 using 10 40 sides. Christiaan Huygens was able to arrive at 10 decimal places in 1654 using a slightly different method equivalent to Richardson extrapolation. \frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots In 1655, John Wallis published what is now known as the Wallis product, also an infinite product: \frac{\pi}{2} = \Big(\frac{2}{1} \cdot \frac{2}{3}\Big) \cdot \Big(\frac{4}{3} \cdot \frac{4}{5}\Big) \cdot \Big(\frac{6}{5} \cdot \frac{6}{7}\Big) \cdot \Big(\frac{8}{7} \cdot \frac{8}{9}\Big) \cdots used infinite series to compute to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations". In the 1660s, the English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz discovered calculus, which led to the development of many infinite series for approximating . Newton himself used an arcsine series to compute a 15-digit approximation of in 1665 or 1666, writing, "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time." In 1671, James Gregory, and independently, Leibniz in 1673, discovered the Taylor series expansion for arctangent: \arctan z = z - \frac {z^3} {3} +\frac {z^5} {5} -\frac {z^7} {7} +\cdots This series, sometimes called the Gregory–Leibniz series, equals \frac{\pi}{4} when evaluated with . But for , it converges impractically slowly (that is, approaches the answer very gradually), taking about ten times as many terms to calculate each additional digit. In 1699, English mathematician Abraham Sharp used the Gregory–Leibniz series for z=\frac{1}{\sqrt{3}} to compute to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm. In 1706, John Machin used the Gregory–Leibniz series to produce an algorithm that converged much faster: \frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}. Machin reached 100 digits of with this formula. Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for calculating digits of . Isaac Newton accelerated the convergence of the Gregory–Leibniz series in 1684 (in an unpublished work; others independently discovered the result): Leonhard Euler popularized this series in his 1755 differential calculus textbook, and later used it with Machin-like formulae, including \tfrac\pi4 = 5\arctan\tfrac17 + 2\arctan\tfrac{3}{79}, with which he computed 20 digits of in one hour. In 1844, a record was set by Zacharias Dase, who employed a Machin-like formula to calculate 200 decimals of in his head at the behest of German mathematician Carl Friedrich Gauss. Machin-like formulae remained the best-known method for calculating well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson the best approximation achieved without the aid of a calculating device. In 1853, British mathematician William Shanks calculated to 607 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect. Though he calculated an additional 100 digits in 1873, bringing the total up to 707, his previous mistake rendered all the new digits incorrect as well. \pi = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \frac{4}{13} - \cdots As individual terms of this infinite series are added to the sum, the total gradually gets closer to , and with a sufficient number of terms can get as close to as desired. It converges quite slowly, though after 500,000 terms, it produces only five correct decimal digits of . An infinite series for (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is: The following table compares the convergence rates of these two series: After five terms, the sum of the Gregory–Leibniz series is within 0.2 of the correct value of , whereas the sum of Nilakantha's series is within 0.002 of the correct value. Nilakantha's series converges faster and is more useful for computing digits of . Series that converge even faster include Machin's series and Chudnovsky's series, the latter producing 14 correct decimal digits per term. \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots Swiss scientist Johann Heinrich Lambert in 1768 proved that is irrational, meaning it is not equal to the quotient of any two integers. Lambert's proof exploited a continued-fraction representation of the tangent function. French mathematician Adrien-Marie Legendre proved in 1794 that 2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that is transcendental, confirming a conjecture made by both Legendre and Euler. where the Greek letters and δ were combined into the fraction for denoting the ratios semiperimeter to semidiameter and perimeter to diameter, that is, what is presently denoted as . (Before then, mathematicians sometimes used letters such as c or p instead. ) Barrow likewise used the same notation, while Gregory instead used \frac \pi \rho to represent . The earliest known use of the Greek letter alone to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in his 1706 work ; or, a New Introduction to the Mathematics. The Greek letter appears on p. 243 in the phrase " \tfrac12 Periphery ( )", calculated for a circle with radius one. However, Jones writes that his equations for are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones. Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767. Euler started using the single-letter form beginning with his 1727 Essay Explaining the Properties of Air, though he used , the ratio of periphery to radius, in this and some later writing. Euler first used in his 1736 work Mechanica, and continued in his widely read 1748 work (he wrote: "for the sake of brevity we will write this number as ; thus is equal to half the circumference of a circle of radius "). Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly, and the practice was universally adopted thereafter in the Western world, though the definition still varied between and as late as 1761. .|upright=1.3 For most numerical calculations involving , a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the circumference of the observable universe with a precision of one atom. Accounting for additional digits needed to compensate for computational round-off errors, Arndt concludes that a few hundred digits would suffice for any scientific application. Despite this, people have worked strenuously to compute to thousands and millions of digits. This effort may be partly ascribed to the human compulsion to break records, and such achievements with often make headlines around the world. They also have practical benefits, such as testing supercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms) and within pure mathematics itself, providing data for evaluating the randomness of the digits of . \frac{1}{\pi} = \frac{2 \sqrt 2}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{k!^4 396^{4k}}. This series converges much more rapidly than most arctan series, including Machin's formula. Bill Gosper was the first to use it for advances in the calculation of , setting a record of 17 million digits in 1985. Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers (Jonathan and Peter) and the Chudnovsky brothers. The Chudnovsky formula developed in 1987 is {{overflow| \frac{1}{\pi} = \frac{\sqrt{10005}}{4270934400} \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!\,k!^3 (-640320)^{3k}}. }} It produces about 14 digits of per term and has been used for several record-setting calculations, including the first to surpass 1 billion (10 9 ) digits in 1989 by the Chudnovsky brothers, 10 trillion (10 13 ) digits in 2011 by Alexander Yee and Shigeru Kondo, and 100 trillion digits by Emma Haruka Iwao in 2022. For similar formulae, see also the Ramanujan–Sato series. In 2006, mathematician Simon Plouffe used the PSLQ integer relation algorithm to generate several new formulae for , conforming to the following template: \pi^k = \sum_{n=1}^\infty \frac{1}{n^k} \left(\frac{a}{q^n-1} + \frac{b}{q^{2n}-1} + \frac{c}{q^{4n}-1}\right), where is (Gelfond's constant), is an odd number, and , , are certain rational numbers that Plouffe computed. \pi \approx \frac{2n\ell}{xt}. Another Monte Carlo method for computing is to draw a circle inscribed in a square, and randomly place dots in the square. The ratio of dots inside the circle to the total number of dots will approximately equal . Another way to calculate using probability is to start with a random walk, generated by a sequence of (fair) coin tosses: independent random variables such that with equal probabilities. The associated random walk is W_n = \sum_{k=1}^n X_k so that, for each , is drawn from a shifted and scaled binomial distribution. As varies, defines a (discrete) stochastic process. Then can be calculated by \pi = \lim_{n\to\infty} \frac{2n}{E[|W_n|]^2}. This Monte Carlo method is independent of any relation to circles, and is a consequence of the central limit theorem, discussed below. These Monte Carlo methods for approximating are very slow compared to other methods, and do not provide any information on the exact number of digits obtained. Thus they are never used to approximate when speed or accuracy is desired. \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. \sin\theta = \sin\left(\theta + 2\pi k \right) \text{ and } \cos\theta = \cos\left(\theta + 2\pi k \right). \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] ). 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. \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, 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. 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. \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. \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 on 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 . \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. \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. 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 is naturally expressed in terms of . 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 predicts that this is the (optimal) upper bound on the volume of a convex body containing only one integer lattice point. \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-theoretic 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. 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 . 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. \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, meaning 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. 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 hence 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. T \approx 2\pi \sqrt\frac{L}{g}. One of the key formulae of quantum mechanics is Heisenberg's uncertainty principle, which shows that the uncertainty in the measurement of a particle's position ( ) and momentum ( ) cannot both be arbitrarily small at the same time (where is the Planck constant): \Delta x\, \Delta p \ge \frac{h}{4\pi}. The fact that is approximately equal to 3 plays a role in the relatively long lifetime of orthopositronium. The inverse lifetime to lowest order in the fine-structure constant is \frac{1}{\tau} = 2\frac{\pi^2 - 9}{9\pi}m_\text{e}\alpha^{6}, where is the mass of the electron. is present in some structural engineering formulae, such as the buckling formula derived by Euler, which gives the maximum axial load that a long, slender column of length , modulus of elasticity , and area moment of inertia can carry without buckling: F =\frac{\pi^2EI}{L^2}. The field of fluid dynamics contains in Stokes' law, which approximates the frictional force exerted on small, spherical objects of radius , moving with velocity in a fluid with dynamic viscosity : F =6\pi\eta Rv. In electromagnetics, the vacuum permeability constant appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation. Before 20 May 2019, it was defined as exactly

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