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4,500 | is an irrational number, meaning that it cannot be written as the ratio of two integers. Fractions such as and are commonly used to approximate , but no common fraction (ratio of whole numbers) can be its exact value. Because is irrational, it has an infinite number of digits in its decimal representation, and does not... | https://en.wikipedia.org/wiki?curid=23601 |
4,501 | The digits of have no apparent pattern and have passed tests for statistical randomness, including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often. The conjecture that is normal has not been proven or disproven. | https://en.wikipedia.org/wiki?curid=23601 |
4,502 | Since the advent of computers, a large number of digits of have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of , and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected t... | https://en.wikipedia.org/wiki?curid=23601 |
4,503 | In addition to being irrational, is also a transcendental number, which means that it is not the solution of any non-constant polynomial equation with rational coefficients, such as . | https://en.wikipedia.org/wiki?curid=23601 |
4,504 | The transcendence of has two important consequences: First, cannot be expressed using any finite combination of rational numbers and square roots or "n"-th roots (such as or ). Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to "square the circle". In other wo... | https://en.wikipedia.org/wiki?curid=23601 |
4,505 | Like all irrational numbers, cannot be represented as a common fraction (also known as a simple or vulgar fraction), by the very definition of irrational number (i.e., not a rational number). But every irrational number, including , can be represented by an infinite series of nested fractions, called a continued fracti... | https://en.wikipedia.org/wiki?curid=23601 |
4,506 | 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 ... | https://en.wikipedia.org/wiki?curid=23601 |
4,507 | Any complex number, say , can be expressed using a pair of real numbers. In the polar coordinate system, one number (radius or "r") is used to represent 's distance from the origin of the complex plane, and the other (angle or ) the counter-clockwise rotation from the positive real line: | https://en.wikipedia.org/wiki?curid=23601 |
4,508 | where is the imaginary unit satisfying = −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: | https://en.wikipedia.org/wiki?curid=23601 |
4,509 | 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 = in Euler's formula results in Euler's identity, celebrated in mathematics due to it containing five important... | https://en.wikipedia.org/wiki?curid=23601 |
4,510 | There are different complex numbers satisfying , and these are called the "-th roots of unity" and are given by the formula: | https://en.wikipedia.org/wiki?curid=23601 |
4,511 | The best-known approximations to dating before the Common Era were accurate to two decimal places; this was improved upon in Chinese mathematics in particular by the mid-first millennium, to an accuracy of seven decimal places. | https://en.wikipedia.org/wiki?curid=23601 |
4,512 | The earliest written approximations of are found in Babylon and Egypt, both within one percent of the true value. In Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats as = 3.125. In Egypt, the Rhind Papyrus, dated around 1650 BC but copied from a document dated to 1850 ... | https://en.wikipedia.org/wiki?curid=23601 |
4,513 | In the Shulba Sutras of Indian mathematics, dating to an oral tradition from the first or second millennium BC, approximations are given which have been variously interpreted as approximately 3.08831, 3.08833, 3.004, 3, or 3.125. | https://en.wikipedia.org/wiki?curid=23601 |
4,514 | The first recorded algorithm for rigorously calculating the value of was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes. This polygonal algorithm dominated for over 1,000 years, and as a result is sometimes referred to as Archimedes's constant. Archimedes computed upp... | https://en.wikipedia.org/wiki?curid=23601 |
4,515 | In ancient China, values for included 3.1547 (around 1 AD), (100 AD, approximately 3.1623), and (3rd century, approximately 3.1556). Around 265 AD, the Wei Kingdom mathematician Liu Hui created a polygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain a value of of 3.1416. Liu later invented ... | https://en.wikipedia.org/wiki?curid=23601 |
4,516 | The Indian astronomer Aryabhata used a value of 3.1416 in his "Āryabhaṭīya" (499 AD). Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes. Italian author Dante apparently employed the value . | https://en.wikipedia.org/wiki?curid=23601 |
4,517 | The Persian astronomer Jamshīd al-Kāshī produced 9 sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424 using a polygon with 3×2 sides, which stood as the world record for about 180 years. French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×2 sides. Flemish mathematicia... | https://en.wikipedia.org/wiki?curid=23601 |
4,518 | The calculation of was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence. Infinite series allowed mathematicians to compute with much greater precision than Archimedes and others who used geometrical techniq... | https://en.wikipedia.org/wiki?curid=23601 |
4,519 | In 1593, François Viète published what is now known as Viète's formula, an infinite product (rather than an infinite sum, which is more typically used in calculations): | https://en.wikipedia.org/wiki?curid=23601 |
4,520 | 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 arcsin series to compute a 15-digit approximation of in 1665 or 1666, writing "I am ashamed to tell you... | https://en.wikipedia.org/wiki?curid=23601 |
4,521 | In 1699, English mathematician Abraham Sharp used the Gregory–Leibniz series for formula_14 to compute to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm. The Gregory–Leibniz series for formula_15 is simple, but converges very slowly (that is, approaches the answer gradual... | https://en.wikipedia.org/wiki?curid=23601 |
4,522 | In 1706, John Machin used the Gregory–Leibniz series to produce an algorithm that converged much faster: | https://en.wikipedia.org/wiki?curid=23601 |
4,523 | 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 . Machin-like formulae remained the best-known method for calculating well into the age of computers, and were used to set ... | https://en.wikipedia.org/wiki?curid=23601 |
4,524 | 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. | https://en.wikipedia.org/wiki?curid=23601 |
4,525 | 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. | https://en.wikipedia.org/wiki?curid=23601 |
4,526 | Some infinite series for converge faster than others. Given the choice of two infinite series for , mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate to any given accuracy. A simple infinite series for is the Gregory–Le... | https://en.wikipedia.org/wiki?curid=23601 |
4,527 | 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 . | https://en.wikipedia.org/wiki?curid=23601 |
4,528 | An infinite series for (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is: | https://en.wikipedia.org/wiki?curid=23601 |
4,529 | 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 Chud... | https://en.wikipedia.org/wiki?curid=23601 |
4,530 | Not all mathematical advances relating to were aimed at increasing the accuracy of approximations. When Euler solved the Basel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between and the prime numbers that later contributed to the development and study of t... | https://en.wikipedia.org/wiki?curid=23601 |
4,531 | 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 is also irrational. In 1882, Germa... | https://en.wikipedia.org/wiki?curid=23601 |
4,532 | In the earliest usages, the Greek letter was used to denote the semiperimeter ("semiperipheria" in Latin) of a circle. and was combined in ratios with δ (for diameter or semidiameter) or ρ (for radius) to form circle constants. (Before then, mathematicians sometimes used letters such as "c" or "p" instead.) The first r... | https://en.wikipedia.org/wiki?curid=23601 |
4,533 | 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 first appears there in the phrase "1/2 Periphery ()" in the discussion of a circle... | https://en.wikipedia.org/wiki?curid=23601 |
4,534 | 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... | https://en.wikipedia.org/wiki?curid=23601 |
4,535 | The development of computers in the mid-20th century again revolutionized the hunt for digits of . Mathematicians John Wrench and Levi Smith reached 1,120 digits in 1949 using a desk calculator. Using an inverse tangent (arctan) infinite series, a team led by George Reitwiesner and John von Neumann that same year achie... | https://en.wikipedia.org/wiki?curid=23601 |
4,536 | Two additional developments around 1980 once again accelerated the ability to compute . First, the discovery of new iterative algorithms for computing , which were much faster than the infinite series; and second, the invention of fast multiplication algorithms that could multiply large numbers very rapidly. Such algor... | https://en.wikipedia.org/wiki?curid=23601 |
4,537 | The iterative algorithms were independently published in 1975–1976 by physicist Eugene Salamin and scientist Richard Brent. These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step ... | https://en.wikipedia.org/wiki?curid=23601 |
4,538 | The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally "multiply" the number of correct digits at each step. For example, the Bren... | https://en.wikipedia.org/wiki?curid=23601 |
4,539 | 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... | https://en.wikipedia.org/wiki?curid=23601 |
4,540 | Modern calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive. The fast iterative algorithms were anticipated in 1914, when Indian mathematician Srinivasa Ramanujan published ... | https://en.wikipedia.org/wiki?curid=23601 |
4,541 | 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 Pete... | https://en.wikipedia.org/wiki?curid=23601 |
4,542 | 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) digits in 1989 by the Chudnovsky brothers, 10 trillion (10) digits in 2011 by Alexander Yee and Shigeru Kondo, and 100 trillion digits by Emma Haruka Iwao in 2022. For simila... | https://en.wikipedia.org/wiki?curid=23601 |
4,543 | In 2006, mathematician Simon Plouffe used the PSLQ integer relation algorithm to generate several new formulas for , conforming to the following template: | https://en.wikipedia.org/wiki?curid=23601 |
4,544 | where is (Gelfond's constant), is an odd number, and are certain rational numbers that Plouffe computed. | https://en.wikipedia.org/wiki?curid=23601 |
4,545 | Monte Carlo methods, which evaluate the results of multiple random trials, can be used to create approximations of . Buffon's needle is one such technique: If a needle of length is dropped times on a surface on which parallel lines are drawn units apart, and if of those times it comes to rest crossing a line ( > 0), th... | https://en.wikipedia.org/wiki?curid=23601 |
4,546 | 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 . | https://en.wikipedia.org/wiki?curid=23601 |
4,547 | 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 | https://en.wikipedia.org/wiki?curid=23601 |
4,548 | 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 | https://en.wikipedia.org/wiki?curid=23601 |
4,549 | This Monte Carlo method is independent of any relation to circles, and is a consequence of the central limit theorem, discussed below. | https://en.wikipedia.org/wiki?curid=23601 |
4,550 | 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 that are obtained. Thus they are never used to approximate when speed or accuracy is desired. | https://en.wikipedia.org/wiki?curid=23601 |
4,551 | Two algorithms were discovered in 1995 that opened up new avenues of research into . They are called spigot algorithms because, like water dripping from a spigot, they produce single digits of that are not reused after they are calculated. This is in contrast to infinite series or iterative algorithms, which retain and... | https://en.wikipedia.org/wiki?curid=23601 |
4,552 | Mathematicians Stan Wagon and Stanley Rabinowitz produced a simple spigot algorithm in 1995. Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms. | https://en.wikipedia.org/wiki?curid=23601 |
4,553 | Another spigot algorithm, the BBP digit extraction algorithm, was discovered in 1995 by Simon Plouffe: | https://en.wikipedia.org/wiki?curid=23601 |
4,554 | This formula, unlike others before it, can produce any individual hexadecimal digit of without calculating all the preceding digits. Individual binary digits may be extracted from individual hexadecimal digits, and octal digits can be extracted from one or two hexadecimal digits. Variations of the algorithm have been d... | https://en.wikipedia.org/wiki?curid=23601 |
4,555 | Between 1998 and 2000, the distributed computing project PiHex used Bellard's formula (a modification of the BBP algorithm) to compute the quadrillionth (10th) bit of , which turned out to be 0. In September 2010, a Yahoo! employee used the company's Hadoop application on one thousand computers over a 23-day period to ... | https://en.wikipedia.org/wiki?curid=23601 |
4,556 | 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. | https://en.wikipedia.org/wiki?curid=23601 |
4,557 | 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 . | https://en.wikipedia.org/wiki?curid=23601 |
4,558 | 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. | https://en.wikipedia.org/wiki?curid=23601 |
4,559 | 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 t... | https://en.wikipedia.org/wiki?curid=23601 |
4,560 | 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: | https://en.wikipedia.org/wiki?curid=23601 |
4,561 | In that integral the function represents the height over the formula_30-axis of a semicircle (the square root is a consequence of the Pythagorean theorem), and the integral computes the area below the semicircle. | https://en.wikipedia.org/wiki?curid=23601 |
4,562 | The trigonometric functions rely on angles, and mathematicians generally use radians as units of measurement. plays an important role in angles measured in radians, which are defined so that a complete circle spans an angle of 2 radians. The angle measure of 180° is equal to radians, and 1° = /180 radians. | https://en.wikipedia.org/wiki?curid=23601 |
4,563 | Common trigonometric functions have periods that are multiples of ; for example, sine and cosine have period 2, so for any angle and any integer , | https://en.wikipedia.org/wiki?curid=23601 |
4,564 | Many of the appearances of in the formulas 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. | https://en.wikipedia.org/wiki?curid=23601 |
4,565 | 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 formula_32, or formula_33. Thus is an eigen... | https://en.wikipedia.org/wiki?curid=23601 |
4,566 | 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 formula_35 with and , both square integrable, we have: | https://en.wikipedia.org/wiki?curid=23601 |
4,567 | 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 function... | https://en.wikipedia.org/wiki?curid=23601 |
4,568 | 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 | https://en.wikipedia.org/wiki?curid=23601 |
4,569 | 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 inequalit... | https://en.wikipedia.org/wiki?curid=23601 |
4,570 | for "f" a smooth function with compact support in , formula_40 is the gradient of "f", and formula_41 and formula_42 refer respectively to the and -norm. The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants. | https://en.wikipedia.org/wiki?curid=23601 |
4,571 | 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 | https://en.wikipedia.org/wiki?curid=23601 |
4,572 | 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. | https://en.wikipedia.org/wiki?curid=23601 |
4,573 | 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: | https://en.wikipedia.org/wiki?curid=23601 |
4,574 | 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 . | https://en.wikipedia.org/wiki?curid=23601 |
4,575 | 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, | https://en.wikipedia.org/wiki?curid=23601 |
4,576 | 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 represent... | https://en.wikipedia.org/wiki?curid=23601 |
4,577 | 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. The Gaussian function, which is the probability density function of the normal... | https://en.wikipedia.org/wiki?curid=23601 |
4,578 | The factor of formula_47 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: | https://en.wikipedia.org/wiki?curid=23601 |
4,579 | 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... | https://en.wikipedia.org/wiki?curid=23601 |
4,580 | Let be the set of all twice differentiable real functions formula_49 that satisfy the ordinary differential equation formula_50. Then is a two-dimensional real vector space, with two parameters corresponding to a pair of initial conditions for the differential equation. For any formula_51, let formula_52 be the evaluat... | https://en.wikipedia.org/wiki?curid=23601 |
4,581 | 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 | https://en.wikipedia.org/wiki?curid=23601 |
4,582 | 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 | https://en.wikipedia.org/wiki?curid=23601 |
4,583 | The constant appears in many other integral formulae in topology, in particular, those involving characteristic classes via the Chern–Weil homomorphism. | https://en.wikipedia.org/wiki?curid=23601 |
4,584 | Vector calculus is a branch of calculus that is concerned with the properties of vector fields, and has many physical applications such as to electricity and magnetism. The Newtonian potential for a point source situated at the origin of a three-dimensional Cartesian coordinate system is | https://en.wikipedia.org/wiki?curid=23601 |
4,585 | which represents the potential energy of a unit mass (or charge) placed a distance from the source, and is a dimensional constant. The field, denoted here by , which may be the (Newtonian) gravitational field or the (Coulomb) electric field, is the negative gradient of the potential: | https://en.wikipedia.org/wiki?curid=23601 |
4,586 | Special cases include Coulomb's law and Newton's law of universal gravitation. Gauss' law states that the outward flux of the field through any smooth, simple, closed, orientable surface containing the origin is equal to : | https://en.wikipedia.org/wiki?curid=23601 |
4,587 | It is standard to absorb this factor of into the constant , in which case it appears in the numerator of the equation for the potential. This argument shows why it must appear "somewhere". Furthermore, is the surface area of the unit sphere, but we have not assumed that is the sphere. However, as a consequence of the d... | https://en.wikipedia.org/wiki?curid=23601 |
4,588 | A consequence of the Gauss law is that the negative Laplacian of the potential is equal to times the Dirac delta function: | https://en.wikipedia.org/wiki?curid=23601 |
4,589 | More general distributions of matter (or charge) are obtained from this by convolution, giving the Poisson equation | https://en.wikipedia.org/wiki?curid=23601 |
4,590 | The constant also plays an analogous role in four-dimensional potentials associated with Einstein's equations, a fundamental formula which forms the basis of the general theory of relativity and describes the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy: | https://en.wikipedia.org/wiki?curid=23601 |
4,591 | where is the Ricci curvature tensor, is the scalar curvature, is the metric tensor, is the cosmological constant, is Newton's gravitational constant, is the speed of light in vacuum, and is the stress–energy tensor. The left-hand side of Einstein's equation is a non-linear analogue of the Laplacian of the metric tensor... | https://en.wikipedia.org/wiki?curid=23601 |
4,592 | 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 | https://en.wikipedia.org/wiki?curid=23601 |
4,593 | 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... | https://en.wikipedia.org/wiki?curid=23601 |
4,594 | 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 : | https://en.wikipedia.org/wiki?curid=23601 |
4,595 | 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 | https://en.wikipedia.org/wiki?curid=23601 |
4,596 | The factorial function formula_69 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 formula_70. When the gamma function is evaluated a... | https://en.wikipedia.org/wiki?curid=23601 |
4,597 | 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. | https://en.wikipedia.org/wiki?curid=23601 |
4,598 | 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: | https://en.wikipedia.org/wiki?curid=23601 |
4,599 | The gamma function can be used to create a simple approximation to the factorial function for large : formula_77 which is known as Stirling's approximation. Equivalently, | https://en.wikipedia.org/wiki?curid=23601 |
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