Gamma function

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For the gamma function of ordinals, see Veblen function. Gamma function_sentence_0

For the gamma distribution in statistics, see Gamma distribution. Gamma function_sentence_1

For the function used in video and image color representations, see Gamma correction. Gamma function_sentence_2

Derived by Daniel Bernoulli for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: Gamma function_sentence_3

The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. Gamma function_sentence_4

Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. Gamma function_sentence_5

It is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics. Gamma function_sentence_6

Motivation Gamma function_section_0

The gamma function can be seen as a solution to the following interpolation problem: Gamma function_sentence_7

Gamma function_description_list_0

  • "Find a smooth curve that connects the points (x, y) given by y = (x − 1)! at the positive integer values for x."Gamma function_item_0_0

A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that precisely describes the curve, in which the number of operations does not depend on the size of x. Gamma function_sentence_8

The simple formula for the factorial, x! Gamma function_sentence_9

= 1 × 2 × ⋯ × x, cannot be used directly for fractional values of x since it is only valid when x is a natural number (or positive integer). Gamma function_sentence_10

There are, relatively speaking, no such simple solutions for factorials; no finite combination of sums, products, powers, exponential functions, or logarithms will suffice to express x! Gamma function_sentence_11

but it is possible to find a general formula for factorials using tools such as integrals and limits from calculus. Gamma function_sentence_12

A good solution to this is the gamma function. Gamma function_sentence_13

There are infinitely many continuous extensions of the factorial to non-integers: infinitely many curves can be drawn through any set of isolated points. Gamma function_sentence_14

The gamma function is the most useful solution in practice, being analytic (except at the non-positive integers), and it can be defined in several equivalent ways. Gamma function_sentence_15

However, it is not the only analytic function which extends the factorial, as adding to it any analytic function which is zero on the positive integers, such as k sin mπx, will give another function with that property. Gamma function_sentence_16

A more restrictive property than satisfying the above interpolation is to satisfy the recurrence relation defining a translated version of the factorial function, Gamma function_sentence_17

for any positive real number x. Gamma function_sentence_18

But this would allow for multiplication by any periodic analytic function which evaluates to 1 on the positive integers, such as e. One of several ways to finally resolve the ambiguity comes from the Bohr–Mollerup theorem. Gamma function_sentence_19

It states that when the condition that f be logarithmically convex (or "super-convex") is added, it uniquely determines f for positive, real inputs. Gamma function_sentence_20

From there, the gamma function can be extended to all real and complex values (except the negative integers and zero) by using the unique analytic continuation of f. Gamma function_sentence_21

Definition Gamma function_section_1

Main definition Gamma function_section_2

converges absolutely, and is known as the Euler integral of the second kind. Gamma function_sentence_22

(Euler's integral of the first kind is the beta function.) Gamma function_sentence_23

Using integration by parts, one sees that: Gamma function_sentence_24

for all positive integers n. This can be seen as an example of proof by induction. Gamma function_sentence_25

Alternative definitions Gamma function_section_3

Euler's definition as an infinite product Gamma function_section_4

Weierstrass's definition Gamma function_section_5

The definition for the gamma function due to Weierstrass is also valid for all complex numbers z except the non-positive integers: Gamma function_sentence_26

In terms of generalized Laguerre polynomials Gamma function_section_6

A representation of the incomplete gamma function in terms of generalized Laguerre polynomials is Gamma function_sentence_27

Properties Gamma function_section_7

General Gamma function_section_8

Other important functional equations for the gamma function are Euler's reflection formula Gamma function_sentence_28

which implies Gamma function_sentence_29

and the Legendre duplication formula Gamma function_sentence_30

The duplication formula is a special case of the multiplication theorem (See, Eq. Gamma function_sentence_31

5.5.6) Gamma function_sentence_32

A simple but useful property, which can be seen from the limit definition, is: Gamma function_sentence_33

In particular, with z = a + bi, this product is Gamma function_sentence_34

If the real part is an integer or a half-integer, this can be finitely expressed in closed form: Gamma function_sentence_35

Perhaps the best-known value of the gamma function at a non-integer argument is Gamma function_sentence_36

Another useful limit for asymptotic approximations is: Gamma function_sentence_37

The derivatives of the gamma function are described in terms of the polygamma function. Gamma function_sentence_38

For example: Gamma function_sentence_39

Using the identity Gamma function_sentence_40

we have in particular Gamma function_sentence_41

Inequalities Gamma function_section_9

When restricted to the positive real numbers, the gamma function is a strictly logarithmically convex function. Gamma function_sentence_42

This property may be stated in any of the following three equivalent ways: Gamma function_sentence_43

Gamma function_unordered_list_1

  • For any two positive real numbers x and y with y > x,Gamma function_item_1_1

There are also bounds on ratios of gamma functions. Gamma function_sentence_44

The best-known is Gautschi's inequality, which says that for any positive real number x and any s ∈ (0, 1), Gamma function_sentence_45

Stirling's formula Gamma function_section_10

Residues Gamma function_section_11

and the denominator Gamma function_sentence_46

So the residues of the gamma function at those points are: Gamma function_sentence_47

Minima Gamma function_section_12

Integral representations Gamma function_section_13

There are many formulas, besides the Euler integral of the second kind, that express the gamma function as an integral. Gamma function_sentence_48

For instance, when the real part of z is positive, Gamma function_sentence_49

Binet's first integral formula for the gamma function states that, when the real part of z is positive, then: Gamma function_sentence_50

The integral on the right-hand side may be interpreted as a Laplace transform. Gamma function_sentence_51

That is, Gamma function_sentence_52

Binet's second integral formula states that, again when the real part of z is positive, then: Gamma function_sentence_53

again valid whenever z is not an integer. Gamma function_sentence_54

Fourier series expansion Gamma function_section_14

which was for a long time attributed to Ernst Kummer, who derived it in 1847. Gamma function_sentence_55

However, Iaroslav Blagouchine discovered that Carl Johan Malmsten first derived this series in 1842. Gamma function_sentence_56

Raabe's formula Gamma function_section_15

In 1840 Joseph Ludwig Raabe proved that Gamma function_sentence_57

Pi function Gamma function_section_16

Using the pi function the reflection formula takes on the form Gamma function_sentence_58

where sinc is the normalized sinc function, while the multiplication theorem takes on the form Gamma function_sentence_59

We also sometimes find Gamma function_sentence_60

The volume of an n-ellipsoid with radii r1, ..., rn can be expressed as Gamma function_sentence_61

Relation to other functions Gamma function_section_17

Gamma function_unordered_list_2

  • In the first integral above, which defines the gamma function, the limits of integration are fixed. The upper and lower incomplete gamma functions are the functions obtained by allowing the lower or upper (respectively) limit of integration to vary.Gamma function_item_2_2
  • The gamma function is related to the beta function by the formulaGamma function_item_2_3

Gamma function_unordered_list_3

Particular values Gamma function_section_18

Main article: Particular values of the gamma function Gamma function_sentence_62

Including up to the first 20 digits after the decimal point, some particular values of the gamma function are: Gamma function_sentence_63

The complex-valued gamma function is undefined for non-positive integers, but in these cases the value can be defined in the Riemann sphere as ∞. Gamma function_sentence_64

The reciprocal gamma function is well defined and analytic at these values (and in the entire complex plane): Gamma function_sentence_65

The log-gamma function Gamma function_section_19

Because the gamma and factorial functions grow so rapidly for moderately large arguments, many computing environments include a function that returns the natural logarithm of the gamma function (often given the name lgamma or lngamma in programming environments or gammaln in spreadsheets); this grows much more slowly, and for combinatorial calculations allows adding and subtracting logs instead of multiplying and dividing very large values. Gamma function_sentence_66

It is often defined as Gamma function_sentence_67

The digamma function, which is the derivative of this function, is also commonly seen. Gamma function_sentence_68

In the context of technical and physical applications, e.g. with wave propagation, the functional equation Gamma function_sentence_69

This can be used to accurately approximate ln(Γ(z)) for z with a smaller Re(z) via (P.E.Böhmer, 1939) Gamma function_sentence_70

A more accurate approximation can be obtained by using more terms from the asymptotic expansions of ln(Γ(z)) and Γ(z), which are based on Stirling's approximation. Gamma function_sentence_71

In a more "natural" presentation: Gamma function_sentence_72

The coefficients of the terms with k > 1 of z in the last expansion are simply Gamma function_sentence_73

where the Bk are the Bernoulli numbers. Gamma function_sentence_74

Properties Gamma function_section_20

The Bohr–Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the gamma function is log-convex, that is, its natural logarithm is convex on the positive real axis. Gamma function_sentence_75

Another characterisation is given by the Wielandt theorem. Gamma function_sentence_76

In a certain sense, the ln(Γ) function is the more natural form; it makes some intrinsic attributes of the function clearer. Gamma function_sentence_77

A striking example is the Taylor series of ln(Γ) around 1: Gamma function_sentence_78

with ζ(k) denoting the Riemann zeta function at k. Gamma function_sentence_79

So, using the following property: Gamma function_sentence_80

we can find an integral representation for the ln(Γ) function: Gamma function_sentence_81

or, setting z = 1 to obtain an integral for γ, we can replace the γ term with its integral and incorporate that into the above formula, to get: Gamma function_sentence_82

see. Gamma function_sentence_83

This formula is sometimes used for numerical computation, since the integrand decreases very quickly. Gamma function_sentence_84

Integration over log-gamma Gamma function_section_21

The integral Gamma function_sentence_85

can be expressed in terms of the Barnes G-function (see Barnes G-function for a proof): Gamma function_sentence_86

where Re(z) > −1. Gamma function_sentence_87

It can also be written in terms of the Hurwitz zeta function: Gamma function_sentence_88

D. H. Bailey and his co-authors gave an evaluation for Gamma function_sentence_89

In addition, it is also known that Gamma function_sentence_90

Approximations Gamma function_section_22

Complex values of the gamma function can be computed numerically with arbitrary precision using Stirling's approximation or the Lanczos approximation. Gamma function_sentence_91

A fast algorithm for calculation of the Euler gamma function for any algebraic argument (including rational) was constructed by E.A. Gamma function_sentence_92

Karatsuba, Gamma function_sentence_93

For arguments that are integer multiples of 1/24, the gamma function can also be evaluated quickly using arithmetic–geometric mean iterations (see particular values of the gamma function and harvtxt error: no target: CITEREFBorweinZucker1992 (help)). Gamma function_sentence_94

Applications Gamma function_section_23

One author describes the gamma function as "Arguably, the most common special function, or the least 'special' of them. Gamma function_sentence_95

The other transcendental functions […] are called 'special' because you could conceivably avoid some of them by staying away from many specialized mathematical topics. Gamma function_sentence_96

On the other hand, the gamma function y = Γ(x) is most difficult to avoid." Gamma function_sentence_97

Integration problems Gamma function_section_24

The gamma function finds application in such diverse areas as quantum physics, astrophysics and fluid dynamics. Gamma function_sentence_98

The gamma distribution, which is formulated in terms of the gamma function, is used in statistics to model a wide range of processes; for example, the time between occurrences of earthquakes. Gamma function_sentence_99

The fact that the integration is performed along the entire positive real line might signify that the gamma function describes the cumulation of a time-dependent process that continues indefinitely, or the value might be the total of a distribution in an infinite space. Gamma function_sentence_100

It is of course frequently useful to take limits of integration other than 0 and ∞ to describe the cumulation of a finite process, in which case the ordinary gamma function is no longer a solution; the solution is then called an incomplete gamma function. Gamma function_sentence_101

(The ordinary gamma function, obtained by integrating across the entire positive real line, is sometimes called the complete gamma function for contrast.) Gamma function_sentence_102

An important category of exponentially decaying functions is that of Gaussian functions Gamma function_sentence_103

The integrals we have discussed so far involve transcendental functions, but the gamma function also arises from integrals of purely algebraic functions. Gamma function_sentence_104

In particular, the arc lengths of ellipses and of the lemniscate, which are curves defined by algebraic equations, are given by elliptic integrals that in special cases can be evaluated in terms of the gamma function. Gamma function_sentence_105

The gamma function can also be used to calculate "volume" and "area" of n-dimensional hyperspheres. Gamma function_sentence_106

Calculating products Gamma function_section_25

The gamma function's ability to generalize factorial products immediately leads to applications in many areas of mathematics; in combinatorics, and by extension in areas such as probability theory and the calculation of power series. Gamma function_sentence_107

Many expressions involving products of successive integers can be written as some combination of factorials, the most important example perhaps being that of the binomial coefficient Gamma function_sentence_108

The example of binomial coefficients motivates why the properties of the gamma function when extended to negative numbers are natural. Gamma function_sentence_109

A binomial coefficient gives the number of ways to choose k elements from a set of n elements; if k > n, there are of course no ways. Gamma function_sentence_110

If k > n, (n − k)! Gamma function_sentence_111

is the factorial of a negative integer and hence infinite if we use the gamma function definition of factorials—dividing by infinity gives the expected value of 0. Gamma function_sentence_112

We can replace the factorial by a gamma function to extend any such formula to the complex numbers. Gamma function_sentence_113

Generally, this works for any product wherein each factor is a rational function of the index variable, by factoring the rational function into linear expressions. Gamma function_sentence_114

If P and Q are monic polynomials of degree m and n with respective roots p1, …, pm and q1, …, qn, we have Gamma function_sentence_115

If we have a way to calculate the gamma function numerically, it is a breeze to calculate numerical values of such products. Gamma function_sentence_116

The number of gamma functions in the right-hand side depends only on the degree of the polynomials, so it does not matter whether b − a equals 5 or 10. Gamma function_sentence_117

By taking the appropriate limits, the equation can also be made to hold even when the left-hand product contains zeros or poles. Gamma function_sentence_118

By taking limits, certain rational products with infinitely many factors can be evaluated in terms of the gamma function as well. Gamma function_sentence_119

Due to the Weierstrass factorization theorem, analytic functions can be written as infinite products, and these can sometimes be represented as finite products or quotients of the gamma function. Gamma function_sentence_120

We have already seen one striking example: the reflection formula essentially represents the sine function as the product of two gamma functions. Gamma function_sentence_121

Starting from this formula, the exponential function as well as all the trigonometric and hyperbolic functions can be expressed in terms of the gamma function. Gamma function_sentence_122

More functions yet, including the hypergeometric function and special cases thereof, can be represented by means of complex contour integrals of products and quotients of the gamma function, called Mellin–Barnes integrals. Gamma function_sentence_123

Analytic number theory Gamma function_section_26

An elegant and deep application of the gamma function is in the study of the Riemann zeta function. Gamma function_sentence_124

A fundamental property of the Riemann zeta function is its functional equation: Gamma function_sentence_125

Among other things, this provides an explicit form for the analytic continuation of the zeta function to a meromorphic function in the complex plane and leads to an immediate proof that the zeta function has infinitely many so-called "trivial" zeros on the real line. Gamma function_sentence_126

Borwein et al. Gamma function_sentence_127

call this formula "one of the most beautiful findings in mathematics". Gamma function_sentence_128

Another champion for that title might be Gamma function_sentence_129

Both formulas were derived by Bernhard Riemann in his seminal 1859 paper "Über die Anzahl der Primzahlen unter einer gegebenen Größe" ("On the Number of Prime Numbers less than a Given Quantity"), one of the milestones in the development of analytic number theory—the branch of mathematics that studies prime numbers using the tools of mathematical analysis. Gamma function_sentence_130

Factorial numbers, considered as discrete objects, are an important concept in classical number theory because they contain many prime factors, but Riemann found a use for their continuous extension that arguably turned out to be even more important. Gamma function_sentence_131

History Gamma function_section_27

The gamma function has caught the interest of some of the most prominent mathematicians of all time. Gamma function_sentence_132

Its history, notably documented by Philip J. Davis in an article that won him the 1963 Chauvenet Prize, reflects many of the major developments within mathematics since the 18th century. Gamma function_sentence_133

In the words of Davis, "each generation has found something of interest to say about the gamma function. Gamma function_sentence_134

Perhaps the next generation will also." Gamma function_sentence_135

18th century: Euler and Stirling Gamma function_section_28

The problem of extending the factorial to non-integer arguments was apparently first considered by Daniel Bernoulli and Christian Goldbach in the 1720s, and was solved at the end of the same decade by Leonhard Euler. Gamma function_sentence_136

Euler gave two different definitions: the first was not his integral but an infinite product, Gamma function_sentence_137

of which he informed Goldbach in a letter dated October 13, 1729. Gamma function_sentence_138

He wrote to Goldbach again on January 8, 1730, to announce his discovery of the integral representation Gamma function_sentence_139

which is valid for n > 0. Gamma function_sentence_140

By the change of variables t = −ln s, this becomes the familiar Euler integral. Gamma function_sentence_141

Euler published his results in the paper "De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt" ("On transcendental progressions, that is, those whose general terms cannot be given algebraically"), submitted to the St. Gamma function_sentence_142 Petersburg Academy on November 28, 1729. Gamma function_sentence_143

Euler further discovered some of the gamma function's important functional properties, including the reflection formula. Gamma function_sentence_144

James Stirling, a contemporary of Euler, also attempted to find a continuous expression for the factorial and came up with what is now known as Stirling's formula. Gamma function_sentence_145

Although Stirling's formula gives a good estimate of n!, also for non-integers, it does not provide the exact value. Gamma function_sentence_146

Extensions of his formula that correct the error were given by Stirling himself and by Jacques Philippe Marie Binet. Gamma function_sentence_147

19th century: Gauss, Weierstrass and Legendre Gamma function_section_29

Carl Friedrich Gauss rewrote Euler's product as Gamma function_sentence_148

and used this formula to discover new properties of the gamma function. Gamma function_sentence_149

Although Euler was a pioneer in the theory of complex variables, he does not appear to have considered the factorial of a complex number, as instead Gauss first did. Gamma function_sentence_150

Gauss also proved the multiplication theorem of the gamma function and investigated the connection between the gamma function and elliptic integrals. Gamma function_sentence_151

Karl Weierstrass further established the role of the gamma function in complex analysis, starting from yet another product representation, Gamma function_sentence_152

where γ is the Euler–Mascheroni constant. Gamma function_sentence_153

Weierstrass originally wrote his product as one for 1/Γ, in which case it is taken over the function's zeros rather than its poles. Gamma function_sentence_154

Inspired by this result, he proved what is known as the Weierstrass factorization theorem—that any entire function can be written as a product over its zeros in the complex plane; a generalization of the fundamental theorem of algebra. Gamma function_sentence_155

The name gamma function and the symbol Γ were introduced by Adrien-Marie Legendre around 1811; Legendre also rewrote Euler's integral definition in its modern form. Gamma function_sentence_156

Although the symbol is an upper-case Greek gamma, there is no accepted standard for whether the function name should be written "gamma function" or "Gamma function" (some authors simply write "Γ-function"). Gamma function_sentence_157

The alternative "pi function" notation Π(z) = z! Gamma function_sentence_158

due to Gauss is sometimes encountered in older literature, but Legendre's notation is dominant in modern works. Gamma function_sentence_159

19th–20th centuries: characterizing the gamma function Gamma function_section_30

It is somewhat problematic that a large number of definitions have been given for the gamma function. Gamma function_sentence_160

Although they describe the same function, it is not entirely straightforward to prove the equivalence. Gamma function_sentence_161

Stirling never proved that his extended formula corresponds exactly to Euler's gamma function; a proof was first given by Charles Hermite in 1900. Gamma function_sentence_162

Instead of finding a specialized proof for each formula, it would be desirable to have a general method of identifying the gamma function. Gamma function_sentence_163

One way to prove would be to find a differential equation that characterizes the gamma function. Gamma function_sentence_164

Most special functions in applied mathematics arise as solutions to differential equations, whose solutions are unique. Gamma function_sentence_165

However, the gamma function does not appear to satisfy any simple differential equation. Gamma function_sentence_166

Otto Hölder proved in 1887 that the gamma function at least does not satisfy any algebraic differential equation by showing that a solution to such an equation could not satisfy the gamma function's recurrence formula, making it a transcendentally transcendental function. Gamma function_sentence_167

This result is known as Hölder's theorem. Gamma function_sentence_168

A definite and generally applicable characterization of the gamma function was not given until 1922. Gamma function_sentence_169

Harald Bohr and Johannes Mollerup then proved what is known as the Bohr–Mollerup theorem: that the gamma function is the unique solution to the factorial recurrence relation that is positive and logarithmically convex for positive z and whose value at 1 is 1 (a function is logarithmically convex if its logarithm is convex). Gamma function_sentence_170

Another characterisation is given by the Wielandt theorem. Gamma function_sentence_171

The Bohr–Mollerup theorem is useful because it is relatively easy to prove logarithmic convexity for any of the different formulas used to define the gamma function. Gamma function_sentence_172

Taking things further, instead of defining the gamma function by any particular formula, we can choose the conditions of the Bohr–Mollerup theorem as the definition, and then pick any formula we like that satisfies the conditions as a starting point for studying the gamma function. Gamma function_sentence_173

This approach was used by the Bourbaki group. Gamma function_sentence_174

Borwein & Corless review three centuries of work on the gamma function. Gamma function_sentence_175

Reference tables and software Gamma function_section_31

Although the gamma function can be calculated virtually as easily as any mathematically simpler function with a modern computer—even with a programmable pocket calculator—this was of course not always the case. Gamma function_sentence_176

Until the mid-20th century, mathematicians relied on hand-made tables; in the case of the gamma function, notably a table computed by Gauss in 1813 and one computed by Legendre in 1825. Gamma function_sentence_177

Tables of complex values of the gamma function, as well as hand-drawn graphs, were given in Tables of Higher Functions by Jahnke and Emde [], first published in Germany in 1909. Gamma function_sentence_178

According to Michael Berry, "the publication in J&E of a three-dimensional graph showing the poles of the gamma function in the complex plane acquired an almost iconic status." Gamma function_sentence_179

There was in fact little practical need for anything but real values of the gamma function until the 1930s, when applications for the complex gamma function were discovered in theoretical physics. Gamma function_sentence_180

As electronic computers became available for the production of tables in the 1950s, several extensive tables for the complex gamma function were published to meet the demand, including a table accurate to 12 decimal places from the U.S. National Bureau of Standards. Gamma function_sentence_181

Abramowitz and Stegun became the standard reference for this and many other special functions after its publication in 1964. Gamma function_sentence_182

Double-precision floating-point implementations of the gamma function and its logarithm are now available in most scientific computing software and special functions libraries, for example TK Solver, Matlab, GNU Octave, and the GNU Scientific Library. Gamma function_sentence_183

The gamma function was also added to the C standard library (math.h). Gamma function_sentence_184

Arbitrary-precision implementations are available in most computer algebra systems, such as Mathematica and Maple. Gamma function_sentence_185

PARI/GP, MPFR and MPFUN contain free arbitrary-precision implementations. Gamma function_sentence_186

A little-known feature of the calculator app included with the Android operating system is that it will accept fractional values as input to the factorial function and return the equivalent gamma function value. Gamma function_sentence_187

The same is true for Windows Calculator (in scientific mode). Gamma function_sentence_188

See also Gamma function_section_32

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: function.