Editing Gamma function (section)

== Applications ==
One author describes the gamma function as "Arguably, the most common special function, or the least 'special' of them. The other transcendental functions […] are called 'special' because you could conceivably avoid some of them by staying away from many specialized mathematical topics. On the other hand, the gamma function {{math|1= Γ(''z'')}} is most difficult to avoid."<ref>Michon, G. P. "[http://home.att.net/~numericana/answer/functions.htm Trigonometry and Basic Functions] {{Webarchive|url=https://web.archive.org/web/20100109035934/http://home.att.net/~numericana/answer/functions.htm |date=9 January 2010 }}". ''Numericana''. Retrieved 5 May 2007.</ref>

=== Integration problems ===
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The gamma function finds application in such diverse areas as [[quantum physics]], [[astrophysics]] and [[fluid dynamics]].<ref>{{cite book |last=Chaudry |first=M. A. |last2=Zubair |first2=S. M. |year=2001 |title=On A Class of Incomplete Gamma Functions with Applications |publisher=CRC Press |location=Boca Raton |isbn=1-58488-143-7 |page=37 }}</ref> 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.<ref>{{cite book |last=Rice |first=J. A. |year=1995 |title=Mathematical Statistics and Data Analysis |edition=Second |publisher=Duxbury Press |location=Belmont |isbn=0-534-20934-3 |pages=52–53 }}</ref>

The primary reason for the gamma function's usefulness in such contexts is the prevalence of expressions of the type <math>f(t)e^{-g(t)}</math>  which describe processes that decay exponentially in time or space. Integrals of such expressions can occasionally be solved in terms of the gamma function when no elementary solution exists. For example, if {{math|''f''}} is a power function and {{math|''g''}} is a linear function, a simple change of variables <math>u:=a\cdot t</math> gives the evaluation

<math display="block">\int_0^\infty t^b e^{-at} \,dt = \frac{1}{a^b} \int_0^\infty u^b e^{-u} d\left(\frac{u}{a}\right) = \frac{\Gamma(b+1)}{a^{b+1}}.</math>

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.

It is of course frequently useful to take limits of integration other than 0 and {{math|∞}} 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]]. (The ordinary gamma function, obtained by integrating across the entire positive real line, is sometimes called the ''complete gamma function'' for contrast.)

An important category of exponentially decaying functions is that of [[Gaussian function]]s
<math display="block">ae^{-\frac{(x-b)^2}{c^2}}</math>
and integrals thereof, such as the [[error function]]. There are many interrelations between these functions and the gamma function; notably, the factor <math>\sqrt{\pi}</math> obtained by evaluating <math display="inline">\Gamma \left( \frac{1}{2} \right)</math> is the "same" as that found in the normalizing factor of the error function and the [[normal distribution]].

The integrals discussed so far involve [[transcendental function]]s, but the gamma function also arises from integrals of purely algebraic functions. In particular, the [[arc length]]s of [[ellipse]]s and of the [[Lemniscate of Bernoulli#Arc length and elliptic functions|lemniscate]], which are curves defined by algebraic equations, are given by [[elliptic integral]]s that in special cases can be evaluated in terms of the gamma function. The gamma function can also be used to [[Volume of an n-ball|calculate "volume" and "area"]] of {{math|''n''}}-dimensional [[hypersphere]]s.

=== Calculating products ===
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]]. 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]].  For example, for any complex numbers {{mvar|z}} and {{mvar|n}}, with {{math|{{abs|''z''}} < 1}}, we can write
<math display="block">(1 + z)^n = \sum_{k=0}^\infty \frac{\Gamma(n+1)}{k!\Gamma(n-k+1)} z^k,</math>
which closely resembles the binomial coefficient when {{mvar|n}} is a non-negative integer,
<math display="block">(1 + z)^n = \sum_{k=0}^n \frac{n!}{k!(n-k)!} z^k = \sum_{k=0}^n \binom{n}{k} z^k.</math>

The example of binomial coefficients motivates why the properties of the gamma function when extended to negative numbers are natural. A binomial coefficient gives the number of ways to choose {{mvar|k}} elements from a set of {{mvar|n}} elements; if {{math|''k'' > ''n''}}, there are of course no ways. If {{math|''k'' > ''n''}}, {{math|(''n'' − ''k'')!}} 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.

We can replace the factorial by a gamma function to extend any such formula to the complex numbers. 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. If {{math|''P''}} and {{math|''Q''}} are monic polynomials of degree {{mvar|m}} and {{mvar|n}} with respective roots {{math|''p''{{sub|1}}, …, ''p{{sub|m}}''}} and {{math|''q''{{sub|1}}, …, ''q{{sub|n}}''}}, we have
<math display="block">\prod_{i=a}^b \frac{P(i)}{Q(i)} = \left( \prod_{j=1}^m \frac{\Gamma(b-p_j+1)}{\Gamma(a-p_j)} \right) \left( \prod_{k=1}^n \frac{\Gamma(a-q_k)}{\Gamma(b-q_k+1)} \right).</math>

If we have a way to calculate the gamma function numerically, it is very simple to calculate numerical values of such products. The number of gamma functions in the right-hand side depends only on the degree of the polynomials, so it does not matter whether {{math|''b'' − ''a''}} equals 5 or 10<sup>5</sup>. By taking the appropriate limits, the equation can also be made to hold even when the left-hand product contains zeros or poles.

By taking limits, certain rational products with infinitely many factors can be evaluated in terms of the gamma function as well. 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. We have already seen one striking example: the reflection formula essentially represents the sine function as the product of two gamma functions. 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.

More functions yet, including the [[hypergeometric function]] and special cases thereof, can be represented by means of complex [[contour integral]]s of products and quotients of the gamma function, called [[Barnes integral|Mellin–Barnes integral]]s.

=== Analytic number theory ===

An application of the gamma function is the study of the [[Riemann zeta function]]. A fundamental property of the Riemann zeta function is its [[functional equation]]:
<math display="block">\Gamma\left(\frac{s}{2}\right)\zeta(s)\pi^{-\frac{s}{2}} = \Gamma\left(\frac{1-s}{2}\right)\zeta(1-s)\pi^{-\frac{1-s}{2}}.</math>

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. Borwein ''et al.'' call this formula "one of the most beautiful findings in mathematics".<ref>{{cite book |author = Borwein, J. |author2 = Bailey, D. H. |author3 = Girgensohn, R. |name-list-style = amp |year = 2003 |title = Experimentation in Mathematics |publisher = A. K. Peters |pages = 133 |isbn = 978-1-56881-136-9 }}</ref> Another contender for that title might be
<math display="block">\zeta(s) \; \Gamma(s) = \int_0^\infty \frac{t^s}{e^t-1} \, \frac{dt}{t}.</math>

Both formulas were derived by [[Bernhard Riemann]] in his seminal 1859 paper "''[[On the Number of Primes Less Than a Given Magnitude|Ueber die Anzahl der Primzahlen unter einer gegebenen Größe]]''" ("On the Number of Primes Less Than a Given Magnitude"), one of the milestones in the development of [[analytic number theory]]—the branch of mathematics that studies [[prime number]]s using the tools of mathematical analysis.