Editing Gamma function (section)

=== 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.