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Return to Part VII of the course APMA0340
Introduction to Linear Algebra with Mathematica
Glossary
Preface
The gamma function Γ(x) is the natural extension of the factorial function n!=∏nk=1k=1⋅2⋅3⋯n from integer n to real or complex x. It was first defined and studied by L. Euler in 18th century, who used the notation Γ(z), the capital letter gamma from the Greek alphabet. It is commonly used in many mathematical problems, including differential equations, but it appears occasionally in physical problems such as the normalization of Coulomb wave functions and the computation of probabilities in statistical mechanics. In general, however, it has less direct physical application and interpretation than, say, the Chebyshev and Bessel functions. Rather, its importance stems from its usefulness in developing other functions that have direct physical application.
Gamma Function of Euler
The first definition, given by Leonhard Euler is
\Gamma (\nu ) = \int_0^{\infty} t^{\nu -1} e^{-t} {\text d} t , \qquad \Re (\nu ) > 0.
\label{EqGamma.1}
\end{equation}
There are known several equivalent definitions of the gamma function:
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Mathematica has a dedicated command for the gamma function -- Gamma[ z ].
You can make a fancy plotof this function:
Plot[Gamma[x], {x, -3, 3}, PlotStyle -> {Purple, Thickness[0.01]}]
The gamma function has a local minimum at +1.46163214496836234126.
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The Gamma function. | Mathematica code |
When ν = ½, Eq.(???) is just the Gauss erroe integral Γ(1/2)=√π. In general, we have
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With Mathematica, one can plot the reciprocal of the gamma functions:
Plot[1/Gamma[x], {x, -3.1, 3}, PlotStyle -> {Purple, Thickness[0.01]}]
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The reciprocal of the Gamma function. | Mathematica code |
Incomplete gamma function
Beta function
Digamma function
The exponential integral
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