Particular values of the gamma function

Particular values of the Gamma function are of significant interest in various fields of mathematics and science, such as complex analysis, number theory, and physics. The Gamma function, denoted as Γ(z), extends the factorial function to complex numbers, except for negative integers and zero. For any positive integer n, Γ(n) = (n-1)!, and for a complex number z with a real part greater than zero, it is defined through an improper integral. This article focuses on the specific values of the Gamma function that have notable importance and applications.

Definition

The Gamma function is defined for complex numbers z with a real part greater than zero by the integral:

\[ \Gamma(z) = \int_0^\infty t^{z-1}e^{-t} dt \]

For other values of z, the function is defined through analytic continuation.

Particular Values

Half-Integer Values

For half-integer values, that is, when z = n + 1/2 where n is a non-negative integer, the Gamma function has a closed-form expression involving the square root of pi (\(\sqrt{\pi}\)):

\[ \Gamma\left(n + \frac{1}{2}\right) = \frac{(2n)!}{4^n n!}\sqrt{\pi} = \frac{(2n-1)!!}{2^n}\sqrt{\pi} \]

where n!! denotes the double factorial.

Euler's Reflection Formula

Euler's reflection formula provides a relationship between Γ(z) and Γ(1−z) for complex number z:

\[ \Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)} \]

This formula is particularly useful for determining the values of the Gamma function for negative arguments.

Gauss's Multiplication Theorem

Gauss's multiplication theorem gives a way to express the Gamma function at values z/n for positive integer n:

\[ \Gamma\left(\frac{z}{n}\right)\Gamma\left(\frac{z+1}{n}\right)\cdots\Gamma\left(\frac{z+n-1}{n}\right) = (2\pi)^{\frac{n-1}{2}}n^{\frac{1}{2}-z} \Gamma(z) \]

Special Values

• \(\Gamma(1) = 0!\) = 1
• \(\Gamma(\frac{1}{2}) = \sqrt{\pi}\)
• \(\Gamma(2) = 1!\) = 1
• For negative integers and zero, the Gamma function is not defined.

Applications

The particular values of the Gamma function have applications in various areas of mathematics and science, including:

• Probability theory and statistics, especially in the definition of distributions such as the gamma distribution and the beta distribution.
• Quantum physics, where the Gamma function appears in the calculation of transition probabilities and in path integral formulations.
• Number theory, in the study of partitions and in the functional equations of the Riemann zeta function and other L-functions.

See Also

• Factorial
• Beta function
• Riemann zeta function
• Euler's reflection formula
• Gauss's multiplication theorem

References

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