Exploring Gamma Functions: Analytic Possibilities & Integrals

In summary, the conversation discusses the possibility of finding the Gamma function analytically for non-integer and non-halfinteger numbers, and how to address the convergence issue in real analysis for the integral representation of the Gamma function. The use of Gauß' formula and analytic continuation are suggested as possible solutions.
  • #1
LagrangeEuler
717
20
I have two questions related Gamma functions

1. Finding ##\Gamma## analytically. Is that possible only for integers and halfintegers? Or is it possible mayble for some other numbers? For example is it possible to find analytically ##\Gamma(\frac{3}{4})##?

2. Integral ##\Gamma(x)=\int^{\infty}_0 \xi^{x-1}e^{-\xi}d \xi ## converge only for ##x>0## in real analysis. How can we then write ##\Gamma(\frac{1}{2})=\Gamma(-\frac{1}{2}+1)## when relation ##\Gamma(x+1)=x\Gamma(x)## is derived from partial integration?
 
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  • #2
How about to use Gauß' formula for ##x \in ℂ \backslash \{0, -1, -2, \dots \}## instead:

$$Γ(x) =\lim_{n→\infty} \frac{n!n^x}{x(x+1) \cdots (x+n)}$$

Edit: It's sufficient to require ##Re(x) > 0## for the integral formula.
 
  • #3
The gamma function has singularities at 0 and negative integers. Using analytic continuation the function can be defined elsewhere.
 

1. What is a gamma function?

A gamma function is a mathematical function that extends the concept of factorial to real and complex numbers. It is denoted by the Greek letter gamma (Γ) and is defined as Γ(n) = (n-1)!, where n is a positive integer.

2. How is the gamma function used in mathematical analysis?

The gamma function is used in mathematical analysis to generalize the concept of factorial to non-integer values. It is also used to solve certain integrals and differential equations, and to study the convergence and divergence of series.

3. What are some key properties of the gamma function?

Some key properties of the gamma function include: it is an increasing function, it has simple poles at negative integer values, it satisfies the recurrence relation Γ(n+1) = nΓ(n), and it has a logarithmic singularity at n = 0.

4. How is the gamma function related to other mathematical functions?

The gamma function is closely related to other mathematical functions, including the factorial function, the beta function, and the incomplete gamma function. It can also be expressed in terms of other special functions, such as the hypergeometric function and the confluent hypergeometric function.

5. What are some applications of the gamma function?

The gamma function has many applications in mathematics, physics, and engineering. It is used in probability theory, number theory, and combinatorics. It also has applications in statistics, quantum mechanics, and fluid mechanics. Additionally, the gamma function is used in the fields of signal processing, image processing, and data compression.

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