How does WolframAlpha use the gamma function to solve this integral?

In summary, the conversation is about finding the integral of e^(ix)/(ix)^(1/5) from -infinity to infinity in terms of a gamma function. One method to solve this is by using the Fourier transform of f(x) in non-unitary, angular frequency form, where f(x) = (ix)^(-1/5). The bottom entry in the table for Fourier transform with \alpha = 1/5 can be used, and at \upsilon = -1, the expression becomes \hat f (-1) = 2π/\Gamma (1/5) = 1.368... which matches with wolframalpha's calculated value.
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I don't see how it produced [itex] \Gamma (\frac{4}{5}) [/itex], but I did find one way to solve it:

The Fourier transform of [itex] f(x) [/itex] in non-unitary, angular frequency form is:

[itex] \hat f (\upsilon) = \int_{-\infty}^{\infty} f(x) e^{-i \upsilon x} dx [/itex]

We have [itex] f(x) = (ix)^{-\frac{1}{5}} [/itex] and will just need to evaluate [itex] \hat f (\upsilon) [/itex] at [itex] \upsilon = -1 [/itex] once we find an expression for it:

You'll note that the bottom entry in this table (http://en.wikipedia.org/wiki/Fourier_transform#Distributions) works for us in this situation, with [itex] \alpha = \frac{1}{5} [/itex]

At [itex] \upsilon = -1 [/itex] the expression reduces to [itex] \hat f (-1) = \frac{2 \pi}{\Gamma (\frac{1}{5})} = 1.368... [/itex] -- agreeing with wolframalpha's calculated value.
 
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1. What is a gamma function?

The 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(x) = (x-1)!, where x is a complex number.

2. What are the applications of the gamma function?

The gamma function has various applications in mathematics, physics, and statistics. It is used in the calculation of probabilities in the fields of statistics and probability theory. It also appears in the solution of differential equations and in the study of special functions in mathematical analysis.

3. How do you form a gamma function?

The gamma function can be formed by integrating the exponential function e^(-t) over the interval from 0 to infinity. This can also be expressed as Γ(x) = ∫e^(-t) * t^(x-1) dt. Additionally, the gamma function can be defined using the properties of the factorial function and the use of the Euler-Mascheroni constant.

4. What are the properties of the gamma function?

The gamma function has several important properties, including the reflection formula, which states that Γ(x) * Γ(1-x) = π / sin(πx). It also satisfies the recurrence formula Γ(x+1) = x * Γ(x), and has a pole at x = 0 and negative integer values. Furthermore, it has connections to other special functions such as the beta function and the digamma function.

5. How is the gamma function related to the factorial function?

The gamma function is an extension of the factorial function and can be seen as a continuous version of it. The factorial of a positive integer n can be expressed as Γ(n+1), and it follows that Γ(n) = (n-1)! for positive integer values of n. However, the gamma function can also be used for non-integer values of x, making it a more versatile tool in mathematical calculations.

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