- #1

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sorry if this is in the wrong section.

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- Thread starter mjk1
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- #1

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sorry if this is in the wrong section.

- #2

mathman

Science Advisor

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Gamma(x)=int(0,oo)t

It can be extended by analytic continuation.

- #3

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yeah the original integral was [tex]\int[/tex][0,4]Y[tex]^{3/2}[/tex](16-Y[tex]^{2}[/tex])[tex]^{1/2}[/tex] dy

which i simplified to

64[tex]\int[/tex](0,1) t[tex]^{(5/4)-1}[/tex](1-t)[tex]^{(3/2)-1}[/tex] dt

using Gamma(x)=int(0,oo)tx-1e-tdt, =>> [tex]\beta[/tex](5/4, 3/2)

hence i am trying to solve [tex]\beta[/tex](5/4, 3/2)

= [tex]\Gamma[/tex](5/4)[tex]\Gamma[/tex](3/2) / [tex]\Gamma[/tex]((5/2)+(3/2))

i am trying to solve this to get an answer in terms of a number.

i dont know how to solve [tex]\Gamma[/tex](5/4)

which i simplified to

64[tex]\int[/tex](0,1) t[tex]^{(5/4)-1}[/tex](1-t)[tex]^{(3/2)-1}[/tex] dt

using Gamma(x)=int(0,oo)tx-1e-tdt, =>> [tex]\beta[/tex](5/4, 3/2)

hence i am trying to solve [tex]\beta[/tex](5/4, 3/2)

= [tex]\Gamma[/tex](5/4)[tex]\Gamma[/tex](3/2) / [tex]\Gamma[/tex]((5/2)+(3/2))

i am trying to solve this to get an answer in terms of a number.

i dont know how to solve [tex]\Gamma[/tex](5/4)

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- #4

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[tex]\Gamma\left(\frac{1}{4}\right)[/tex] has no known basic expression, but is known to be transcendental.

- #5

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thanks for your help

- #6

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I read that (1/q)! has no representation for integer q>2, except decimal form. But, if you are concerned with gamma(1/4), you can see it to 1,000,000 decimals at:

http://www.dd.chalmers.se/~frejohl/math/gamma14_1_000_000.txt

Working with Euler's reflection formula, I get

(1/4)!(3/4)! = [tex]\frac{3\pi\sqrt2}{16}=.833041[/tex]

http://www.dd.chalmers.se/~frejohl/math/gamma14_1_000_000.txt

Working with Euler's reflection formula, I get

(1/4)!(3/4)! = [tex]\frac{3\pi\sqrt2}{16}=.833041[/tex]

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- #7

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- #8

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mjk1: how about (0)

Well, when you have a form like[tex]\int_{0}^{\infty} \frac{e^-x}{x} dx[/tex], you have trouble.

If you check out the Euler's reflection formula, we have

[tex]\Gamma(Z)\Gamma(1-Z) = \frac{\pi}{sin\pi(x)} for Z=0.[/tex]

Well, when you have a form like[tex]\int_{0}^{\infty} \frac{e^-x}{x} dx[/tex], you have trouble.

If you check out the Euler's reflection formula, we have

[tex]\Gamma(Z)\Gamma(1-Z) = \frac{\pi}{sin\pi(x)} for Z=0.[/tex]

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