What is Beta function: Definition and 38 Discussions

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral





B

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1



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1


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t


{\displaystyle \mathrm {B} (x,y)=\int _{0}^{1}t^{x-1}(1-t)^{y-1}\,dt}
for complex number inputs x, y such that Re x > 0, Re y > 0.
The beta function was studied by Euler and Legendre and was given its name by Jacques Binet; its symbol Β is a Greek capital beta.

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  1. S

    A Relationship between bare and renormalized beta functions

    I'm looking at a proof of beta function universality in ##\phi^4## theory, and at one point they do the following step: after imposing that the renormalized coupling ##\lambda## is independent of the cutoff ##\Lambda##, we have $$0= \Lambda \frac {\text{d} \lambda}{\text{d} \Lambda} = \Lambda...
  2. PLAGUE

    I Relation Between Beta and Gamma functions

    So, my teacher showed me this proof and unfortunately it is vacation now. I don't understand what just happened in the marked line. Can someone please explain?
  3. PhysicsRock

    Exploring an Unexplored Expression for the Beta Function

    So, I've recently played around a little with the Gamma Function and eventually managed to find an expression for the Beta Function I have not yet seen. So I'm asking you guys, if you've ever seen this expression somewhere or if this is a new thing. Would be cool if it was, so here's the...
  4. A

    Break a Stick Example: Random Variables

    Hello, I would like to confirm my answers to the following random variables question. Would anyone be willing to provide feedback and see if I'm on the right track? Thank you in advance. My attempt:
  5. P

    A $\phi^4$ in $4 - \epsilon$ dimension renormalization beta function

    Hi all, I am currently studying renormalization group and beta functions. Since I'm not in school there is no one to fix my mis-understandings if any, so I'd really appreciate some feedback. PART I: I wrote this short summary of what I understand of the beta function: Is this reasoning...
  6. fenyutanchan

    Beta Function for Scalar QCD

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  7. DaniV

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  8. L

    A Integral -- Beta function, Bessel function?

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  9. tomdodd4598

    I QED - running of coupling (beta function)

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  10. J

    Is this manipulation "evil" or simply not intuitive due to unfamiliarity?

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  11. diegzumillo

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  12. C

    I Two Loop Beta Function for Strong Coupling in QCD

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  13. S

    A ΛQCDΛQCD from the beta function of QCD coupling constant

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  14. N

    A Perturbative expansion of Beta function - Renormalization

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  15. dykuma

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  16. S

    Beta function of electroweak theory

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  17. nikosbak

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  18. WannabeNewton

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  19. J

    Beta Function used in Beta distribution

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  20. A

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  21. A

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  22. alyafey22

    MHB An integral related to beta function

    Prove the following \int^1_0 \frac{x^{r-1} (1-x)^{s-1}}{\left(ax+b(1-x)+c \right)^{r+s}}\, dx =\frac{\beta(r,s)}{ (a+c)^{r}(b+c)^{s}}
  23. alyafey22

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  24. J

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    Hi, We have: \beta(a,b)=\int_0^1 t^{a-1}(1-t)^{b-1}dt,\quad Re(a)>0, Re(b)>0 and according to Wikipedia: http://en.wikipedia.org/wiki/Pochhammer_contour we can write: \left(1-e^{2\pi ia}\right)\left(1-e^{2\pi ib}\right)\beta(a,b)= \int_P t^{a-1}(1-t)^{b-1}dt valid for all a and b where P...
  25. FeDeX_LaTeX

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  26. C

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  27. T

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  28. A

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  29. M

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  30. M

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  31. M

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  32. S

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  33. N

    Beta Function in String Theory

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  34. R

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  35. S

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  36. M

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  37. W

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    I watched the elegant universe and one scienetist noticed the euler beta function seemed to explain what he was doing and it had properties of a vibrating string. So I'm wondering if I could find his original paper, explanation or some additional info.
  38. E

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