In mathematics, the beta function (occasionally written as Beta function), also called the Euler integral of the first kind, is a special function defined by

$$

\mathrm{\Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,dt \!

for Re(x), Re(y) > 0.

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet.

Properties

---

The beta function is symmetric, meaning that

$$

\mathrm{\Beta}(x,y) = \mathrm{\Beta}(y,x). \!

It has many other forms, including:

$$

\mathrm{\Beta}(x,y)=\frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!

$$

\mathrm{\Beta}(x,y) = 2\int_0^{\pi/2}\sin^{2x-1}\theta\cos^{2y-1}\theta\,d\theta, \qquad \Re(x)>0,\ \Re(y)>0 \!

$$

\mathrm{\Beta}(x,y) = \int_0^\infty\frac{t^{x-1}}{(1+t)^{x+y}}\,dt, \qquad \Re(x)>0,\ \Re(y)>0 \!

$$

\mathrm{\Beta}(x,y) = \frac{1}{y}\sum_{n=0}^\infty(-1)^n\frac{(y)_{n+1}}{n!(x+n)} \!

where $\backslash Gamma(x)$ is the gamma function and (x)_n is the falling factorial; i.e., $x(x\; -\; 1)(x\; -\; 2)\; ...\; (x\; -\; n\; +\; 1)$. The second identity shows in particular $\backslash Gamma(1/2)\; =\; \backslash sqrt\; \backslash pi$.

Just as the gamma function for integers describes factorials, the beta function can define a binomial coefficient after adjusting indices:

$\{n\; \backslash choose\; k\}\; =\; \backslash frac1\{(n+1)\; \backslash mathrm\{B\}(n-k+1,\; k+1)\}$

The beta function was the first known scattering amplitude in string theory, first conjectured by Gabriele Veneziano.

Derivatives

---

The derivatives follow:

$\{\backslash partial\; \backslash over\; \backslash partial\; x\}\; \backslash mathrm\{B\}(x,\; y)\; =\; \backslash mathrm\{B\}(x,\; y)\; \backslash left(\; \{\backslash Gamma\text{'}(x)\; \backslash over\; \backslash Gamma(x)\}\; -\; \{\backslash Gamma\text{'}(x\; +\; y)\; \backslash over\; \backslash Gamma(x\; +\; y)\}\; \backslash right)\; =\; \backslash mathrm\{B\}(x,\; y)\; (\backslash psi(x)\; -\; \backslash psi(x\; +\; y))$

where $\backslash psi(x)$ is the digamma function.

Integrals

---

The Nörlund-Rice integral is a contour integral involving the beta function.

Incomplete beta function

---

The incomplete beta function is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral. The situation is analogous to the incomplete gamma function being a generalization of the gamma function.

The incomplete beta function is defined as

$\backslash mathrm\{B\}\_x(a,b)\; =\; \backslash int\_0^x\; t^\{a-1\}\backslash ,(1-t)^\{b-1\}\backslash ,\backslash mathrm\{d\}t\; \backslash !$

For x = 1, the incomplete beta function coincides with the complete beta function.

The regularized incomplete beta function (or regularized beta function for short) is defined in terms of the incomplete beta function and the complete beta function:

$I\_x(a,b)\; =\; \backslash frac\{\backslash mathrm\{B\}\_x(a,b)\}\{\backslash mathrm\{B\}(a,b)\}\; \backslash !$

Working out the integral for integer values of a and b, one finds:

$I\_x(a,b)\; =\; \backslash sum\_\{j=a\}^\{a+b-1\}\; \{(a+b-1)!\; \backslash over\; j!(a+b-1-j)!\}\; x^j\; (1-x)^\{a+b-1-j\}$

Properties

$I\_0(a,b)\; =\; 0\; \backslash ,$

$I\_1(a,b)\; =\; 1\; \backslash ,$

$I\_x(a,b)\; =\; 1\; -\; I\_\{1-x\}(b,a)\; \backslash ,$

(Many other properties could be listed here.)

See also

---

• Beta distribution
• binomial distribution
• Yule-Simon distribution

References

---

• Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See §6.2, 6.6, and 26.5)
• W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1992. Second edition. (See section 6.4)
• Arbitrarily accurate values can be obtained from The Wolfram Functions Site, Evaluate Beta Regularized Incomplete beta

Gamma and related functions | Special hypergeometric functions

Betafunktion | Fonction_Beta | Betafunctie | ベータ関数 | Funkcja beta | Бета-функция | Funkcija beta | Бета-функција | Fungsi béta