Definition:Beta Function/Definition 1

Definition

The beta function $\Beta: \C \times \C \to \C$ is defined for $\map \Re x, \map \Re y > 0$ as:

$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$

Also rendered as

In a frequently-seen abuse of notation, the improper nature of the integral defining the beta function is often ignored, and the expression is rendered:

$\ds \map \Beta {x, y} := \int_0^1 t^{x - 1} \paren {1 - t}^{y - 1} \rd t$

Also see

• Equivalence of Definitions of Beta Function

• Results about the beta function can be found here.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $17.1$: Definition of the Beta Function $\map \Beta {m, n}$
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 17.7 \ (4)$
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $40$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): beta function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): beta function