Definition:Beta Function/Definition 1/Also rendered as

Beta Function: Definition $1$: 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:

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

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 17$: Definition of the Beta Function $\map \Beta {m, n}$: $17.1$
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $40$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 26$: The Beta Function: Definition of the Beta Function $\map \Beta {m, n}$: $26.1.$