Beta Functions and Incomplete Beta Functions
Let $a,b,x$ be positive real numbers and $0 < x < 1$, this calculator evaluates the following six functions.
- Beta function $\displaystyle\mathbf{B}(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$
- Non-normalized incomplete beta function $\displaystyle\mathbf{B}_x(a,b)=\int_0^x t^{a-1}(1-t)^{b-1}\;dt$
- Non-normalized incomplete beta function $\mathbf{B}(a,b)-\mathbf{B}_x(a,b)$
- Normalized incomplete beta function $\displaystyle\mathbf{I}_x(a,b)=\frac{1}{\mathbf{B}(a,b)}\int_0^x t^{a-1}(1-t)^{b-1}\;dt$
- Normalized incomplete beta function $1-\mathbf{I}_x(a,b)$
- Derivative of normalized incomplete beta function $\displaystyle \frac{d}{dx}\mathbf{I}_x(a,b)$