guys,
I encountered a function like $$ f(x) = B(x;1-x,x) $$ where $B(\cdot)$ is the incomplete beta function and input $0 < x < 1$ is some positive small real value close to zero . I want to evaluate the value of $f(x)$ and its derivative. I am wondering is there any special property that I can make use of to approximate the function.
Thanks
Using these $\text B_z(a,b)$ identities. There is this special proprety:
$$\text B_x(1-x,x)=(-1)^{-x}\left(\pi(\cot(\pi x)+i)-\left(1-\frac1x\right)^x\Phi\left(1-\frac1x,1,x\right)\right)$$
shown here with Lerch Phi $\Phi(z,s,a)$
Now only 2 arguments of $\Phi(z,s,a)$ have variables instead of all three of $\text B_z(a,b)$. The derivative of $\text B_x(1-x,x)$ has generalized hypergeometric functions, but differentiating the right hand side uses simpler functions:
$$\frac d{dx} (-1)^{-x}\left(\pi(\cot(\pi x)+i)-\left(1-\frac1x\right)^x\Phi\left(1-\frac1x,1,x\right)\right)=\left(\frac1x-1\right)^x\left(\pi(-1)^{\frac52}\Phi+\Phi-\ln\left(1-\frac1x\right)\Phi+\frac1{1-x}+\pi^2\left((-1)^\frac32\cot(\pi x)-\csc^2(\pi x)+1\right)\right),\Phi=\Phi\left(1-\frac1x,1,x\right)$$
