c++ - 超大数的运算问题

Question

在一个问题中遇到的需要求这样的一个表达式：

应该如何计算可以得到结果？如果用龙贝格积分计算是算不出来的，伽马函数用定义算也求不出来。请问应该使用什么算法？

Answer 1

The front is easy to say, but the points in the back are not easy to calculate

Answer 2

I suspect there is a clerical error in the question. The number in the lower left corner should be 251528. In this way, the fraction part is exactly the definition of the Beta function:

$$frac{Gamma(m+n)}{Gamma(m) Gamma(n)}=frac{1}{B(n,m)}$$

The integral part is the definition of incomplete Beta function:

$$int_0^{frac{1}{2}} theta ^{n-1} (1-theta )^{m-1} , dtheta=B_{frac{1}{2}}(n, m)$$

The product of the two is called the regularized incomplete Beta function (Regularized Beta Function).

$$frac{B_{frac{1}{2}}(n,m)}{B(n,m)}=I_{frac{1}{2}}(n,m)$$

This function happens to be the cumulative distribution function (CDF) of the Beta distribution, so many software packages can calculate it. For example Excel:

The result does not exceed the representation range of computer floating point numbers.