伽馬函數(shù)

2023-06-07 23:30 作者:~Sakuno醬 0人讀過 | 我要投稿

最近發(fā)現(xiàn)用伽馬函數(shù)很容易出錯(cuò) 原因就是沒有完全理解伽馬函數(shù)

首先給出伽馬函數(shù)的定義

$%5CGamma(s)%3D%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%7Bx%5E%7Bs-1%7De%5E%7B-x%7D%5Cmathrm%7Bd%7Dx%7D%20$

伽馬函數(shù)的性質(zhì)有

$%5CGamma(s)%3D(s-1)%5CGamma(s-1)$

如果讓你計(jì)算?

$%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%5Clambda%20xe%5E%7B-%5Clambda%20x%7D%20%5Cmathrm%7Bd%7Dx%20$ ?其中? $%5Clambda%20%3E0$

很容易計(jì)算成

$%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%20xe%5E%7B-%5Clambda%20x%7D%20%5Cmathrm%7Bd%7D%5Clambda%20x$

$%3D%5CGamma(2)%3D1$

這個(gè)是肯定算錯(cuò)了

正確做法是

$%5Cfrac%7B1%7D%7B%5Clambda%7D%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%20(%5Clambda%20x)e%5E%7B-%5Clambda%20x%7D%20%5Cmathrm%7Bd%7D%5Clambda%20x%20%3D%20%5Cfrac%7B1%7D%7B%5Clambda%7D%20%5CGamma(2)%20$

類似的如果要計(jì)算

$%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%5Clambda%20x%5E2e%5E%7B-%5Clambda%20x%7D%20%5Cmathrm%7Bd%7Dx%20$ ? 其中? $%5Clambda%20%3E0$

則有

$%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%5Clambda%20x%5E2e%5E%7B-%5Clambda%20x%7D%20%5Cmathrm%7Bd%7Dx%20$

$%3D%5Cfrac%7B1%7D%7B%5Clambda%20%5E2%7D%5Cint_%7B0%7D%5E%7B%5Cinfty%7D(%5Clambda%20x)%5E2e%5E%7B-%5Clambda%20x%7D%20%5Cmathrm%7Bd%7D%20%5Clambda%20x%20$