[Calculus] Euler-Mascheroni Constant

2021-11-02 21:20 作者:AoiSTZ23 0人讀過 | 我要投稿

By: Tao Steven Zheng (鄭濤)

【Problem】

Part 1: Differentiate the gamma function

$%5CGamma(n)%20%3D%20%5Cint%20_%7B%200%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%7B%20t%20%7D%5E%7B%20n-1%20%7D%7B%20e%20%7D%5E%7B%20-t%20%7D%20%7D%20dt$

Part 2: The Euler-Mascheroni constant is defined

$%20%5Clim_%7Bn%5Crightarrow%20%5Cinfty%7D%7B%5Cleft%5B%7B%20H%20%7D_%7B%20n%2B1%20%7D%20-%5Cln%7B(n)%7D%5Cright%5D%7D%20%3D%20%5Cgamma$

Use the result in part 1 to show that $%7B%5CGamma%7D%5E%7B'%7D%20(1)%20%3D%20-%5Cgamma$ .

【Solution】

Part 1
We begin with the integral definition of the Gamma function

$%5Clim%20_%7B%20x%5Crightarrow%20%5Cinfty%20%20%7D%7B%20%5Cint%20_%7B%200%20%7D%5E%7B%20x%20%7D%7B%20%7B%20e%20%7D%5E%7B%20-t%20%7D%7B%20t%20%7D%5E%7B%20n-1%20%7D%20%7D%20dt%20%7D%20$

To differentiate under the integral, use the Leibniz Rule.

$%5Cfrac%7Bd%7D%7Bdx%7D%5Cint_%7Ba(x)%7D%5E%7Bb(x)%7D%20f(x%2Ct)dt%20%3D%20%5Cint_%7Ba(x)%7D%5E%7Bb(x)%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20x%7D%20f(x%2Ct)%20dt%20%2B%20f%5Cbig(x%2Cb(x)%5Cbig)%5Ccdot%20b'(x)%20-%20f%5Cbig(x%2Ca(x)%5Cbig)%5Ccdot%20a'(x)$

For this problem, replace $x%20$ with $%20n%20$ .

$%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20n%7D%20%3D%20%7Bt%7D%5E%7Bn-1%7D%7Be%7D%5E%7B-t%7D%20%5Cln%7B(t)%7D$

Thus,

$%5Cbegin%7Balign%7D%0A%5CGamma%20'(n%2Ct)%20%26%3D%20%5Clim%20_%7B%20x%5Crightarrow%20%5Cinfty%20%20%7D%7B%20%5Cleft%5B%5Cint%20_%7B%200%20%7D%5E%7B%20x%20%20%7D%7B%7Bt%7D%5E%7Bn-1%7D%5Cln(t)%7Be%7D%5E%7B-t%7D%7D%20dt%20%2B%20%5Cleft(%7B%20x%20%7D%5E%7B%20n-1%20%7D%7B%20e%20%7D%5E%7B-x%7D%5Cright)%20%5Ccdot%201%20-%200%20%5Ccdot%200%20%5Cright%5D%20%7D%20%20%5C%5C%0A%26%3D%20%5Clim%20_%7B%20x%5Crightarrow%20%5Cinfty%20%20%7D%7B%20%5Cleft%5B%5Cint%20_%7B%200%20%7D%5E%7B%20x%20%20%7D%7B%7Bt%7D%5E%7Bn-1%7D%5Cln(t)%7Be%7D%5E%7B-t%7D%7D%20dt%20%2B%20%7B%20x%20%7D%5E%7B%20n-1%20%7D%7B%20e%20%7D%5E%7B-x%7D%5Cright%5D%20%7D%20%5C%5C%0A%26%3D%20%5Cint%20_%7B%200%20%7D%5E%7B%20%5Cinfty%20%7D%7B%20%7B%20e%20%7D%5E%7B%20-t%20%7D%7B%20t%20%7D%5E%7B%20n-1%20%7D%20%5Cln%5Cleft(t%5Cright)%7D%20dt%20%5C%5C%0A%5Cend%7Balign%7D$

Part 2
To evaluate $%7B%5CGamma%7D%5E%7B'%7D%20(1)$ , we set $n%3D1$ ; thus,

$%5Cint%20_%7B%200%20%7D%5E%7B%20%5Cinfty%20%7D%7B%20%7B%20e%20%7D%5E%7B%20-t%20%7D%5Cln%5Cleft(t%5Cright)%7D%20dt%20$

We replace $%20%7Be%7D%5E%7B-t%7D$ with $%5Clim%20_%7B%20n%5Crightarrow%20%5Cinfty%20%20%7D%7B%20%5Cleft(1-%5Cfrac%7Bt%7D%7Bn%7D%20%5Cright)%5E%7Bn%7D%20%7D$ .

Let $s%20%3D%201%20-%20%5Cfrac%7Bt%7D%7Bn%7D%20$ and $-nds%20%3D%20dt%20$ , we get

$%20%5Cbegin%7Balign%7D%0A%5Clim%20_%7B%20n%5Crightarrow%20%5Cinfty%20%20%7D%7B%20%5Cleft%5B%20%5Cint%20_%7B%201%7D%5E%7B%200%7D%7B%20%7B%20s%20%7D%5E%7B%20n%20%7D%20%7D%20%5Cleft%5B%5Cln(n)%2B%5Cln(1-s)%20%5Cright%5D%20(-n)ds%20%5Cright%5D%20%20%7D%20%20%5C%5C%0A%20%26%3D%20%5Clim%20_%7B%20n%5Crightarrow%20%5Cinfty%20%20%7D%7B%20%5Cleft%5B%20n%5Cln%7B(n)%7D%5Cint%20_%7B%200%20%7D%5E%7B%201%20%7D%7B%20%7B%20s%20%7D%5E%7B%20n%20%7D%20%7D%20ds%2B%5Cint%20_%7B%200%20%7D%5E%7B%201%20%7D%7B%20%7B%20s%20%7D%5E%7B%20n%20%7D%20%7D%20%5Cln(1-s)ds%20%5Cright%5D%20%20%7D%20%20%5C%5C%0A%20%26%3D%5Clim%20_%7B%20n%5Crightarrow%20%5Cinfty%20%20%7D%7B%20%5Cleft%5B%20%5Cfrac%20%7B%20n%20%7D%7B%20n%2B1%20%7D%20%5Cln%7B(n)%7D%20%2B%5Cfrac%20%7B%20-1%20%7D%7B%20n%2B1%20%7D%20%5Cint%20_%7B%200%20%7D%5E%7B%201%20%7D%7B%20%5Cfrac%20%7B%20%7B%20s%20%7D%5E%7B%20n%2B1%20%7D-1%20%7D%7B%20s-1%20%7D%20%20%7D%20ds%20%5Cright%5D%20%20%7D%20%20%20%5C%5C%0A%20%26%3D%5Clim%20_%7B%20n%5Crightarrow%20%5Cinfty%20%20%7D%7B%5Cfrac%20%7B%20n%20%7D%7B%20n%2B1%20%7D%20%5Cleft%5B%5Cln%7B(n)%7D-%7B%20H%20%7D_%7B%20n%2B1%20%7D%20%5Cright%5D%7D%20%5C%5C%0A%5Cend%7Balign%7D%20$

Since the Euler-Mascheroni constant is defined $%5Clim_%7Bn%5Crightarrow%20%5Cinfty%7D%7B%5Cleft%5B%7B%20H%20%7D_%7B%20n%2B1%20%7D%20-%5Cln%7B(n)%7D%20%5Cright%5D%7D%20$ ,