Derivative of the Gamma Function

Show that Γ(1)=γ{\Gamma}^{'} (1) = -\gamma where γ\gamma is the Euler-Mascheroni constant.

Solution

We begin with the integral definition of the Gamma function limx0xettn1dt.\lim _{ x\rightarrow \infty }{ \int _{ 0 }^{ x }{ { e }^{ -t }{ t }^{ n-1 } } dt } .

Let f(n,t)=ettn1f(n,t) = {e}^{-t}{t}^{n-1} and fn=ettn1ln(t).\frac{\partial f}{\partial n} = {e}^{-t}{t}^{n-1} ln\left(t\right).

By the Leibniz rule, limx[0xettn1ln(t)dtexxn10]\lim _{ x\rightarrow \infty }{ \left[ \int _{ 0 }^{ x }{ { e }^{ -t }{ t }^{ n-1 }ln(t) } dt-{ e }^{ -x }{ x }^{ n-1 }-0 \right] } which reduces to 0ettn1ln(t)dt. \int _{ 0 }^{ \infty }{ { e }^{ -t }{ t }^{ n-1 } ln\left(t\right)} dt .

To evaluate Γ(1){\Gamma}^{'} (1) , we set n=1n=1 thus 0etln(t)dt. \int _{ 0 }^{ \infty }{ { e }^{ -t }ln\left(t\right)} dt .

We replace et{e}^{-t} with limn(1tn)n.\lim _{ n\rightarrow \infty }{ \left(1-\frac{t}{n} \right)^{n} } .

Let s=1tns = 1 - \frac{t}{n} and nds=dt,-nds = dt, we get limn[01sn[ln(n)+ln(1s)](n)ds]=limn[nln(n)01snds+01snln(1s)ds]=limn[nn+1ln(n)+1n+101sn+11s1ds]=limnnn+1[ln(n)Hn+1]. \begin{aligned} \lim _{ n\rightarrow \infty }{ \left[ \int _{ 0 }^{ 1 }{ { s }^{ n } } \left[ ln(n)+ln(1-s) \right] (-n)ds \right] } \\ &= \lim _{ n\rightarrow \infty }{ \left[ nln(n)\int _{ 0 }^{ 1 }{ { s }^{ n } } ds+\int _{ 0 }^{ 1 }{ { s }^{ n } } ln(1-s)ds \right] } \\ &=\lim _{ n\rightarrow \infty }{ \left[ \frac { n }{ n+1 } ln(n)+\frac { -1 }{ n+1 } \int _{ 0 }^{ 1 }{ \frac { { s }^{ n+1 }-1 }{ s-1 } } ds \right] } \\ &=\lim _{ n\rightarrow \infty }{\frac { n }{ n+1 } \left[ ln(n)-{ H }_{ n+1 } \right]} . \end{aligned}

Hn+1{H}_{n+1} is the harmonic number. By definition limn[Hn+1ln(n)] \lim_{n\rightarrow \infty}{\left[{ H }_{ n+1 } -ln(n)\right]} is the Euler-Mascheroni constant; therefore, Γ(1)=γ{\Gamma}^{'} (1) = -\gamma .

Note by Steven Zheng
4 years, 9 months ago

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How do u add alignment?

Aneesh Kundu - 4 years, 9 months ago

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Consider this page: LaTex Align equations

Steven Zheng - 4 years, 9 months ago

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Thnx

Aneesh Kundu - 4 years, 9 months ago

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I just noticed I wrote a note with the exact title. Might as well move the content to that note.

Steven Zheng - 4 years, 9 months ago

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