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Derivative of the Gamma Function

Problem 1. Differentiate the gamma function \[\Gamma(n) = \int _{ 0 }^{ \infty }{ { t }^{ n-1 }{ e }^{ -t } } dt.\]

Solution

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

To differentiate under the integral, we use Leibniz Rule .

First we find \[\frac{\partial f}{\partial n} = {t}^{n-1}{e}^{-t} ln\left(t\right).\]

Then we evaluate the limit

\[ \begin{align*} \Gamma '(n,t) &= \lim _{ x\rightarrow \infty }{ \left[\int _{ 0 }^{ x }{{t}^{n-1}(ln t){e}^{-t}} dt - { x }^{ n-1 }{ e }^{-x} - 0 \right] } \\ &= \lim _{ x\rightarrow \infty }{ \left[\int _{ 0 }^{ x }{{t}^{n-1}(ln t){e}^{-t}} dt - { x }^{ n-1 }{ e }^{-x}\right] } \\ &= \int _{ 0 }^{ \infty }{ { e }^{ -t }{ t }^{ n-1 } ln\left(t\right)} dt . \end{align*}\]

Problem 2. Show that \({\Gamma}^{'} (1) = -\gamma \) where \(\gamma\) is the Euler-Mascheroni constant.

Solution

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

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

Let \(s = 1 - \frac{t}{n} \) and \(-nds = dt, \) we get \[ \begin{align*} \lim _{ n\rightarrow \infty }{ \left[ \int _{ 1}^{ 0}{ { s }^{ n } } \left[ ln(n)+ln(1-s) \right] (-n)ds \right] } \\ &= \lim _{ n\rightarrow \infty }{ \left[ nln(n)\int _{ 0 }^{ 1 }{ { s }^{ n } } ds+\int _{ 0 }^{ 1 }{ { s }^{ n } } ln(1-s)ds \right] } \\ &=\lim _{ n\rightarrow \infty }{ \left[ \frac { n }{ n+1 } ln(n)+\frac { -1 }{ n+1 } \int _{ 0 }^{ 1 }{ \frac { { s }^{ n+1 }-1 }{ s-1 } } ds \right] } \\ &=\lim _{ n\rightarrow \infty }{\frac { n }{ n+1 } \left[ ln(n)-{ H }_{ n+1 } \right]} . \end{align*}\]

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

Check out my other notes at Proof, Disproof, and Derivation

Note by Steven Zheng
3 years, 1 month ago

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Why would you write \(\Gamma(n,t)\) ? the \(t\) is a dummy variable.

Haroun Meghaichi - 3 years, 1 month ago

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We don't?

Steven Zheng - 3 years, 1 month ago

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@Steven Zheng Yes, you probably got mixed up with the incomplete gamma function.

Haroun Meghaichi - 3 years, 1 month ago

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