Solving Local Minimum of Gamma Function with Integral Calculation

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SUMMARY

The discussion centers on the local minimum of the Gamma function, specifically at n=1.461632, where the function achieves a value of approximately 0.885603. The derivative of the Gamma function is expressed as the integral of t^(n-1)*e^(-t)*ln(t) dt. This calculation was verified using a calculator, confirming the significance of the local minimum in the context of the Gamma function's behavior.

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My physics professor introduce us to the gamma function a few weeks ago, and I've been thinking about it quite a bit. Is there some sort of significance to the fact that the Gamma function has a local minimum value at about n=1.462?

I worked out that the derivative of the gamma function can be shown to be

Integral t^(n-1)*e^(-t)*ln(t) dtand found the zero of that function using my calculator.
 
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A better value for this is n=1.461632

And gamma(1.461632)=.885603
 

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