Solving Local Minimum of Gamma Function with Integral Calculation

[Doom of Doom]

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.

 

[Doom of Doom]

A better value for this is n=1.461632

And gamma(1.461632)=.885603

 

[tacman]

The local minimum value of the Gamma function at n=1.462 does indeed have significance, as it represents the point where the function has its smallest value before increasing again. This can be seen in the graph of the Gamma function, where the curve reaches a minimum before sharply increasing towards infinity.

The fact that the derivative of the Gamma function can be expressed as an integral is also interesting, as it allows for a more analytical approach to finding the local minimum. By setting the derivative to zero and solving for n, we can find the exact value of the local minimum rather than estimating it using a calculator.

Furthermore, the presence of the natural logarithm in the integral indicates that there may be a connection between the Gamma function and exponential functions, which can be explored further. Overall, the local minimum of the Gamma function at n=1.462 presents an interesting mathematical problem that can be approached in different ways, providing a deeper understanding of the function and its properties.