The English defined many things. They said 0! is 1. And also that gamma functions for negative numbers exist. They said as the negative number increases in modulus value more, the gamma function approaches towards 0. So basically gamma functions of negative numbers range from 1 to 0.
e and log functions give gamma functions for n greater than or equal to zero only.
Both my gamma functions one, (n/(n+1)! - (n-1)/n!) and two, the integral with 4.666 work, for negatives, 0 and positive values. The integral with 4.666 is approximate; while (n/(n+1)! - (n-1)/n!) gives accurate results.
But my gamma function based on (arithmetic progression)/(geometric progression) work only for greater than or equal to zero values.
While talking to IAS, ministers and all one should have competence. I am actually only an E-5. But I have pi expressions, golden ratio expressions of my own. So, I believe that I have the necessary competence to try and influence a minister, for example.
Coming back to gamma function, It is indeed a privilege that my gamma functions cover the entire spectrum of numbers as English defined them. Negatives, zero and positives.
Have a nice day !!
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