02/09/2023, 08:09 AM
(02/08/2023, 06:03 AM)JmsNxn Wrote: These are questions I have asked myself for 12 years
STUDY RAMANUJAN'S MASTER THEOREM AND THE MELLIN TRANSFORM!
The answer is so banal, that you may not like it, but I'll give a shot. I'll start with, why does math choose \(\Gamma(z+1)\) from \(n!\).
Because:
\[
\frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \Gamma(z) x^{-z} = e^{-x}\\
\]
And:
\[
\int_0^\infty e^{-x} x^{z-1} \, dx = \Gamma(z)\\
\]
Where Ramanujan showed in a rough handed way; where \(H\) is a linear operator (or a function):
\[
\int_0^\infty e^{-Hx} x^{z-1}\,dx = \Gamma(z) H(z)\\
\]
Where then:
\[
e^{-Hx} = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \Gamma(z) H(z) x^{-z}\,dz\\
\]
And the ONLY SOLUTIONS WHICH WORK ARE THE ONES WHICH HAVE CONVERGENT INTEGRALS!
This relates deeply to \(\sin(\pi z)\) and Carlson's theorem. Carlson's theorem is actually, in an historical perspective, a manner of justifying Ramanujan's Master Theorem in a much much much broader sense. He wanted to justify things Ramanujan just took for granted, lmao. In perfect Ramanujan form
When I think about this some more, I think
Quote:"And the ONLY SOLUTIONS WHICH WORK ARE THE ONES WHICH HAVE CONVERGENT INTEGRALS!"Is not a principal which seems to work in general. If \( \Gamma \) is defined in terms of unique thing that has convergent integrals, then \( \frac{1}{n!} \) grows pretty large in the imaginary direction, but \( \frac{1}{n! + 2\sin(\pi z)}\) is quite small and so it will lead to convergent integrals. Are there some other conditions you have in mind in general?
To be more particular, if I have a specific sequence \( a_n \) on the integers, what conditions are you proposing determines the unique analytical continuation? Or perhaps, if I have a sequence \( a_n \) and an analytical continuation \( A(z) \), what specific conditions determine if this is the right analytical continuation? In the case of \( a_n = \frac{1}{n!} \), what do your conditions say is the right answer? (Or, perhaps is more information needed besides just the sequence to determine the right analytical continuation?)