How Ramanujan proved his master theorem
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- Опубликовано: 6 сен 2024
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• Ramanujan's master the...
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The normal proof (whatever that is) that I know is based on the operational form of Taylor's theorem: f(x+ε) = e^(ε(d/dx)) f. A gap in your proof is that you have to show, since, in the proof, g is given first, that given any φ satisfying the hypotheses, we can solve for an analytic g satisfying g(a + hr^s) = φ(s)
Not a big fan of that either. Hardy's proof using residues is brilliant beyond measure.
Ramanujan seriously needs some more love. Hope yo see you cover more of his amazing work.
Thanks. I have been tackling with this for a while and it is nice to see a video about it from a high quality channel like you.
Hey, I've been following your videos for a long time now. I spent last summer watching your videos, and it has been a blast getting better at solving integrals. I do wonder if you have any recommendations for books that I can use to get familiar with the techniques you use? It would be a great pleasure, thank you so much.
I'll mention some in a Q/A video I plan on doing later this month
we want g.h.hardys proof as soon as possible ❤
Thanks for your effort
Not a big fan of just interchanging series and integrals, at least comment on why it's possible (Fubini, dominated convergence, and why the conditions are met)
I can't wait to see Hardy's proof
I'm sorry but your r's look a lot like gammas.
I have to see the proof by hardy!
Why is it that important when it have this many conditions ?
The conditions aren't all that bad tho.
I suppose we need them so that we can safely swap the integral and summmation signs (with the dominated convergence theorem, perhaps? I'm not sure)
What kind of transformation?
Mellin transform
@@maths_505 Thanks. That's great
Wanna better minature?
I sended you.
Where?
@@maths_505 i sended png on gmail
@@maths_505 I sended png on gmail
first
Can someone provide a context?
Context for what mate?
@@maths_505 what is this theorem about?
@@nullvoid12 it's primarily used to derived melin transforms of functions.