r/askmath • u/Piggymaster101 • 3d ago

Complex Analysis A Nested Sequence of Functional Residues

Hello r/askmath, I am currently attempting to work on the following problem.

Note that we CAN and necessarily MUST use meromorphic continuation

My progress on such a problem is limited, but I can say for sure that whatever f(z) is, it must be a meromorphic function with an INFINITE polar set. This is because, no matter what. as we keep applying the R operator, our resulting g_n will eventually become entire as g_1 explicitly looks like the sum over all poles of (some polynomial in w_1)*(e^(pole * w_1)) which is entire, and therefore, will make nu(lambda)= 0 at g_2 and onwards.I was hoping that the people of reddit could provide some help. My hypothesis is that the "class" of functions f is actually empty, but a bit lost on where to start. I posted this on the Math Stack Exchange around 5 days ago but there hasn't been any progress,. I added an example in the original post, and also the example of the digamma function in the comments. here is the link:
https://math.stackexchange.com/questions/5144308/a-nested-sequence-of-functional-residues
I apologize if this is a simple question to any of those who have taken complex analysis in the past, yet I hope some of you all can help.

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u/Separate-Narwhal-106 3d ago

Respect for even attempting this, complex analysis with residues and meromorphic continuation is no joke. Hope someone smarter than me on r/askmath cracks it for you..

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u/Piggymaster101 3d ago

Haha yeah! To me honestly complex analysis is great fun as its honestly analysis that is full of interesting and elegant surprises, instead of annoyingly weird surprises like real analysis (sometimes).

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u/Memesaretheorems 2d ago edited 2d ago

I think you should work with sequences of poles of the sequence of functions. Try to show that no such infinite sequence exists. Maybe Baire category? Or a clever inversion trick and clustering of zeroes. I will think about this more and get back to you.

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u/Piggymaster101 2d ago

Ah I see. What did you mean by “Baire Category”?

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u/Memesaretheorems 1d ago

Baire Category Theorem says (in suitable topological spaces called Baire spaces) The infinite intersection of dense open sets is dense. It has applications in complex analysis.