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Results tagged with cv.complex-variables
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user 86142
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
8
votes
4
answers
1k
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Analytic continuation of the double sum $\sum_{n,m\ge0}x^ny^mt^{nm}$
Define function $f(x,y,t)$ as the analytic continuation of the series
$$f(x,y,t)=\sum_{n,m\ge0}x^ny^mt^{nm}$$
This series definitely converges when all the arguments are small enough. I would like to …
- 319
3
votes
1
answer
127
views
Decomposition of a sum of holomorphic squares into modular forms
Say there is a (non-holomorphic) real function $Z(\tau,\bar{\tau})$ which obeys the (non-holomorphic) modularity conditions of some weight $k$
$$Z(\tau+1,\bar{\tau}+1)=Z(\tau,\bar{\tau}),\qquad Z(-1/\ …
- 319
3
votes
2
answers
382
views
Theta-function in the lower half-plane
Standard theta function
$$\vartheta(q)=\sum_{n=-\infty}^\infty q^{n^2} \qquad\qquad(1)$$
has a natural boundary of analyticity at $|q|=1$. This means that it can not be used to regularize expressions …
- 319
1
vote
Analytic continuation of the double sum $\sum_{n,m\ge0}x^ny^mt^{nm}$
I though it is worth pointing out a closely related case which appears to be solvable. The following formula is due to paper http://arxiv.org/abs/math/9904126 (formula (11) with slight relabeling and …
- 319