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Jul
10
comment Turning a Line Integral into a Contour one
@DanielFischer Yes, this agrees with my computation. I think I could try to compute the additional pole (and relative residue) by brute force, but it's gonna be long and not elegant. While the observation of the authors of the paper on the possibility of turning it into a contour integral with only one pole is very tantalizing. Do you think it is possible to do that? On the other hand, according to the suggestions of Avitus I was trying to play a bit with different contours to exclude this additional pole. Thanks for your interest!
Jul
10
revised Turning a Line Integral into a Contour one
added 1 characters in body
Jul
10
comment Turning a Line Integral into a Contour one
Problems arise if I try some "classic" transformation. $z:=e^{i\theta}$ gives rise to a function holomorphic in $\mathbb{C}\setminus\{0\}$, but the integration path is not closed (it is the right semi-circle). For the transformation $z:=e^{2i\theta}$ the integration path is closed (the unit circle) but $i\sin(\theta)= \frac{1}{2}(\sqrt{z} -\frac{1}{\sqrt{z}})$ which is not holomorphic on the whole unit circle. I think those are just two expressions of the same pathology, i.e. the original integrand is not $\pi$-periodic. What do you think? Thanks again!
Jul
10
comment Turning a Line Integral into a Contour one
@Avitus Thanks for your response! Actually we are supposed to consider both $B$ and $\lambda$ as parameters. As you say the imaginary part of the denominator is killed by $\theta=k\pi$, but the real part is always $\neq0$ for any $\theta$ (excluding at most 2 "annoying" choices of $\lambda$, but we are allowed to do it). In specific the integrand, as a function of $\theta$, is non singular.
Jul
10
asked Turning a Line Integral into a Contour one
Jul
1
comment Finding the integer solutions of $246x + 217y = 3$
Ooops, Clearly $7+1+2\neq 9$ then! I apologize for the mistake... ...in this case we simply observe that $3/3$ and $3/246$ implies that $3/y$ for any solution $y$. So all the solutions are of the form $(x_0,3y_0)$ where $(x_0,y_0)$ is a solution of $\frac{246}{3}x +217y =\frac{3}{3}$. The strategy of the comment above applies to this equation. PS @Tomas, thanks for pointing it out!
Jul
1
comment Finding the integer solutions of $246x + 217y = 3$
You can simplify the equation observing that all the coefficients are divisible by 3. Then apply the Euclidean algorithm en.wikipedia.org/wiki/Extended_Euclidean_algorithm to compute a solution, and finally use Bezout's identity en.wikipedia.org/wiki/Bézout's_identity to compute all of them.
Jun
25
comment How important are automorphic representations among admissible ones?
Thanks! This is the kind of answer I was looking for! In specific I find extremely enlightening the observation about "factorization of automorphic forms" vs "factorization of automorphic representation", I completely oversaw it while studying!
Jun
25
accepted How important are automorphic representations among admissible ones?
Jun
20
asked How important are automorphic representations among admissible ones?
Jun
20
comment Definition of Automorphic Representation
What is the definition of weakly contained in $L^2(G_F\backslash G_A)$?
May
13
comment Calculate an integral in a measurable space
Could you post your proof that it is zero for any $\alpha>1$? I suspect that rearranging it one should be able to prove that the integral actually diverges for any $0<\alpha<1$.
May
7
awarded  Caucus
Apr
23
revised Interpretation for the Functional Determinant
deleted 2 characters in body
Apr
23
asked Interpretation for the Functional Determinant
Apr
22
comment Continuous dependence of zeros on a parameter
Forcing a fixed number of solutions for any $\lambda$, keeping the interval $I$ open, should give you the result!
Apr
20
awarded  Yearling
Apr
18
answered Continuous dependence of zeros on a parameter
Apr
18
comment Continuous dependence of zeros on a parameter
Could you be a bit more explicit in what do you mean by "Assume its solutions are at least 1 and at most $n$ (not dependent on $\lambda$)"? Do you mean that for every $\lambda$ there are at most $n$ solutions? Or there is a finite and fixed number of solutions for every $\lambda$? Thank you!
Apr
16
comment Realization of Bessel functions
You could be interested in what the DLMF (Digital Library of Mathematical Functions) has to say for the Bessel functions. Here is the link: dlmf.nist.gov/10 .