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17h
awarded  Necromancer
2d
comment An alternative way to determine when $\int_{0}^{\infty} \cos(\alpha x) \prod_{m=1}^{n} J_{0}(\beta_{m} x) \, dx =0$
@Dr.MV Correct. I stated in the question that the function being integrated is entire. And I thought perhaps there could be another way to explain why those conditions cause the integral to evaluate to $0$.
2d
comment An alternative way to determine when $\int_{0}^{\infty} \cos(\alpha x) \prod_{m=1}^{n} J_{0}(\beta_{m} x) \, dx =0$
@Dr.MV Yes. If $\int_{-\infty}^{\infty} e^{i\alpha x} \prod_{m=1}^{n} J_{0}(\beta_{m} x) \, dx =0$ under the conditions on the parameters stated, then we also know that $\int_{0}^{\infty} \cos(\alpha x)\prod_{m=1}^{n} J_{0}(\beta_{m} x) \, dx =0$ under the same conditions since $J_{0}(x)$ is an even function that is real-valued along the real axis.
2d
comment An alternative way to determine when $\int_{0}^{\infty} \cos(\alpha x) \prod_{m=1}^{n} J_{0}(\beta_{m} x) \, dx =0$
@Dr.MV $J_{0}(x)$ is an even function
2d
asked An alternative way to determine when $\int_{0}^{\infty} \cos(\alpha x) \prod_{m=1}^{n} J_{0}(\beta_{m} x) \, dx =0$
Feb
10
awarded  Popular Question
Feb
8
awarded  Necromancer
Feb
8
revised How to find branch points
I restated parts of my answer more clearly.
Feb
1
answered Evaluating $\int^{\pi}_0\arctan\left(\frac{p\sin x}{1-p\cos x}\right)\sin(nx) dx$ by differentiation under integral?
Jan
31
awarded  Revival
Jan
31
revised Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function
added 2 characters in body
Jan
31
revised Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function
corrected typos
Jan
30
answered Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function
Jan
30
revised Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function
I deleted the answer to my question and posted it as an answer.
Jan
30
comment Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function
@epimorphic Even though the stuff after the "EDIT:" was posted a long time ago, it's probably a good idea to do that. I don't just want to cut and paste it, though. It needs some words of explanation.
Jan
29
revised Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function
I edited the title so that my question is more clear.
Jan
29
awarded  Popular Question
Jan
26
awarded  Revival
Jan
26
revised How to choose a contour in order to use the residue theorem to sum up a series from Ryzhik?
added 24 characters in body
Jan
26
revised How to choose a contour in order to use the residue theorem to sum up a series from Ryzhik?
added 382 characters in body