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awarded  Popular Question
Dec
19
accepted Simpler proof of an integral representation of Bessel function of the first kind $J_n(x)$
Dec
19
comment Simpler proof of an integral representation of Bessel function of the first kind $J_n(x)$
Yup, I had the same substitutions in mind, except into (2) to show (1). It was really simple once recalled that $e^{in\tau}$ has a period of $2\pi$ over angle $\tau$ for integer $n$. Didn't get around to updating my post, but you should still get the credit for reminding me of that. :)
Dec
19
comment Simpler proof of an integral representation of Bessel function of the first kind $J_n(x)$
Ahhh -- I think I got it! Of course, $e^{in\tau}$ has a period of $2\pi$ as long as $n$ is an integer. Since cosine also has a periodic of $2\pi$, everything works! Thank you, I'll update my post either tonight or tomorrow.
Dec
18
comment Simpler proof of an integral representation of Bessel function of the first kind $J_n(x)$
Unfortunately, I am not very strong at solving differential equations (having never taken a course on them), but I'll look into this one. I tried the substitution that you mentioned, but couldn't get it to work, since I couldn't find a way to get the right limits of the integral...
Dec
18
asked Simpler proof of an integral representation of Bessel function of the first kind $J_n(x)$
Nov
17
awarded  Popular Question
Nov
4
awarded  Popular Question
Oct
16
revised Approximating two-dimensional convolution
solved the problem
Oct
14
asked Approximating two-dimensional convolution
Oct
12
awarded  Popular Question
Oct
5
accepted Equation involving an error and Gaussian functions
Oct
5
accepted Extension of the convolution theorem
Oct
3
awarded  Nice Question
Oct
2
revised Fourier transform of a truncated Gaussian function
typo
Sep
30
comment Equation involving an error and Gaussian functions
You are absolutely correct, the resolution on my graph wasn't high enough. The positive solution is clearly not $x=1$, and, yes, differentiating both sides makes no sense...
Sep
30
revised Equation involving an error and Gaussian functions
updated per deleted comment
Sep
30
asked Equation involving an error and Gaussian functions
Sep
30
revised Fourier transform of a truncated Gaussian function
linked RonGordon
Sep
30
comment Fourier transform of a truncated Gaussian function
@RonGordon Yup, that's the answer to the question that I mention at the end of my update. I'll put in an explicit link to your answer (which I really enjoyed reading, btw, it's quite a tour de force).