That's pretty funny actually, one of the other ways to do it is using Feynman's trick on I(t) = int(-inf,inf) exp(-t²(x²+1))/(x²+1)dx, and that integrand appears at the very end of your calculations. In a sense the series expansion method explains why it's natural to use that function for Feynman's trick.
More complicated than the usual approach, but on the other hand, one does not need to know polar coordinates... Very nice way! :) Did you come up with that on your own?
By the way, do you know the RUclips channel of Dr Peyam? He has a whole playlist on the Gaussian integral, with 12 different methods. But yours was not included there, so apparently it's really a new one?
Yes, I'm a fan of Dr Peyam! This approach is adapted from/related to something I remember seeing a few years ago - I can't seem to find a reference though for where I found it. I'll see if I can find it again.
@@DrBarker I've actually seen this exact approach to the Gaussian integral before. As far as I know, it originated as an answer by user Stefan Lafon to a popular Math Stack Exchange question asking to give various methods of evaluating the Gaussian integral. I remember being intrigued by seeing this clever and elegant solution for the first time and wondering if it could be adapted into a proof of the Gaussian integral's twin sisters, the Fresnel integrals (as many proofs of the Gaussian integral can). So, a year or two ago I figured out a way to adapt this method to evaluate the Fresnel integrals as well, without too much difficulty. I tried before to share a link to Stefan Lafon's answer on MSE detailing this method of attacking the Gaussian integral, but my comment was instantly deleted by the comment section's spam filter, so I hope you can find it without the link.
@@violintegral I found the Stack Exchange post - I've put a link in the description. I'd be interested to know if the Stack Exchange answer is the original source for this method, or if it also appears in a book? I'm curious about the analogous approach for the Fresnel integrals. My gut feeling is that it could be simpler than the Gaussian integral, but I haven't tried writing out the calculations. Did this require any "tricks" beyond exchanging summation/integration, or the "1/(2k+1) = int_0^1 x^{2k} dx" trick?
I hadn't seen this one before but it is straightforward one you break it down as you have done. Paul Levy's original functional analysis treatise which Norbert Wiener used for his development of Brownian motion actually developed the Gaussian from volumes of n dimensional spheres. As usual you have done a very clear and succinct exposition.
I thought it was overcomplicated until the mid part where I really liked how many little surprises it had! I enjoyed it. The ending with the "ok, and this messy part will go to zero" was kinda underwhelming tbh, but good video overall.
Out of interest, what prevents applying the power series method without squaring the integral? Edit: I'm going to try it out and see where I hit issues
@@DrBarker I hit an issue very quickly; interchanging the integral and sum doesn't work since the conditions of Fubini aren't satisfied. Oh well, worth an exploration.
WOW!!! I never knew this was possible with infinite series!
Check yer email/spam
That's pretty funny actually, one of the other ways to do it is using Feynman's trick on I(t) = int(-inf,inf) exp(-t²(x²+1))/(x²+1)dx, and that integrand appears at the very end of your calculations.
In a sense the series expansion method explains why it's natural to use that function for Feynman's trick.
More complicated than the usual approach, but on the other hand, one does not need to know polar coordinates... Very nice way! :) Did you come up with that on your own?
By the way, do you know the RUclips channel of Dr Peyam? He has a whole playlist on the Gaussian integral, with 12 different methods. But yours was not included there, so apparently it's really a new one?
Yes, I'm a fan of Dr Peyam! This approach is adapted from/related to something I remember seeing a few years ago - I can't seem to find a reference though for where I found it. I'll see if I can find it again.
@@DrBarker I've actually seen this exact approach to the Gaussian integral before. As far as I know, it originated as an answer by user Stefan Lafon to a popular Math Stack Exchange question asking to give various methods of evaluating the Gaussian integral. I remember being intrigued by seeing this clever and elegant solution for the first time and wondering if it could be adapted into a proof of the Gaussian integral's twin sisters, the Fresnel integrals (as many proofs of the Gaussian integral can). So, a year or two ago I figured out a way to adapt this method to evaluate the Fresnel integrals as well, without too much difficulty. I tried before to share a link to Stefan Lafon's answer on MSE detailing this method of attacking the Gaussian integral, but my comment was instantly deleted by the comment section's spam filter, so I hope you can find it without the link.
@@violintegral I found the Stack Exchange post - I've put a link in the description. I'd be interested to know if the Stack Exchange answer is the original source for this method, or if it also appears in a book?
I'm curious about the analogous approach for the Fresnel integrals. My gut feeling is that it could be simpler than the Gaussian integral, but I haven't tried writing out the calculations. Did this require any "tricks" beyond exchanging summation/integration, or the "1/(2k+1) = int_0^1 x^{2k} dx" trick?
I hadn't seen this one before but it is straightforward one you break it down as you have done. Paul Levy's original functional analysis treatise which Norbert Wiener used for his development of Brownian motion actually developed the Gaussian from volumes of n dimensional spheres. As usual you have done a very clear and succinct exposition.
I know it precisely the other way round - volumes of n-dimensional spheres are derived with previous knowledge of the Gaussian integral.
@@bjornfeuerbacher5514 yep.
Brilliant!!
Ok, ok, you convinced me. I'll learn polar coordinates.
Nice video! :)
Proper good job there mate.
Thank you so much for this interesting approach other than the polar coordinates one.
There are other methods that are shorter, but this one certainly does the job.
I thought it was overcomplicated until the mid part where I really liked how many little surprises it had! I enjoyed it. The ending with the "ok, and this messy part will go to zero" was kinda underwhelming tbh, but good video overall.
❤
Heavy duty!
Out of interest, what prevents applying the power series method without squaring the integral?
Edit: I'm going to try it out and see where I hit issues
Let me know if you get anywhere with this approach, I'd be keen to hear if this gives another alternative way of evaluating the integral.
@@DrBarker I will. I'll see what progress I can make this evening
@@DrBarker I hit an issue very quickly; interchanging the integral and sum doesn't work since the conditions of Fubini aren't satisfied. Oh well, worth an exploration.
1:14 ☕️☕️☕️
80% of the math in one video!😀
Always squaring... not that I am complaining, but it is - I guess - a highly surprising trick.
suprise! arctan(1) saves the day