I'm at the 1st semester Condensed Matter of Physics, graduate student. really found this helpful before my mid exam "mathematical physics " Thank you so much
Thanks a lot! But.. can anyone tell me where does this original differential equation come from? Why would anyone even try to solve this in 17xx? It feels like it just appears out of thin air, and then like - btw let's use it to solve Schrodinger equation...
I understand the reason behind the polynomials, but I can`t seem to find the reason _why_ you want to solve it in the first place? Or in other words, in which practical situations does the legende DE occur?
Ah, good question! Legendre polynomials show up in a lot of physics applications involving spherical symmetry, including finding the electric potential due to a sphere, determining the diffusion of heat involving spherically symmetric objects, and solving the Schrodinger equation in quantum mechanics for spherically symmetric potentials, such as the Hydrogen atom. More generally, there are many different "special functions" out there that can be used to construct analogous Fourier series for other physical scenarios (cylindrical symmetry, spring-like potentials, etc...). These all work in a similar way as the Legendre Polynomials, so in a sense once you've learned Legendre, you've got the main ideas for all the others.
+Mobashshir Feroz In order to make a Fourier Series of a function, one has to be able to compute the Fourier Coefficients (see one of my other videos on this). The Fourier Coefficients involve computing some integrals of the function times sines or cosines. If these integrals don't converge - usually because the function has some sort of singular behavior, like 1/(1-x) at x = 1 - then it can't be represented by a standard Fourier Series!
Dr. Underwood's Physics RUclips Page Thanks for the info. My teacher said that we can take the function and we can make its Fourier series in the period pi-3pi or any period that does not involves the singularity. And I got your point, but can we make Fourier series of (Sinx)/x. Heard its a very funny function.
+Mobashshir Feroz Indeed, if you just "work around" the singularity, you'll be fine. Note that sin(x)/x doesn't actually have a singularity - it's regular and differentiable everywhere. Thanks for the comments!
Are you sure your Fourier series is correct? I think you're lacking either terms b_n*cos(n*pi*x/L), or a phase term in the sin. I mean, how would you represent functions f with f(0) =/= 0? I think an important point in both of these sets of base functions ({sin, cos} and {P_0(x), P_1(x), ...}) is that there have to be even and odd functions inside the set.
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The part you said "Im not going to check that" to was literally the only part I actually needed you to check...
you could use the rodrigues method which is pretty simple
I like the explanation. Very concise. Thanks Professor
can you please do a video on spherical harmonics expansion
Can you do a video on how you got the solution for when L =2 ? The rodrigues method seems pretty obnoxious compared to this
can you upload the lengdre eq derivation of singular point then regular/irregular and then series solution and last polynomial ??
I'm at the 1st semester Condensed Matter of Physics, graduate student. really found this helpful before my mid exam "mathematical physics " Thank you so much
Thanks a lot! But.. can anyone tell me where does this original differential equation come from? Why would anyone even try to solve this in 17xx? It feels like it just appears out of thin air, and then like - btw let's use it to solve Schrodinger equation...
greetings from a nuclear scientist , math give you wings , thank you very much .
cool.
what does L=1 and L=2 mean? Does L=2 mean y=x^2?
How did solve for l=1??
Thank you for this compliment to Jackson
I understand the reason behind the polynomials, but I can`t seem to find the reason _why_ you want to solve it in the first place? Or in other words, in which practical situations does the legende DE occur?
Ah, good question! Legendre polynomials show up in a lot of physics applications involving spherical symmetry, including finding the electric potential due to a sphere, determining the diffusion of heat involving spherically symmetric objects, and solving the Schrodinger equation in quantum mechanics for spherically symmetric potentials, such as the Hydrogen atom.
More generally, there are many different "special functions" out there that can be used to construct analogous Fourier series for other physical scenarios (cylindrical symmetry, spring-like potentials, etc...). These all work in a similar way as the Legendre Polynomials, so in a sense once you've learned Legendre, you've got the main ideas for all the others.
What is the name of the software you used? Can you please tell me.
You've probably already found something by now, but some programs off the top of my head that work similarly to this are rnote and xournal++
And what function can't be evaluated using fourier series?
+Mobashshir Feroz In order to make a Fourier Series of a function, one has to be able to compute the Fourier Coefficients (see one of my other videos on this). The Fourier Coefficients involve computing some integrals of the function times sines or cosines. If these integrals don't converge - usually because the function has some sort of singular behavior, like 1/(1-x) at x = 1 - then it can't be represented by a standard Fourier Series!
Dr. Underwood's Physics RUclips Page Thanks for the info. My teacher said that we can take the function and we can make its Fourier series in the period pi-3pi or any period that does not involves the singularity. And I got your point, but can we make Fourier series of (Sinx)/x. Heard its a very funny function.
+Mobashshir Feroz Indeed, if you just "work around" the singularity, you'll be fine. Note that sin(x)/x doesn't actually have a singularity - it's regular and differentiable everywhere. Thanks for the comments!
Dr. Underwood's Physics RUclips Page Thanks vice versa sir. I never thought I'll talk to a this much intelligent person. THANKS AGAIN Sir.
Are you sure your Fourier series is correct? I think you're lacking either terms b_n*cos(n*pi*x/L), or a phase term in the sin. I mean, how would you represent functions f with f(0) =/= 0? I think an important point in both of these sets of base functions ({sin, cos} and {P_0(x), P_1(x), ...}) is that there have to be even and odd functions inside the set.
From a first semester Nuclear Engineering graduate student, thanks a lot!
I have never come across that constraint "-1
It´s on the book "Fourier Series and Boundary Values" of Churchill
After watching this and other mathematical videos, I know that mathematics equations needs memorizing and if you don't, you'll fail
2 = ∂
Thank you
This wouldve been a great tutorial, if your handwriting was a little understandable.
and the '2' didn't look like '∂'
would be hell if he was dealing with partial derivatives
Thqnks sir but TWO is 2
great thnx
thanks professor👏👏👏👏👍👍👌👌
Hi sir
how the fuck do you write the number 2? it looks like the partial derivative
Your saying Legendre wrong!! That is my born last name
900 pp