Please continue your videos! I'm a physics and CS major and I always come to this channel when I don't understand something from my courses! This channel is very useful!
why the 2:01 recursion relation is terminate at x^k, I know that the recursion relation have n-k so n = k wil terminate the part of k is odd or even, and another part of even and odd will not terminate? how to explain? thx
I think the recurrence relation is a_n+2 = - ((n-k)(n+k+1) / [ (n+2)(n+1) ] ) a_n. I guess you drop the minus sign. By the way, You made a great video. It really helped me a lot.
probably a silly question, but in the beginning of the video when you defined the a_k value you made sure to choose a value related to the k constant in the legendre equation, and you made sure to point out that this k shouldn't be confused with the running summation. Yet when you did the change of index you used k for the running sum and used this running sum k to cancel with the constant k terms. What is the insight I am missing here? It feels like these should be two different values unable to cancel with each other
hi.what is the radius of convergence legendre polynomials?.....and university ask us to tell them how we find a-k.....how can i tell them???they told us us a(n+2) formula.....thanks
Hi! Thanks for the great explanation. Using your videos to learn maths for second year physics. I was wondering, how did you arrive at the expression for ak in the red cloud? I couldn't find it given anywhere in the preceding working.
Is the final solution in this video that is a finite series really equal to our sollution? I mean, even if we use the recursion relation backward to go from the highest order to the lowest, hence we get a finite series, the recursion relation itself still indicates that there are more terms that are higher than the highest order that we chose. In a previous video, you said that either odd or even will terminate but not both. Can using the recursion relation backward really change a infinite series into a finite series? Can we really use an equal sign?
When you say that 'the recursion relation itself still indicates that there are more terms that are higher than the highest order that we chose', that's not an entirely accurate description. The even or odd series will still terminate, depending on the value of the parameter k in the Legendre ODE shown in 0:57. In the previous video, I said that if your Legendre ODE had k = 3, then the odd series in the recursion relation would terminate (all odd-indexed coefficients after a3). If k = 4, then the even series would terminate (all even indexed coefficient after a4). So in this case, when we're going backwards in the recursion relation from a_k, we're actually going backwards from the final coefficient of the Legendre Polynomial (i.e. the terminating solution to the ODE). Anything beyond the final coefficient of a_k (e.g. a_(k+2)) is zero. If you go back to my previous video (ruclips.net/video/3e5BUrtUKZc/видео.html), you'll see this illustrated starting from 7:15. When one of the coefficients is zero (e.g. a3 = 0 when k = 1), all subsequent coefficients (e.g. a5, a7, etc) would also be zero, since those subsequent coefficients depend on a3. In this video, we're basically going backwards from ak, if that makes any sense (a_(k+2) etc are all zero). That's a long explanation, and I hope it addresses your question. If you need any more clarification, let me know!
Each legendre polynomials contains only odd powers or even powers. However can the power series solution of the legendre differential equation can have both odd and even powers right? Are one of the branches necessarily zero? If yes, then why?.. does one of the branches diverge unless set equal to zero or something?
Hey man amazing video, quick question though, how do we actually show that P(x=1) =1? I mean i plug it in the final eqn and we are only left with is a_(k-2m)... and the only conclusion is that it must equal 1. But how do we show that a_(k-2m) summed over m is 1? does it simplify to a known series or...? thanks
A doubt.. why are legendre polynomials confined to region [-1,+1]. I can see that there is a singularity at x=1,-1... how do the solutions of Legendre differential equation look outside of the region [-1,+1]
The expansion around x=0 is defined/converges up to the closest singular point, which in this case is 1 and -1 as they are both the same distance away from 0
Generally, any value of a_k is fine, but for the purposes of computing the Legendre polynomials, a_k isn't set to an arbitrary constant; it's set to a constant which depends on the value of k which is found in Legendre's ODE. Setting a_k to the value I set it as makes the computation of the polynomial much easier.
You can probably still find a solution, just that you wouldn't be able to find a nice pattern using the recursion relation because your formulas would be really cumbersome to deal with. If you can't find a pattern, then your formulas, by extension, wouldn't be 'good'.
and once again a work of brilliance thanks your great work and effort to educate others like us thanks once again sir but sir why calculate from asubk and not from a asub0 is there anything to it
Hi Vishnu, Thank you again for the kind words! The reason we calculate from asubk is that it makes our calculations simpler. I've personally tried, and I found it much more difficult to come up with a Legendre polynomial formula by setting asub0 to a constant like 1. Most books calculate it from asubk as well. Also, we want all the Legendre polynomials to maintain P_k(1)=1, and it's tougher to do that if we set asub0 to a constant. Hope that helps!
Sir plz plz day after tomorrow is my exam....Can u solve this problem: The relation between a small horizontal deflection 'theta' of a bar magnet under the action of earth's magnetic field is A d2theta/dt2+MHtheta=0.....e Where A is the moment of inertia of the magnet about the axis,M is the magnetic moment of the magnet and H is the horizontal component of intensity of field due to earth...Find the time required for a complete vibration????
Please continue your videos! I'm a physics and CS major and I always come to this channel when I don't understand something from my courses! This channel is very useful!
Thank you!
you saved my life..
a physics major knows.. how much hectic it is to differentiate to recieve polynomials.. 😢😢
Glad you found it useful!
Thank you for your lectures and your way of explaining things.
Super precise! Amazingly condensed in a professional way!
It's sad to see so much less views on such a video. Your work is really awesome man, you just saved my sem.
Would love a similar video on Tchebychev polynomials. Thank you for all your hard work and keep it up!
3:51 But..., where it comes from? How do you know, that this actually works?
I'm confused in that too! 😕 If you get to know, pls let me know too!
Saved my day. Thanks for making such helpful videos
i dont know to express of my feeling really thank you very much
There is a mistake on 6:28 of video. (K-2)! in the denominator Was supposed to be to the power 2. What about that?
Not really, because the k-1 remaining in the denominator combines with one of the (k-2)!'s, giving (k-1)!(k-2)!
@C Malb yeah I was wondering the same thing as well
ahhh ok after a bit thinking i realised that the power of 2 disappeared but to kompensate this he wrote another (k-1)!
So awesome, I am excited to understanding Dyadic Greens functions so I can see how wave guides are sized.
You are a life saver❤❤
9:57 Here floor function would simplify notation
why the 2:01 recursion relation is terminate at x^k, I know that the recursion relation have n-k so n = k wil terminate the part of k is odd or even, and another part of even and odd will not terminate?
how to explain? thx
I think the recurrence relation is a_n+2 = - ((n-k)(n+k+1) / [ (n+2)(n+1) ] ) a_n. I guess you drop the minus sign. By the way, You made a great video. It really helped me a lot.
What is the infinite series solution in 3:17
You're awesome! I'm going to devour your whole channel. I'm giving thumbs ups in return.
probably a silly question, but in the beginning of the video when you defined the a_k value you made sure to choose a value related to the k constant in the legendre equation, and you made sure to point out that this k shouldn't be confused with the running summation. Yet when you did the change of index you used k for the running sum and used this running sum k to cancel with the constant k terms. What is the insight I am missing here? It feels like these should be two different values unable to cancel with each other
Thank you sir this was very helpful for me
arre waah
hi.what is the radius of convergence legendre polynomials?.....and university ask us to tell them how we find a-k.....how can i tell them???they told us us a(n+2) formula.....thanks
Please sir, make video on leguerre and hermite differential equation
Hi! Thanks for the great explanation. Using your videos to learn maths for second year physics. I was wondering, how did you arrive at the expression for ak in the red cloud? I couldn't find it given anywhere in the preceding working.
Can you explain where did you get that Ak, it just pop right there
i saw from the other books, that it came from constant coefficient of the solutions.
When k is an even number lets say, the even series terminates, so does the odd series still count as a solution to the ODE?
Thanks for the tutorials, could you please tell me what program do you use to write such beautiful lectures.
Is the final solution in this video that is a finite series really equal to our sollution? I mean, even if we use the recursion relation backward to go from the highest order to the lowest, hence we get a finite series, the recursion relation itself still indicates that there are more terms that are higher than the highest order that we chose. In a previous video, you said that either odd or even will terminate but not both. Can using the recursion relation backward really change a infinite series into a finite series? Can we really use an equal sign?
When you say that 'the recursion relation itself still indicates that there are more terms that are higher than the highest order that we chose', that's not an entirely accurate description. The even or odd series will still terminate, depending on the value of the parameter k in the Legendre ODE shown in 0:57.
In the previous video, I said that if your Legendre ODE had k = 3, then the odd series in the recursion relation would terminate (all odd-indexed coefficients after a3). If k = 4, then the even series would terminate (all even indexed coefficient after a4). So in this case, when we're going backwards in the recursion relation from a_k, we're actually going backwards from the final coefficient of the Legendre Polynomial (i.e. the terminating solution to the ODE). Anything beyond the final coefficient of a_k (e.g. a_(k+2)) is zero.
If you go back to my previous video (ruclips.net/video/3e5BUrtUKZc/видео.html), you'll see this illustrated starting from 7:15. When one of the coefficients is zero (e.g. a3 = 0 when k = 1), all subsequent coefficients (e.g. a5, a7, etc) would also be zero, since those subsequent coefficients depend on a3. In this video, we're basically going backwards from ak, if that makes any sense (a_(k+2) etc are all zero).
That's a long explanation, and I hope it addresses your question. If you need any more clarification, let me know!
The infinite series solution is when k is odd the even series does not terminate part, k is even when odd series does not terminate?
Each legendre polynomials contains only odd powers or even powers. However can the power series solution of the legendre differential equation can have both odd and even powers right? Are one of the branches necessarily zero? If yes, then why?.. does one of the branches diverge unless set equal to zero or something?
Hey man amazing video, quick question though, how do we actually show that P(x=1) =1? I mean i plug it in the final eqn and we are only left with is a_(k-2m)... and the only conclusion is that it must equal 1. But how do we show that a_(k-2m) summed over m is 1? does it simplify to a known series or...? thanks
A doubt.. why are legendre polynomials confined to region [-1,+1]. I can see that there is a singularity at x=1,-1... how do the solutions of Legendre differential equation look outside of the region [-1,+1]
The expansion around x=0 is defined/converges up to the closest singular point, which in this case is 1 and -1 as they are both the same distance away from 0
so helpful, thanks.
Hi! When the a_k is set to arbitrary constant, does that mean we are choosing one of many possible solutions of Legendre's ODE and any constant is ok?
Generally, any value of a_k is fine, but for the purposes of computing the Legendre polynomials, a_k isn't set to an arbitrary constant; it's set to a constant which depends on the value of k which is found in Legendre's ODE. Setting a_k to the value I set it as makes the computation of the polynomial much easier.
Thank you for answering! To make it clear - would it be (theoretically) possible to obtain good solution by setting a_k = k and if not, why?
You can probably still find a solution, just that you wouldn't be able to find a nice pattern using the recursion relation because your formulas would be really cumbersome to deal with. If you can't find a pattern, then your formulas, by extension, wouldn't be 'good'.
Thank you!
and once again a work of brilliance thanks your great work and effort to educate others like us thanks once again sir but sir why calculate from asubk and not from a asub0 is there anything to it
Hi Vishnu,
Thank you again for the kind words! The reason we calculate from asubk is that it makes our calculations simpler. I've personally tried, and I found it much more difficult to come up with a Legendre polynomial formula by setting asub0 to a constant like 1. Most books calculate it from asubk as well. Also, we want all the Legendre polynomials to maintain P_k(1)=1, and it's tougher to do that if we set asub0 to a constant. Hope that helps!
ok thanks for the video and the comment
on 4: 07 of video how i can find a formula a sub k equal to 2 k factorial over 2 to the power k times square k factorial
I'm confused in that too! 😕
If you get to know, pls let me know too!
Bro which book is best for mathematical physics?
Sir plz plz day after tomorrow is my exam....Can u solve this problem:
The relation between a small horizontal deflection 'theta' of a bar magnet under the action of earth's magnetic field is
A d2theta/dt2+MHtheta=0.....e
Where A is the moment of inertia of the magnet about the axis,M is the magnetic moment of the magnet and H is the horizontal component of intensity of field due to earth...Find the time required for a complete vibration????
That's actually a very simple question
Find angular frequency from equation and set T = 2π/angular frequency
thanks a ton sir!!!
No problem! Glad you liked it!
In my opinion he nreplaced n = k - 2 too early
See what happens for Chebyshov polynomials or for Hermite polynomials
more parts
❤
Why do you sound like a robot?
Still cheating you are not explaining throughly on a_k