Legendre polynomials

Constructing the Legendre polynomials, which are an orthonormal basis for the set of polynomials. Example of Gram-Schmidt to inner product spaces.

[ssdd]

blackpenredpen said that writing an integral without dx is like driving without seatbelt

[pitreason]

Legendre polynomials are the part of solutions of Shrodinger equation for hydrogen atom

[garrytalaroc]

Ohhh boy. How long I have waited for this. I love this channel.

[apostoloskountouris]

"It looks scary but this is not that scary" . Thanx Dr Peyam for sharing your wisdom, simple but to the point.

[benjaminbrat]

Orthogonal polynomials also appear in the spherical harmonics, analytical solutions of the angular part of Schrödinger's equation for a potential V = blabla/r. Section 4.1.2 in the Griffith, if you want a nice read =)

[dominicellis]

These polynomials are used for the angular part of the laplacian in spherical coordinates. The time independent Schrödinger equation in spherical coordinates is a great application of spherical harmonics. The legrende function is the product of legrende polynomials with a sinusoid.

[BorisNVM]

This is very interesant, this basis is important in electromagnetism in the matter. Thank u

[goodplacetostart]

I like the way he first says thanks
This is what DIFFERENTIATES him from others

[noahschulz]

Do more about these and polynomial solutions like it are often introduced as solutions to differential equations but never explained.

[TheRedfire]

Hey Dr.Peyam I got an interesting linear algebra problem that I read when revisting Splinder book
-Snow White distributed 21 liters of milk among the seven dwarfs.
The first dwarf then distributed the contents of his pail evenly to the pails of other six dwarfs. Then

the second did the same, and so on. After the seventh dwarf distributed the contents
of his pail evenly to the other six dwarfs, it was found that each dwarf had exactly as
much milk in his pail as at the start.
What was the initial distribution of the milk?
Generalize to N dwarfs

[regulus]

Legen... wait for it... drery!!! Legendrery!

[ffggddss]

I must have missed something, because you kept saying that each polynomial needed to evaluate to 1 at x=1.
Isn't it rather that in an orthonormal basis, each vector (polynomial) dotted with itself has to = 1?
("Ortho" means mutually orthogonal; "normal" means normalized to unit length.)

Fred

[shayanmoosavi]

wow. This was a much more straightforward approach to legendre polynomials. How is making an orthonormal set from the basis {1, x, x^2, ...} in the interval [-1, 1] relate to the solution of the legendre equation? They seem completely unrelated to me.

[HAL-ojjb]

I feel oddly referenced at 0:40 ...
Great video btw, as always! ^^

[weinihao]

Hello Dr. Peyam, this again was a very interesting video. The dot product of functions (or rather inner product) here exists, because the integral always converges, since the functions are bounded and their support is finite. Do you also plan to make a video on the space of bounded functions with infinite support that possesses an inner product (L2 space on all R^n or C^n) and how it is expanded by distributions (Gelfandsches Raumtripel) to create the expanded Hilbert space that forms the very heart of signal processing (in which the Fourier transform is still an automorphism, but cool functions like non-decaying complex exponentials exist whose integrals don't converge in standard L2)?
Furthermore, the orthogonal basis you create contains infinitely many elements. It would be very nice if you added a video about the difference between a Schauder basis and a Hamel basis (apologies, if you did already and I didn't find it)!

[martinepstein]

Hey Dr. Peyam, I've seen some weird closed form expressions for the n'th Legendre polynomial. It would be cool to see you derive one in a video!

[Royvan]

is there any motivation behind the p_n(1) = 1 condition? the first normalization condition that comes to my mind is ||p_n|| = 1

[danielmilyutin]

Legendre polynomials and other orthogonal polynomials are used for numeric approximation and integration. They particularly arise from eigenproblems for pde.
Chebyshev polynomials rule for interpolation. 😁

[holyshit]

We have generating function
f(x, t) = 1/sqrt(1 - 2xt + t^2)
If someone like odes
Legendre polynomial is particular solution of following ode
(1-x^2)y'' -2xy' + n(n+1)y = 0
which satisfies condition y(1) = 1
I found in tables integral which gives not quite the Legendre polynomial but Legendre polynomial is easy to get from this integral
-
with assumption that theta is in interval where cos(\frac{\theta}{2}) > 0, f. e. (-\pi ;\pi)
There are connections with Chebyshov polynomials
= U_{n}(x)
= (n+1)P_{n}(x)

[ozzyfromspace]

This was awesome! I knew about GS in linear algebra but never thought about it for functional spaces. Does this generalize to differential operators? I ask because they too are elements of a vector space. Thank you for enlightening me! I tried deriving this based on the behavior of polynomials I was seeking, but kept getting P1(x)=0 and thinking it didn’t make sense. You showed that everything is okay 🙏🏽😊