Question

(PDE)(Unknown) My instructor is posting (and testing us on) a lot of material outside of our textbook as standalone notes, and I don't understand what's going on in some of this, more info posted below.

Answer 1

The first thing is this right here: http://i.imgur.com/0eHCeCi.png

I have no idea what I_n generally represents. Yeah, it's an integral, but what's the point of it? Does it have a name? Is this part of some general process? This is related to Gram-Schmidt, but my book once more doesn't cover it, and the sources I can find Gram-Schmidt mostly cover vector orthonormalization/orthogonalization, not *function* orthogonalization. @Kainui

Answer 2

The second thing is this, because all I have are these notes: How are f_1, f_2, f_3, and so on, related to f_n, between the previous and the following image? http://i.imgur.com/QiRiN8j.png

Answer 3

@iambatman , are you any good with this?

Answer 4

(And sorry, f_0, f_1, and f_2, not f_1, f_2, and f_3.)

Answer 5

@jim_thompson5910

Answer 6

Alright, I get what (in the second image) happened in I_0, now trying to figure out whatever is happening in I_n.

Answer 7

plug n=0

Answer 8

I'm understanding bits and pieces of what the expressi-yeah, I said I got that, n=0 isn't the issue, it's some of the other stuff. In I_n is where I'm a bit confused now, he has an expression, and he sets it equal to the integral of that expression's derivative plus something else, the original term evaluated from 0 to infinity.

Answer 9

where

Answer 10

Yeah, I just have....I_n. At I_n.

Answer 11

I_n=n*I_n-1

Answer 12

seen from integration by parts

Answer 13

I get integration by parts now on my own looking at that, but I don't understand what you just said, lmao. Thanks. I'm starting to get this.

Answer 14

\[I_n=n*I_{n-1}\]

Answer 15

so it cascades

Answer 16

\[I_n=n*I_{n-1}=n*(n-1)I_{n-2}=n*(n-1)(n-2)(n-3)...I_{n-n}\]
and we know I_o =1

Answer 17

so n!

Answer 18

Yeah, I got it, it was the way you first wrote it without LaTeX; thanks so much, this makes sense now. I guess I'm still asking: How is this relevant to Gram-Schmidt, or is this? Let me just post the whole notes example to see if this makes sense in context, one sec.

Answer 19

(2nd and 3rd solution are omitted, just dealing with this right now) http://i.imgur.com/eBcqfEy.png

Answer 20

look at the inner product of the 2 functions and see if it is 0, that is the definition of functional orthogonality

Answer 21

Parts II and III seem very relevant, but I don't understand how part I is related to Gram-Scmhidt.

Answer 22

Yeah, just the first part. I mean.

And there are three functions, does that mean (if they were orthogonal) I would be taking the inner product of f_0f_1, f_0f_2, and f_1f_2?

It's still just the first part with the integrals, I don't see the relevance.

Answer 23

?

Answer 24

@dan815

Answer 25

yea

Answer 26

Yeah, just what I asked earlier, I fail to see the relevance of Part I to the Gram-Schmidt procedure. That's all I want to know atm, is how that is relevant. We found that I_n = n!, and how does that affect or change the process whatsoever.

Answer 27

(Or is it not apparently relevant, and the document was just...poorly named and labeled, overall, you think.)

Answer 28

it shows u relationship betwen f_ns

Answer 29

you just have an integral representation of n!

Answer 30

now u can see what n! means if n is not integer if such a thing exists

Answer 31

I've gotten this far, but I don't understand this last question. Either way, thank you very much, I'm going to have to think about this.

Answer 32

ok :)