Exponential input theorem

Hello, I am puzzled about this operation (1) with polynomials in a general proof of Exponential input theorem and I don't understand the the concept of *a* being s-fold zero.

* p(D) = q(D)(D - a)*^{s } q(a) = nonzero

The above implies

p^{(s)}(a) = q(a)s!

The proof lets k be the degree of q(D) and writes it in powers of (D-a)

(1) q(D) = q(a) + c_{1}(D - a) + ... c_{k}(D - a)^{k }

*(2) q(D) = **q(a)(D−a)*^{s}+c_{1}(D−a)^{s+1 }+...+c_{k}(D−a)^{s+k }then substitute D with a^{ }How did they figure (1) out?

if k = 2, I get

D^{2} + c_{1}D +c_{2} = A^{2} + c_{1}A + c_{2} + c_{1}D - c_{1}A + c_{2}D^{2 }- c_{2}2DA + c_{2}A^{2 }

I feel totally confused and as if I missed a lot :(

Could you please point out what I do not understand?

Thank you

Re: linear differential operators

Hey likelylad.

Can you clarify whether D is just a variable or whether it is a differential operator please?

Re: linear differential operators

I think it is a differential operator..

the proof is on page 5 of and is very short

http://ocw.mit.edu/courses/mathemati...18_03S10_o.pdf

Then the middle terms of (D-a)^k disappear becease Da is zero, right? Still do not see what they did there :/

Re: linear differential operators

If you are wondering how they got q(D) out then you should think about a Taylor expansion type result around the centre of a: In other words, they are expanding something around a instead of 0.

Re: linear differential operators

hey I can see it ! :)

I am still wondering about a few things.

Can you rewrite

p^{(s)}(D) = q(a)s! + powers of (D-a)

D=a

p^{(s)}(a) = q(a)s!

into

D^{s}p(D) = q(a)s! + powers of (D-a)

D=a

p(a) = q(a)s!/a^{s}

?

It really got me confused:

Is there a proof that you can treat d/dx as a variable in some instances? It obviously works but..

Do people think about this (like in topology or some Hilbert spaces, I had a look into operator theory and it looks years

-away hard - I am noob at math) ? Or is it clear how to use it to everyone except for me? :D

Thank you for the enlightenment:)