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Math Help - linear differential operators

  1. #1
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    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) + c1(D - a) + ... ck(D - a)k

    (2) q(D) = q(a)(D−a)s+c1(D−a)s+1 +...+ck(D−a)s+k

    then substitute D with a

    How did they figure (1) out?

    if k = 2, I get

    D2 + c1D +c2 = A2 + c1A + c2 + c1D - c1A + c2D2 - c22DA + c2A2

    I feel totally confused and as if I missed a lot
    Could you please point out what I do not understand?
    Thank you
    Last edited by likelylad; February 21st 2013 at 02:45 PM.
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  2. #2
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    Re: linear differential operators

    Hey likelylad.

    Can you clarify whether D is just a variable or whether it is a differential operator please?
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  3. #3
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    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 :/
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  4. #4
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    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.
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  5. #5
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    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
    Dsp(D) = q(a)s! + powers of (D-a)
    D=a
    p(a) = q(a)s!/as
    ?

    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?

    Thank you for the enlightenment
    Last edited by likelylad; February 23rd 2013 at 06:45 AM.
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