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Math Help - Functions of Matrices

  1. #1
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    Functions of Matrices

    Let  D: \mathbb {R} [X] \to \mathbb {R} [X] be the differentiation operator  D(f(X)) = f'(X) . Prove that  e^{tD} (f(X)) = f(X+t) for a real number  t \in \mathbb {R} .

    I really don't know where to start on this and would appreciate if anyone could give me hints on what actually needs to be shown in a more concrete way. Anything that could start me off would be great! thanks
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Hint :

    e^{tD}=1+tD+\frac{t^2D^2}{2!}+...

    e^{tD}f(x)=1+tf'(x)+\frac{t^2f''(x)}{2!}+...

    Now think in terms of Taylor series...
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  3. #3
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    Quote Originally Posted by Bruno J. View Post

    e^{tD}f(x)=1+tf'(x)+\frac{t^2f''(x)}{2!}+...
    that 1 should actually be f(x).
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  4. #4
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    Indeed Thanks!
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