- Shift theorem
In
mathematics , the (exponential) shift theorem is atheorem aboutpolynomial differential operators ("D"-operators) andexponential function s. It permits one to eliminate, in certain cases, the exponential from under the "D"-operators.The theorem states that, if "P"("D") is a polynomial "D"-operator, then, for any sufficiently
differentiable function "y",:P(D)(e^{ax}y)equiv e^{ax}P(D+a)y.,
To prove the result, proceed by induction. Note that only the special case :P(D)=D^n,
needs to be proved, since the general result then follows by linearity of "D"-operators.
The result is clearly true for "n" = 1 since
:D(e^{ax}y)=e^{ax}(D+a)y.,
Now suppose the result true for "n" = "k", that is,
:D^k(e^{ak}y)=e^{ax}(D+a)^k y.,
Then,
:egin{align}D^{k+1}(e^{ax}y)&equivfrac{d}{dx}{e^{ax}(D+a)^ky}\&{}=e^{ax}frac{d}{dx}{(D+a)^k y}+ae^{ax}{(D+a)^ky}\&{}=e^{ax}left{left(frac{d}{dx}+a ight)(D+a)^ky ight}\&{}=e^{ax}(D+a)^{k+1}y.end{align}
This completes the proof.
The shift theorem applied equally well to inverse operators:
:frac{1}{P(D)}(e^{ax}y)=e^{ax}frac{1}{P(D+a)}y.,
There is a similar version of the shift theorem for Laplace transforms (t):
:e^{ax}scriptstylemathcal{L}(f(t))=scriptstylemathcal{L}(f(t-a)).,
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