Shift theorem

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In mathematics, the (exponential) shift theorem is a theorem about polynomial differential operators (D-operators) and exponential functions. 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


needs to be proved, since the general result then follows by linearity of D-operators.

The result is clearly true for n = 1 since


Now suppose the result true for n = k, that is,

D^k(e^{ax}y)=e^{ax}(D+a)^k y.\,


&{}=e^{ax}\frac{d}{dx}\{(D+a)^k y\}+ae^{ax}\{(D+a)^ky\}\\

This completes the proof.

The shift theorem applied equally well to inverse operators:


There is a similar version of the shift theorem for Laplace transforms (t<a):

\scriptstyle\mathcal{L}(e^{at} f(t))=\scriptstyle\mathcal{L}(f(t-a)).\,