Shift Theorem

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,

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,

Then,

$\begin{align}D^{k+1}(e^{ax}y)&\equiv\frac{d}{dx}\{e^{ax}(D+a)^ky\}\\ &{}=e^{ax}\frac{d}{dx}\{(D+a)^k y\}+ae^{ax}\{(D+a)^ky\}\\ &{}=e^{ax}\left\{\left(\frac{d}{dx}+a\right)(D+a)^ky\right\}\\ &{}=e^{ax}(D+a)^{k+1}y.\end{align}$

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 :

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