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

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

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:

\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<a):

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