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",

:$P(D)(e^\{ax\}y)equiv\; e^\{ax\}P(D+a)y.,$

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,

:

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 (

:$e^\{ax\}scriptstylemathcal\{L\}(f(t))=scriptstylemathcal\{L\}(f(t-a)).,$