Girish Mahajan (Editor)

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 a x y ) e a x 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 a x y ) = e a x ( D + a ) y .

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

D k ( e a x y ) = e a x ( D + a ) k y .

Then,

D k + 1 ( e a x y ) d d x { e a x ( D + a ) k y } = e a x d d x { ( D + a ) k y } + a e a x { ( D + a ) k y } = e a x { ( d d x + a ) ( D + a ) k y } = e a x ( D + a ) k + 1 y .

This completes the proof.

The shift theorem applied equally well to inverse operators:

1 P ( D ) ( e a x y ) = e a x 1 P ( D + a ) y .

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

L ( e a t f ( t ) ) = L ( f ( t a ) ) .

References

Shift theorem Wikipedia