mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the (exponential) shift theorem is a

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

about

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

differential operators In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

(''D''-operators) and exponential functions. It permits one to eliminate, in certain cases, the exponential from under the ''D''-operators.

Statement

The theorem states that, if ''P''(''D'') is a polynomial of the ''D''-operator, then, for any sufficiently

differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...

function ''y'', :

P(D)(e^y)\equiv e^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 In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...

of ''D''-operators. The result is clearly true for ''n'' = 1 since :

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

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

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

Then, :

\begin
D^(e^y)&\equiv\frac\left\\\
&=e^\frac\left\ + ae^\left\\\
&=e^\left\\\
&=e^(D+a)^y.
\end

This completes the proof. The shift theorem can be applied equally well to inverse operators: :

\frac(e^y)=e^\fracy.

There is a similar version of the shift theorem for

Laplace transforms In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the complex-valued fre ...

(

t):

: e^\mathcal\ = \mathcal\.

Examples

The exponential shift theorem can be used to speed the calculation of higher derivatives of functions that is given by the product of an exponential and another function. For instance, if

f(x) = \sin(x) e^x

, one has that

\begin
D^3 f &= D^3 (e^x\sin(x)) = e^x (D+1)^3 \sin (x) \\
&= e^x \left(D^3 + 3D^2 + 3D + 1\right) \sin(x) \\
&= e^x\left(-\cos(x)-3\sin(x)+3\cos(x)+\sin(x)\right)
\end

Another application of the exponential shift theorem is to solve

linear differential equation In mathematics, a linear differential equation is a differential equation that is linear equation, linear in the unknown function and its derivatives, so it can be written in the form a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x) wher ...

s whose

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

has repeated roots.See the article homogeneous equation with constant coefficients for more details.

Notes

References

* {{Cite book, url=https://archive.org/details/ordinarydifferen00tene_0, title=Ordinary differential equations : an elementary textbook for students of mathematics, engineering, and the sciences, last=Morris, first=Tenenbaum, last2=Pollard, first2=Harry, date=1985, publisher=Dover Publications, isbn=0486649407, location=New York, oclc=12188701, url-access=registration Multivariable calculus Shift theorem Theorems in mathematical analysis

Statement

Related

Examples

Notes

References