mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the (exponential) shift theorem is a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

about

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

differential operators (''D''-operators) and

exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...

s. 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 ''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 its ...

function ''y'', :

P(D)(e^y)\equiv e^P(D+a)y.

To prove the result, proceed by

induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...

. Note that only the special case :

P(D)=D^n

needs to be proved, since the general result then follows by

linearity Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...

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 (

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 defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b( ...

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 chara ...

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 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. Statem ...

Theorems in analysis

Statement

Related

Examples

Notes

References