Exponentiation

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The exponentiation is an operation definable on an algebra over a complete normed field or Banach algebra (complete normed vector space that is also a ring) that generalizes the exponential function of the real numbers.

When a and b are two whole numbers the operation ${displaystyle scriptstyle a^{b}}$ can be defined in elementary algebraic terms as equivalent to potency. However, a number of specific physical problems led to the attempt to generalize the formula prior to values of b Not whole. When b = 1/2 operation equals a square root. Finally the exponentiation tries to generalize the operation ${displaystyle scriptstyle a^{b}}$ at values b Whatever. Usually such operation can be reduced to the calculation of the operation ${displaystyle scriptstyle e^{bln(a)}}$ . This article generalizes this operation to cases where the exponent is not necessarily a real number, but a complex number, a quaternionic number or more generally an element of a Banach space.

Formal definition

Given an element of a Banach algebra, we have defined a commutative operation of addition and another of multiplication, which allows us to define the ring of polynomials on said algebra. In addition, by having a norm, a notion of convergence and therefore of limit can be defined for some formal power series. Under these conditions, the following operation can be defined:

${displaystyle e^{A}=lim _{nto infty }sum _{k=0}^{n}{A^{k} over k!}=1+A+{A^{2} over 2!}+{A^{3} over 3!}+{A^{4} over 4!}+cdots }$

Note that:

If the body on which the algebra is defined does not contain a $R$ the previous limit could not converge, in fact the algebra could not be an algebra of Banach.
If algebra is not a standard vector space there is no way to establish whether the previous limit converges.

Exponentiation of real numbers

Exponentiation of real numbers is done using the exponential function. Given a real number, its exponentiation is always well defined and has the following properties:

${displaystyle e^{a+b}=e^{a}e^{b};}$
${displaystyle e^{ab}=(e^{a})^{b};}$
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Exponentiation of complex numbers

Exponentiation of complex numbers is easily defined by power series just as it is for real numbers. Given a complex number separated into its real and imaginary parts z = a + bi its exponentiation turns out to be:

${displaystyle e^{z}=e^{a+bi}=e^{a}(cos b+isin b);}$

The properties of exponentiation of complex numbers are similar to those of real numbers (although the properties involving order do not extend to complex numbers):

${displaystyle e^{z_{1}+z_{2}}=e^{z_{1}}e^{z_{2}};}$
${displaystyle neg exists z:e^{z}=0}$

Exponentiation of quaternions

The exponentiation of quaternion numbers is computationally more complicated although it is unambiguously defined. Given a quaternion written in canonical form q = a + bi + cj + dk its exponentiation turns out to be:

$e^q = e^{a+bi+cj+dk} =e^a left(cos sqrt{b^2+c^2+d^2} +frac{sin sqrt{b^2+c^2+d^2}}{sqrt{b^2+c^2+d^2}}(bi+cj+dk)right)$

Since the product of quaternions is not commutative, it is not true that the exponentiation of a sum is equal to the product of exponentials of the addends. For example if we consider q₁ = πi and q₂ = πj we have:

${displaystyle {begin{cases}e^{ipi }=-1\e^{jpi }=-1\e^{ipi +jpi }=cos {{sqrt {2}}pi }+{cfrac {sin {{sqrt {2}}pi }}{sqrt {2}}}(i+j)&neq e^{ipi }e^{jpi }=(-1)(-1)=1end{cases}}}$

The exponentiation of a quaternionic number with zero real part, allows to represent in a very convenient way the rotations in three dimensions in the same way that the exponentiation of a pure imaginary number allows to represent rotations in the plane.

Exponentiation of transfinite numbers

Given two finite sets M and Nsuch that the first contains m elements and the second n elements, one can think of how many mathematical functions can be defined between both. In the case of finite assemblies such as the previous ones, the number of functions is precisely ${displaystyle scriptstyle m^{n}}$ . The problem can also arise in the case of inifinite assemblies whose cardinal is a transfinite number. So the operation ${displaystyle scriptstyle alpha ^{beta }}$ where now $scriptstyle alpha$ and $scriptstyle beta$ are two transfinite cardinals defined as:

${displaystyle alpha ^{beta }={mbox{card}}(F_{Ato B}),qquad {begin{cases}alpha ={mbox{card}}(A),&beta ={mbox{card}}(B)\F_{Ato B}={f|f:Ato B}end{cases}}}$

Some concrete examples turn out to be:

${displaystyle 2^{aleph _{0}}=aleph _{1},aleph _{0}^{aleph _{0}}=aleph _{1}, forall bleq aleph _{n}:b^{aleph _{n}}=aleph _{n+1}}$

Matrix exponentiation

Main article: Exponential matrices

Real or complex square matrices can be interpreted as expressions in a given base of a linear application. This fact can be used to more easily compute the exponential of a matrix. Yeah. A represents the matrix of a certain linear application ${displaystyle f:Eto E}$ then the exponentiation of a matrix can be obtained from the canonical form of Jordan J_f of such endomorphism and the matrix change base C between the original base and the base of Jordan:

${displaystyle A=C^{-1}J_{f}Cto e^{A}=e^{C^{-1}J_{f}C}=sum _{k=0}^{infty }{frac {(C^{-1}J_{f}C)^{k}}{k!}}=sum _{k=0}^{infty }{frac {C^{-1}(J_{f})^{k}C}{k!}}=C^{-1}e^{J_{f}}C}$

The exponential of the Jordan canonical form is very simple, given a Jordan block B_J, submatrix nx n, which performs the linear map in one of the invariant subspaces associated with the Jordan map, we have:

${displaystyle B_{J}={begin{bmatrix}lambda &1&0&cdots &0\0&lambda &1&cdots &0\0&0&lambda &cdots &0\vdots &&&ddots &vdots \0&0&0&cdots &lambda end{bmatrix}}Rightarrow qquad e^{B_{J}}={begin{bmatrix}e^{lambda }&{frac {e^{lambda }}{1!}}&{frac {e^{lambda }}{2!}}&cdots &{frac {e^{lambda }}{(n-1)!}}\0&e^{lambda }&{frac {e^{lambda }}{1!}}&cdots &{frac {e^{lambda }}{(n-2)!}}\0&0&e^{lambda }&cdots &{frac {e^{lambda }}{(n-3)!}}\vdots &&&ddots &vdots \0&0&0&cdots &e^{lambda }end{bmatrix}}}$

Matrix exponentiation has these other properties similar to real numbers:

Approval of the rule: ${displaystyle |e^{A}|leq e^{|A|}}$
Matrix identity: ${displaystyle e^{0}=I;}$
Inverse: ${displaystyle (e^{A})^{-1}=e^{-A};}$
Trace-determining ratio: ${displaystyle det(e^{A})=e^{{mbox{tr}}(A)};}$

An important property of matrix exponentiation is that in general, unlike with real numbers, the exponentiation of a sum of matrices is not the product of matrix exponentials:

${displaystyle ABneq BARightarrow e^{A}e^{B}neq e^{A+B}neq e^{B}e^{A}}$

Although when the commutator is canceled the equality is satisfied:

${displaystyle AB=BAiff e^{A}e^{B}=e^{A+B}=e^{B}e^{A}}$

Exponentiation of operators

The exponentiation of linear operators defined on a normed vector space is a generalization of the case of exponentiation of matrices. Since the fact that the vector space is normed implies that the space of operators is a Banach space.

The exponentiation of operators can be used to solve the Schrödinger equation:

${displaystyle {hat {H}}left|psi (t)rightrangle =ihbar {partial over partial t}left|psi (t)rightrangle }$

A formal solution of this equation is obtained by exponentiation of the Hamiltonian operator:

${displaystyle left|psi (t)rightrangle =e^{-{frac {i}{hbar }}int _{0}^{t}{hat {H}} dt}left|psi _{0}rightrangle }$

However, in many cases the computation of the exponentiation of the Hamiltonian operator can be computationally very complex. In addition, since the Hamiltonian is normally an unbounded operator, the exponential could only be defined over a domain of the Hilbert space and then define an extension of the previously obtained operator.

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