Exponential series - Aetherwisp

The exponential series is the series that defines the exponential function $e^{x}$ :

\sum\limits_{n=0}^{\infty} \frac{x^n}{n!}=e^x

It converges only for $x=1$ . The first terms are

e^{x}=1+ x+ \frac{x^{2}}{2}+ \frac{x^{3}}{6}+\ldots

Exponential of a matrix or operator#

This definition can also be used to extend the exponential to the exponential of a matrix or operator. For a matrix or linear operator $A$ , its exponential is

e^{A}=\sum_{n=0}^{\infty} \frac{A^{n}}{n!}

which is computable by recalling that the power of an operator is simply the repeated application of that operator:

A^{2}=AA,\quad A^{n}=\underbrace{ AA\ldots A }_{ n\text{ volte} }

Then the first terms are

e^{A}=1+ A+ \frac{AA}{2}+ \frac{AAA}{6}+\ldots

As with all operators, the exponential of an operator (itself an operator) only makes sense when applied to something.