Exponential function
Matrices and Banach algebras

The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any unital Banach algebra B. In this setting, e = 1, and e is invertible with inverse e for any x in B. If xy = yx, then e = e e, but this identity can fail for noncommuting x and y.

Some alternative definitions lead to the same function. For instance, e can be defined as lim n → ∞ (1 + x n) n. {\displaystyle \lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.}

Or e can be defined as f (1), where f : R → B is the solution to the differential equation df / dt (t) = x f (t), with initial condition f (0) = 1; it follows that f (t) = e for every t in R.
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