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Here we employ a method called "indirect expansion" to expand the given function. This method uses the known Taylor expansion of the exponential function. In order to expand (1 + x)e x as a Taylor series in x, we use the known Taylor series of function e x:
We begin with the properties that are immediate consequences of the definition as a power series: e 0 = I; exp(X T) = (exp X) T, where X T denotes the transpose of X. exp(X ∗) = (exp X) ∗, where X ∗ denotes the conjugate transpose of X. If Y is invertible then e YXY −1 = Ye X Y −1. The next key result is this one:
For real non-zero values of x, the exponential integral Ei(x) is defined as = =. The Risch algorithm shows that Ei is not an elementary function.The definition above can be used for positive values of x, but the integral has to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero.
Power(x, −n) = Power(x −1, n), Power(x, −n) = (Power(x, n)) −1. The approach also works in non-commutative semigroups and is often used to compute powers of matrices. More generally, the approach works with positive integer exponents in every magma for which the binary operation is power associative.
In this setting, e 0 = 1, and e x is invertible with inverse e −x for any x in B. If xy = yx, then e x + y = e x e y, but this identity can fail for noncommuting x and y. Some alternative definitions lead to the same function. For instance, e x can be defined as (+).
φ is the argument of z, i.e., the angle between the x axis and the vector z measured counterclockwise in radians, which is defined up to addition of 2π. Many texts write φ = tan −1 y / x instead of φ = atan2(y, x), but the first equation needs adjustment when x ≤ 0.
The equation above constitutes the Magnus expansion, or Magnus series, for the solution of matrix linear initial-value problem. The first four terms of this series read The first four terms of this series read
The exponential function is an E-function, in its case c n = 1 for all of the n. If λ is an algebraic number then the Bessel function J λ is an E-function. The sum or product of two E-functions is an E-function. In particular E-functions form a ring. If a is an algebraic number and f(x) is an E-function then f(ax) will be an E-function.