# Eikonal Blog

## 2010.01.28

### The Zassenhaus formula

Filed under: mathematics — Tags: , — sandokan65 @ 16:49

$e^{A+B} = e^{A}e^{B}e^{C_2}e^{C_3}\cdots$
where

• $C_2 = -\frac12 [A,B] = -\frac12 a B$,
• $C_3 = -\frac13 [[A,B],B] - \frac16 [[A,B],A] = -\frac13 b^2 A + \frac16 a^2 B$, etc.

Here, as elsewhere on this site, I am using notation $a:\equiv ad(A)$ etc.

Sources:

• T1264 = W. Magnus “On the exponential solution of differential equations for a linear operator”; Communications of Pure and Applied mathematcis, Vol VII, 6490673 (1954).
• T1269 = A.N.Richmond “Expansion for the exponential of a sum of matrices”; Int. Journal of Control (issue and year unknown).