# Campbell-Hausdorff formula

The

**Campbell-Hausdorff formula** (also called the

**Campbell-Baker-Hausdorff formula**) is the solution to

*z *= ln

*(e*^{x}e^{y}) for non-commuting

*x* and

*y*.

Specifically, let *G* be a simply-connected Lie group with Lie algebra . Let the map exp: be an exponential map, defining,

The general formula is given by:
.

Here ad(*A*) *B* = [*A,B*]
The first few terms are well-known:

.

There is no expression in closed form.

For a matrix Lie algebra the Lie algebra is the tangent space of the identity *I*, and the commutator is simply [*X,Y*] = *XY - YX*; the exponential map is the standard exponential map of matrices,

.

When we solve for *Z* in *e*^{Z} = e^{X} e^{Y}, we obtain a simpler formula:

.

We note first, second, third and fourth order terms are:

## References and external links

" class="external">http://mathworld.wolfram.com/Baker-Campbell-HausdorffSeries.html

- L. Corwin & F.P Greenleaf (1990)
*Representation of nilpotent Lie groups and their applications, Part 1: Basic theory and examples*,