Analytic functions of operators (matrices) A are defined via their Taylor expansion about A = 0. Consider the function

g(x) = exp(x A) B exp(−x A)

where x is real and A, B are operators.

• Compute the derivatives d^{n}g(x)/dx^{n}|_{x=0}for integer n.

• Show that

e^{A}Be^{-A}=B+[A,B]+1/2[A,[A,B]]+1/6[A,[A,[A,B]]]+···