Infinite products and $T$ products of exponentials

Simple explicit expressions are obtained for the logarithm of a $T$ product and finite and infinite products of exponentials; these generalize the Kampbell–Hausdorff–Baker–Dynkin formula. The proofs do not use series expansions and are based on the general calculus of functions of noncommuting operators. The analog of the Wick formula in an arbitrary Lie algebra is obtained.