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Branch Differences and Lambert W

The Lambert W function possesses branches labelled by an index k. The value of W therefore depends upon the value of its argument z and the value of its branch index. Given two branches, labelled n and m, the branch difference is the difference between the two branches, when both are evaluated at the same argument z. It is shown that elementary inverse functions have trivial branch differences, but Lambert W has nontrivial differences. The inverse sine function has real-valued branch differences for real arguments, and the natural logarithm function has purely imaginary branch differences. The Lambert W function, however, has both real-valued differences and complex-valued differences. Applications and representations of the branch differences of W are given.