 Much of classical function theory depends on the properties of the integral

The holomorphic functional calculus is similar in that the resolvent mapping plays a crucial role in obtaining properties one requires from a nice functional calculus.This subsection outlines properties of the resolvent map that are essential in this context.Direct calculation shows, for "z"1, "z2" ∈ ρ("T"),

Therefore,

This equation is called the first resolvent formula.