Paths, Integrals, Derivatives

A simple subdivision Simple subdivision of an interval [a, b] is a nondecreasing sequence { t p } p = 0 n $$\{t_p\}_{p=0}^n$$ with values in [a, b] such that t 0 = a and t n = b. If [a, b] is in the domain of a complex function x, then x is of bounded variation on [a, b] Complex function of bounded variation if there is a number v such that for every simple subdivision { t p } p = 0 n $$\{t_p\}_{p=0}^n$$ , we have ∑ p = 1 n | x ( t p ) − x ( t p − 1 ) | ≤ v $$\sum _{p=1}^n|x(t_p)-x(t_{p-1})|\leq v$$ . The least upper bound of such v is called the total variation of x on [a, b] Complex function total variation of and denoted by ∫ a b | d x | $$\int _a^b|dx|$$ ∫ a b | d x | $$\int _a^b\vert dx\rvert $$ .