Relation to complex analysis
Riemann integration [ edit ] Main article: Riemann integral The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. Let [ � , � ] be a closed interval of the real line; then a tagged partition � of [ � , � ] is a finite sequence � = � 0 ≤ � 1 ≤ � 1 ≤ � 2 ≤ � 2 ≤ ⋯ ≤ � � − 1 ≤ � � ≤ � � = � . This partitions the interval [ � , � ] into � sub-intervals [ � � − 1 , � � ] indexed by � = 1 , … , � , each of which is "tagged" with a distinguished point � � ∈ [ � � − 1 , � � ] . For a function � bounded on [ � , � ] , we define the Riemann sum of � with respect to tagged partition � as ∑ � = 1 � � ( � � ) Δ � , where Δ � = � � − � � − 1 is the width of sub-interval � . Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. The mesh of such a tagge