Branch: Analysis
Analysis is a broad area of mathematics studying the special properties of realvalued or complexvalued functions under the basic ideas of calculus like limits, continuity, differentiation, integration, or holomorphy. The key common feature of calculus is to combine infinitely many infinitely small (i.e. infinitesimal) quantities to get a finite answer. As an example, suppose we want to calculate the area of a circle. Doing it using analytical techniques, we might divide the circle into segments and approximate the area of each segment with the area of a triangle. As the number of segments gets higher and higher, the sum of the areas of all triangles will approximate the area of the circle. The following figures demonstrate the idea of creating infinitely many vanishingly small triangles to get the (finite) total area of a circle.
Theoretical minimum (in a nutshell)
You should be acquainted with set theory, especially the set operations and basics about functions.
Concepts you will learn in this part of BookofProofs
 real analysis of one variable, addressing basic analytical concepts for the calculus of one real variable.
 real analysis of multiple variables, addressing basic analytical concepts for the calculus of more than one real variable.
 complex analysis, dealing with complex numbers and also known as the _theory of functions
 differential equations, dealing with equations containing an unknown function as a derivative and providing a theory capable to describe many physical phenomena,
 linear integral equations, dealing with equations containing the unknown function under the integral
 functional analysis, discussing modern theories of differentiation and integration and the principal problems and methods of handling integral and linear functionals
 vector analysis, concerned with differentiation and integration of vector fields
 calculus of variations, which is concerned with the problem of "extremising" functionals.
Table of Contents
 Part: Historical Development of Analysis
 Part: Real Analysis of One Variable and Elements of Complex Analysis
 Part: Real Analysis of Multiple Variables
 Part: Complex Analysis
 Part: Differential Equations
 Part: Linear Integral Equations
 Part: Functional Analysis
 Part: Calculus of Variations
 Part: Vector Analysis
Mentioned in:
Definitions: 1
Parts: 2 3 4 5
Thank you to the contributors under CC BYSA 4.0!
 Github:

References
Bibliography
 Govers, Timothy: "The Princeton Companion to Mathematics", Princeton University Press, 2008,