Branch: Analysis

Analysis is a broad area of mathematics studying the special properties of real-valued or complex-valued 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.

sectors1 sectors2 sectors3 sectors4

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

  1. Part: Historical Development of Analysis
  2. Part: Real Analysis of One Variable and Elements of Complex Analysis
  3. Part: Real Analysis of Multiple Variables
  4. Part: Complex Analysis
  5. Part: Differential Equations
  6. Part: Linear Integral Equations
  7. Part: Functional Analysis
  8. Part: Calculus of Variations
  9. Part: Vector Analysis

Definitions: 1
Parts: 2 3 4 5

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  1. Govers, Timothy: "The Princeton Companion to Mathematics", Princeton University Press, 2008,