Mathematical analysis

branch of mathematics

Mathematical analysis is a part of mathematics. It is often shortened to analysis. It looks at functions, sequences and series. These have useful properties and characteristics that can be used in engineering. Mathematical analysis provides a rigorous logical foundation to calculus,[1] which studies continuous functions, differentiation and integration.[2] Mathematical analysis is a short version of its old name "infinitesimal analysis",[3] with some of its key subfields including real analysis, complex analysis, differentiation equation and functional analysis.[4]

Gottfried Wilhelm Leibniz and Isaac Newton developed most of the basis of mathematical analysis.

Parts of mathematical analysis




A foundational concept in mathematical analysis is the concept of limit. Limits are used to see what happens very close to things. Limits can also be used to see what happens when things get very big. For example,   is never zero, but as   gets bigger,   gets closer and closer to zero. The limit of   as   gets bigger is exactly zero. This is described by "The limit of   as   goes to infinity is zero", and written as  .

The counterpart would be  . When the   gets bigger, the limit goes to infinity. It is written as  .

The fundamental theorem of algebra can be proven from some basic results in complex analysis. It says that every polynomial   with real or complex coefficients has a complex root (where a root is a number   satisfying the equation  , and some of these roots may be the same).

Differential calculus


The function   is a line. The   shows the slope of the function and the   shows the position of the function on the ordinate. With two points on the line, it is possible to calculate the slope   with:


A function of the form  , which is not linear, cannot be calculated like above. It is only possible to calculate the slope by using tangents and secants. The secant passes through two points and when the two points get closer, it turns into a tangent.

The new formula is  .

This is called difference quotient. The   now gets closer to  . This can be expressed with the following formula:


The result is called derivative or slope of f at the point  .



The integration is about the calculation of areas.

The symbol  

is read as "the integral of   with respect to   from   to  ",[1] and refers to the area between the x-axis, the graph of function  , and the lines   and  . The   is the point where the area should start, and the   where the area should end.


Topics in analysis


Concepts in analysis



  1. 1.0 1.1 "List of Calculus and Analysis Symbols". Math Vault. 2020-05-11. Retrieved 2020-09-19.
  2. Hartmut Seeger. Mathematik. Königswinter: Tandem Verlag. p. 17. ISBN 9-783833-107870.
  3. "Mathematical analysis - Encyclopedia of Mathematics". Retrieved 2020-09-19.
  4. Weisstein, Eric W. "Analysis". Retrieved 2020-09-19.