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Mathematical analysis

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. The mathematical analysis is about continuous functions, differential calculus and integration.[1]

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

Parts of mathematical analysisEdit

LimitsEdit

An example for mathematical analysis is limits. 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 n gets bigger   gets close to zero. The limit of   as n gets bigger is zero. It is usually said "The limit of   as n goes to infinity is zero". It is 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. A root is a number x which gives a solution  . Some of these roots may be the same.

Differential calculusEdit

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   gets now closer to  . This can be expressed with the following formula:

 .

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

IntegrationEdit

The integration is about the calculation of areas.

The symbol  

is read as "the integral of f, from a to b" and refers to the area between the x-axis, the graph of function f, and the lines x=a and x=b. The   is the point where the area should start and the   where the area ends.

Related pagesEdit

Some topics in analysis are:

Some useful ideas in analysis are:

ReferencesEdit

  1. Hartmut Seeger. Mathematik. Königswinter: Tandem Verlag. p. 17. ISBN 9 783833 107870.