Mathematical analysis

2016-01-15T23:27:55Z

76.68.49.128: /* Differential calculus */ typo

'''Mathematical analysis''' is a part of [[mathematics]]. It is often shortened to '''analysis'''. It looks at [[Function (mathematics)|functions]], [[sequence]]s and [[series]]. These have useful properties and characteristics that can be used in [[engineering]]. The mathematical analysis is about [[continuous function]]s, [[differential calculus]] and [[Integral|integration]].<ref>{{Cite book|author=Hartmut Seeger|title=Mathematik|isbn=9 783833 107870|page=17|publisher=Tandem Verlag|location=Königswinter}}</ref>

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

== Parts of mathematical analysis ==
=== Limits ===
An example for mathematical analysis is [[Limit (mathematics)|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, <math>\frac{1}{n}</math> is never zero, but as n gets bigger <math>\frac{1}{n}</math> gets close to zero. The limit of <math>\frac{1}{n}</math> as n gets bigger is zero. It is usually said "The limit of <math>\frac{1}{n}</math> as n goes to [[infinity]] is zero". It is written as <math>\lim_{n\to\infty} \frac{1}{n}=0</math>.

The counterpart would be <math>{2} \times {n}</math>. When the <math>{n}</math> gets bigger, the limit goes to [[infinity]]. It is written as <math>\lim_{n\to\infty} {{2}} \times{n}=\infty</math>.

The [[fundamental theorem of algebra]] can be proven from some basic results in [[complex analysis]]. It says that every polynomial <math>f(x)</math> with [[real]] or [[complex number|complex]] coefficients has a complex root. A root is a number ''x'' which gives a solution <math>f(x)=0</math>. Some of these roots may be the same.

=== Differential calculus ===
{{mainarticle|Differential calculus}}
The function <math>f(x) = {m}{x} + {c}</math> is a [[line (geometry)|line]]. The <math>{m}</math> shows the [[slope]] of the function and the <math>{c}</math> shows the position of the function on the [[ordinate]]. With two points on the line, it is possible to calculate the slope <math>{m}</math> with:

<math>m = \frac{y_1 - y_0}{x_1 - x_0}</math>.

A function of the form <math>f(x) = x^2</math>, which is not linear, cannot be calculated like above. It is only possible to calculate the slope by using [[tangent]]s and [[secant line|secants]]. The secant passes through two points and when the two points get closer, it turns into a tangent.

The new [[formula]] is <math>m = \frac{f(x_1) - f(x_0)}{x_1 - x_0}</math>.

This is called [[difference quotient]]. The <math>x_1</math> gets now closer to <math>x_0</math>. This can be expressed with the following formula:

<math>f'(x) = \lim_{x\rightarrow x_0}\frac{f(x) - f(x_0)}{x - x_0}</math>.

The result is called [[derivative]] or slope of f at the point <math>{x}</math>.

=== Integration ===
{{mainarticle|Integration}}
The integration is about the calculation of [[area]]s.

The symbol <math>\int_{a}^{b} f(x)\, \mathrm{d}x</math>

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 <math>a</math> is the point where the area should start and the <math>b</math> where the area ends.

== Other pages ==
Some topics in analysis are:
* [[Calculus]]
* [[Functional analysis]]
* [[Complex analysis]]

Some useful ideas in analysis are:
* [[Series]]
* [[Sequence]]s
* [[Derivative]]s
* [[Integral]]s

== References ==
{{reflist}}

[[Category:Calculus]]

TCS Wiki - User contributions [en]

Mathematical analysis