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. Mathematical analysis provides a rigorous logical foundation to calculus, which studies continuous functions, differentiation and integration. 
Overview
Mathematical analysis is a short version of its old name "infinitesimal analysis", with some of its key subfields including real analysis, complex analysis, differentiation equation and functional analysis.
Origin
Gottfried Wilhelm Leibniz and Isaac Newton developed most of the basis of mathematical analysis.

Topics
Limits
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).
Differentiation

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 .
Integration

The integration is about the calculation of areas.

The symbol 

is read as "the integral of  with respect to  from  to ", 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.

Related pages
Concepts
 Series
 Integral
 Algebra
 Sequence
 Derivative
Topics
 Calculus
 Complex analysis
 Numerical analysis
 Functional analysis
 Multivariable calculus

 References 

Category:Mathematical analysis