# 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,^{[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[change | change source]

### Limits[change | change source]

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 *n* gets bigger, gets closer and closer to zero. The limit of as *n* gets bigger is zero. This is described by "The limit of as *n* 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 *x* satisfying the equation , and some of these roots may be the same).

### Differential calculus[change | change source]

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 .

### Integration[change | change source]

The integration is about the calculation of areas.

The symbol

is read as "the integral of *f* with respect to *x* from *a* to *b*"^{[1]}, 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 should ends.

## Related pages[change | change source]

### Topics in analysis[change | change source]

### Concepts in analysis[change | change source]

## References[change | change source]

- ↑
^{1.0}^{1.1}"List of Calculus and Analysis Symbols".*Math Vault*. 2020-05-11. Retrieved 2020-09-19. - ↑ Hartmut Seeger.
*Mathematik*. Königswinter: Tandem Verlag. p. 17. ISBN 9 783833 107870. - ↑ "Mathematical analysis - Encyclopedia of Mathematics".
*encyclopediaofmath.org*. Retrieved 2020-09-19. - ↑ Weisstein, Eric W. "Analysis".
*mathworld.wolfram.com*. Retrieved 2020-09-19.