# 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

### 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, ${\displaystyle {\frac {1}{n}}}$ is never zero, but as ${\displaystyle n}$ gets bigger, ${\displaystyle {\frac {1}{n}}}$ gets closer and closer to zero. The limit of ${\displaystyle {\frac {1}{n}}}$ as ${\displaystyle n}$ gets bigger is exactly zero. This is described by "The limit of ${\displaystyle {\frac {1}{n}}}$ as ${\displaystyle n}$ goes to infinity is zero", and written as ${\displaystyle \textstyle \lim _{n\to \infty }{\frac {1}{n}}=0}$.

The counterpart would be ${\displaystyle 2\times n}$. When the ${\displaystyle n}$ gets bigger, the limit goes to infinity. It is written as ${\displaystyle \lim _{n\to \infty }2\times n=\infty }$.

The fundamental theorem of algebra can be proven from some basic results in complex analysis. It says that every polynomial ${\displaystyle f(x)}$ with real or complex coefficients has a complex root (where a root is a number ${\displaystyle x}$ satisfying the equation ${\displaystyle f(x)=0}$, and some of these roots may be the same).

### Differential calculus

The function ${\displaystyle f(x)=mx+c}$ is a line. The ${\displaystyle m}$ shows the slope of the function and the ${\displaystyle c}$ shows the position of the function on the ordinate. With two points on the line, it is possible to calculate the slope ${\displaystyle m}$ with:

${\displaystyle m={\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}}$.

A function of the form ${\displaystyle f(x)=x^{2}}$, 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 ${\displaystyle m={\frac {f(x_{1})-f(x_{0})}{x_{1}-x_{0}}}}$.

This is called difference quotient. The ${\displaystyle x_{1}}$ now gets closer to ${\displaystyle x_{0}}$. This can be expressed with the following formula:

${\displaystyle f'(x)=\lim _{x\rightarrow x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}}$.

The result is called derivative or slope of f at the point ${\displaystyle x}$.

### Integration

The integration is about the calculation of areas.

The symbol ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x}$

is read as "the integral of ${\displaystyle f}$ with respect to ${\displaystyle x}$ from ${\displaystyle a}$ to ${\displaystyle b}$",[1] and refers to the area between the x-axis, the graph of function ${\displaystyle f}$, and the lines ${\displaystyle x=a}$ and ${\displaystyle x=b}$. The ${\displaystyle a}$ is the point where the area should start, and the ${\displaystyle b}$ where the area should end.

## References

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". encyclopediaofmath.org. Retrieved 2020-09-19.
4. Weisstein, Eric W. "Analysis". mathworld.wolfram.com. Retrieved 2020-09-19.