# Analysis

The Analysis [ anaːlyzɪs ] ( Greek ανάλυσις Analysis , German , Resolution ' , ancient Greek ἀναλύειν analýein dissolve') is a branch of mathematics whose basics of Gottfried Wilhelm Leibniz and Isaac Newton as a calculus developed independently. Analysis has existed since Leonhard Euler as an independent sub-area of ​​mathematics alongside the classic sub-areas of geometry and algebra .

The two bodies (the body of real numbers ) and (the body of complex numbers ), along with their geometric, arithmetic , algebraic and topological properties, are fundamental to all analysis . Central concepts of analysis are those of the limit value , the sequence , the series and, in particular, the concept of function . The investigation of real and complex functions with regard to continuity , differentiability and integrability is one of the main subjects of analysis. The methods developed for this are of great importance in all natural and engineering sciences . ${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {C}}$

## Sub-areas of analysis

Gottfried Wilhelm Leibniz
Isaac Newton
Leonhard Euler

Analysis has developed into a very general, not clearly delineated generic term for a variety of areas. In addition to differential and integral calculus, analysis includes other areas that build on it. These include the theory of ordinary and partial differential equations , the calculus of variations , vector analysis , measurement and integration theory and functional analysis .

Function theory also has one of its roots in analysis. The question of which functions the Cauchy-Riemann differential equations fulfill can be understood as a question of the theory of partial differential equations.

Depending on the view, the areas of harmonic analysis , differential geometry with the sub-areas of differential topology and global analysis , analytical number theory , non-standard analysis , distribution theory and micro-local analysis can be included in whole or in part.

### One-dimensional real analysis

#### Differential calculus

For a linear function or a straight line

${\ displaystyle g (x) = mx + c}$

is m is the slope and c of the y-axis intercept or intercept of the straight line. If you only have 2 points and on a straight line, the slope can be calculated by ${\ displaystyle (x_ {0}, y_ {0})}$${\ displaystyle (x_ {1}, y_ {1})}$

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

For non-linear functions such as B. the slope can no longer be calculated as these describe curves and are therefore not straight lines. However, you can place a tangent at a point , which again represents a straight line. The question now is how one can calculate the slope of such a tangent at one point . If you choose a point very close to it and place a straight line through the points and , the slope of this secant is almost the slope of the tangent. The slope of the secant is (see above) ${\ displaystyle f (x) = x ^ {2}}$${\ displaystyle (x_ {0}, f (x_ {0}))}$${\ displaystyle x_ {0}}$${\ displaystyle x_ {1}}$${\ displaystyle x_ {0}}$${\ displaystyle (x_ {0}, f (x_ {0}))}$${\ displaystyle (x_ {1}, f (x_ {1}))}$

${\ displaystyle m = {\ frac {f (x_ {1}) - f (x_ {0})} {x_ {1} -x_ {0}}}.}$

This quotient is called the difference quotient or mean rate of change. If we now come closer and closer to the point , we obtain the gradient of the tangent by means of the difference quotient. We write ${\ displaystyle x_ {1}}$${\ displaystyle x_ {0}}$

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

and call this the derivative or the differential quotient of f in . The expression means that x is increasingly approximated, or that the distance between x and becomes arbitrarily small. We also say: " x goes against ". The name stands for Limes . ${\ displaystyle x_ {0}}$${\ displaystyle \ lim _ {x \ rightarrow x_ {0}}}$${\ displaystyle x_ {0}}$${\ displaystyle x_ {0}}$${\ displaystyle x_ {0}}$${\ displaystyle \ lim}$

${\ displaystyle f ^ {\ prime} (x_ {0})}$is the limit of the difference quotient .

There are also cases where this limit does not exist. That is why the term differentiability was introduced. A function f is called differentiable at the point when the limit value exists. ${\ displaystyle x_ {0}}$${\ displaystyle \ lim _ {x \ rightarrow x_ {0}} {\ frac {f (x) -f (x_ {0})} {x-x_ {0}}}}$

#### Integral calculus

The integral calculus deals clearly with the calculation of areas under function graphs. This area can be approximated by a sum of partial areas and goes over into the integral in the limit value.

${\ displaystyle \ int _ {a} ^ {b} f (x) \, \ mathrm {d} x: = \ lim _ {n \ to \ infty} {\ frac {ba} {n}} \ sum _ {i = 0} ^ {n-1} f \ left (a + i {\ frac {ba} {n}} \ right).}$

The above sequence converges if f satisfies certain conditions (e.g. continuity ). This clear representation (approximation by means of upper and lower sums) corresponds to the so-called Riemann integral that is taught in schools.

In the so-called higher analysis , further integral terms, such as B. considered the Lebesgue integral .

#### Main theorem of integral and differential calculus

According to the main theorem of analysis, differential calculus and integral calculus are "inversely" related to one another in the following way.

If f is a real function continuous on a compact interval , then for : ${\ displaystyle [a, b]}$${\ displaystyle x \ in (a, b)}$

${\ displaystyle {\ mathrm {d} \ over \ mathrm {d} x} \ left (\ int _ {a} ^ {x} f ({\ bar {x}}) \ mathrm {d} {\ bar { x}} \ right) = f (x)}$

and, if f is additionally differentiable to uniformly continuous, ${\ displaystyle (a, b)}$

${\ displaystyle \ int _ {a} ^ {x} \ left ({\ mathrm {d} \ over \ mathrm {d} {\ bar {x}}} f ({\ bar {x}}) \ right) \ mathrm {d} {\ bar {x}} = f (x) -f (a).}$

Therefore, the set of all antiderivatives of a function is also referred to as the indefinite integral and symbolized by. ${\ displaystyle f}$${\ displaystyle \ textstyle \ int f (x) \ mathrm {d} x}$

### Multidimensional real analysis

Example of a multi-dimensional function: ${\ displaystyle f (x, y) = y \ cdot \ sin (x ^ {2})}$

Many textbooks distinguish between analysis in one and analysis in several dimensions. This differentiation does not affect the basic concepts, but there is greater mathematical diversity in several dimensions. Multi-dimensional analysis considers functions of several real variables, which are often represented as a vector or n -tuple. ${\ displaystyle \ textstyle f \ colon D \ subseteq \ mathbb {R} ^ {m} \ to \ mathbb {R} ^ {n}}$

The terms of norm (as a generalization of the amount), convergence , continuity and limit values can easily be generalized from one dimension to several.

The differentiation of functions of several variables differs from one-dimensional differentiation. Important concepts are directional and partial derivatives , which are derivatives in one direction and one variable, respectively. The set of black determines when partial or directional derivatives of different directions may be reversed. In addition, the concept of total differentiation is important. This can be interpreted as the local adaptation of a linear mapping to the course of the multi-dimensional function and is the multi-dimensional analog of the (one-dimensional) derivation. The theorem of the implicit function about the local, unambiguous resolution of implicit equations is an important statement of multidimensional analysis and can be understood as a basis of differential geometry.

In multi-dimensional analysis there are different integral terms such as the curve integral , the surface integral and the space integral . However, from a more abstract point of view from vector analysis, these terms do not differ. To solve these integrals, the transformation theorem as a generalization of the substitution rule and Fubini's theorem , which allows integrals over n -dimensional sets to be converted into iterated integrals, are of particular importance. The integral theorems from the vector analysis of Gauss , Green and Stokes are important in multidimensional analysis. They can be understood as a generalization of the main theorem of integral and differential calculus.

### Functional analysis

Functional analysis is one of the most important areas of analysis. The decisive idea in the development of functional analysis was the development of a coordinate and dimension-free theory. This not only brought a formal gain, but also made it possible to investigate functions on infinite-dimensional topological vector spaces . Here not only the real analysis and the topology are linked, but also methods of algebra play an important role. Key methods for the theory of partial differential equations can be derived from important results of functional analysis, such as the Fréchet-Riesz theorem . In addition, functional analysis, especially with spectral theory , is the appropriate framework for the mathematical formulation of quantum mechanics and theories based on it.

### Theory of differential equations

A differential equation is an equation that contains an unknown function and its derivatives. If only ordinary derivatives appear in the equation, the differential equation is called ordinary. One example is the differential equation

${\ displaystyle y '' (t) + \ omega _ {0} ^ {2} y (t) = 0}$

of the harmonic oscillator . One speaks of a partial differential equation when partial derivatives occur in the differential equation . An example of this class is the Laplace equation

${\ displaystyle \ Delta u (x) = 0}$.

The aim of the theory of differential equations is to find solutions, solution methods and other properties of such equations. A comprehensive theory was developed for ordinary differential equations, with which it is possible to give solutions to given equations, insofar as they exist. Because partial differential equations are more complicated in structure, there is less theory that can be applied to a large class of partial differential equations. Therefore, in the area of ​​partial differential equations, only individual or smaller classes of equations are examined. To find solutions and properties of such equations, methods from functional analysis and also from distribution theory and micro-local analysis are used. However, there are many partial differential equations for which only little information about the structure of the solution could be obtained with the help of these analytical methods. An example of such a complex partial differential equation that is important in physics is the system of Navier-Stokes equations . For these and other partial differential equations one tries to find approximate solutions in numerical mathematics .

### Function theory

In contrast to real analysis, which only deals with functions of real variables, functions of complex variables are examined in function theory (also called complex analysis). Function theory has set itself apart from real analysis with its own methods and different questions. However, some phenomena of real analysis can only be properly understood with the help of function theory. Transferring questions from real analysis to function theory can therefore lead to simplifications.