Mathematical analysis AFrom Wikipedia, the free encyclopedia

Contents

1 Arithmetization of analysis 11.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Research program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Legacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Quotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Asymptote 32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Asymptotes of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Vertical asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Horizontal asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.3 Oblique asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Elementary methods for identifying asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.1 General computation of oblique asymptotes for functions . . . . . . . . . . . . . . . . . . 82.3.2 Asymptotes for rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.3 Transformations of known functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 General definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Curvilinear asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Asymptotes and curve sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 Algebraic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.8 Asymptotic cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Asymptotic expansion 163.1 Formal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Examples of asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Detailed example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

i

ii CONTENTS

4 Constructive analysis 194.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.1 The intermediate value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.2 The least upper bound principle and compact sets . . . . . . . . . . . . . . . . . . . . . . 204.1.3 Uncountability of the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Mathematical analysis 215.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Important concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2.2 Sequences and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.3 Main branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3.1 Real analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3.2 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3.3 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3.4 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3.5 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3.6 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.4 Other topics in mathematical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.5.1 Physical sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.5.2 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.5.3 Other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.10 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.10.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.10.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.10.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Chapter 1

Arithmetization of analysis

The arithmetization of analysiswas a research program in the foundations of mathematics carried out in the secondhalf of the 19th century.

1.1 History

Kronecker originally introduced the term arithmetization of analysis, by which he meant its constructivization in thecontext of the natural numbers (see quotation at bottom of page). The meaning of the term later shifted to signify theset-theoretic construction of the real line. Its main proponent was Weierstrass, who argued the geometric foundationsof calculus were not solid enough for rigorous work.

1.2 Research program

The highlights of this research program are:

• the various (but equivalent) constructions of the real numbers by Dedekind and Cantor resulting in the modernaxiomatic definition of the real number field;

• the epsilon-delta definition of limit; and

• the naïve set-theoretic definition of function.

1.3 Legacy

An important spinoff of the arithmetization of analysis is set theory. Naive set theory was created by Cantor andothers after arithmetization was completed as a way to study the singularities of functions appearing in calculus.The arithmetization of analysis had several important consequences:

• the widely held belief in the banishment of infinitesimals from mathematics until the creation of non-standardanalysis by Abraham Robinson in the 1960s, whereas in reality the work on non-Archimedean systems con-tinued unabated, as documented by P. Ehrlich;

• the shift of the emphasis from geometric to algebraic reasoning: this has had important consequences in theway mathematics is taught today;

• it made possible the development of modern measure theory by Lebesgue and the rudiments of functionalanalysis by Hilbert;

• it motivated the currently prevalent philosophical position that all of mathematics should be derivable fromlogic and set theory, ultimately leading to Hilbert’s program, Gödel's theorems and non-standard analysis.

1

2 CHAPTER 1. ARITHMETIZATION OF ANALYSIS

1.4 Quotations• “God created the natural numbers, all else is the work of man.” -- Kronecker

1.5 References• Torina Dechaune Lewis (2006) The Arithmetization of Analysis: From Eudoxus to Dedekind, Southern Univer-sity.

• Carl B. Boyer, Uta C. Merzbach (2011) A History of Mathematics John Wiley & Sons.

Chapter 2

Asymptote

For other uses, see Asymptote (disambiguation).In analytic geometry, an asymptote (/ˈæsɪmptoʊt/) of a curve is a line such that the distance between the curve and

The graph of a function with a horizontal (y = 0), vertical (x = 0), and oblique asymptote (blue line).

the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross

3

4 CHAPTER 2. ASYMPTOTE

A curve intersecting an asymptote infinitely many times.

the line infinitely often, but this is unusual for modern authors.[1] In some contexts, such as algebraic geometry, anasymptote is defined as a line which is tangent to a curve at infinity.[2][3]

The word asymptote is derived from the Greek ἀσύμπτωτος (asumptōtos) which means “not falling together”, fromἀ priv. + σύν “together” + πτωτ-ός “fallen”.[4] The term was introduced by Apollonius of Perga in his work on conicsections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.[5]

There are potentially three kinds of asymptotes: horizontal, vertical and oblique asymptotes. For curves given by thegraph of a function y = ƒ(x), horizontal asymptotes are horizontal lines that the graph of the function approaches asx tends to +∞ or −∞. Vertical asymptotes are vertical lines near which the function grows without bound.More generally, one curve is a curvilinear asymptote of another (as opposed to a linear asymptote) if the distancebetween the two curves tends to zero as they tend to infinity, although the term asymptote by itself is usually reservedfor linear asymptotes.Asymptotes convey information about the behavior of curves in the large, and determining the asymptotes of a func-tion is an important step in sketching its graph.[6] The study of asymptotes of functions, construed in a broad sense,forms a part of the subject of asymptotic analysis.

2.1 Introduction

The idea that a curve may come arbitrarily close to a line without actually becoming the same may seem to countereveryday experience. The representations of a line and a curve as marks on a piece of paper or as pixels on a computerscreen have a positive width. So if they were to be extended far enough they would seem to merge, at least as far asthe eye could discern. But these are physical representations of the corresponding mathematical entities; the line andthe curve are idealized concepts whose width is 0 (see Line). Therefore the understanding of the idea of an asymptoterequires an effort of reason rather than experience.Consider the graph of the function f(x) = 1

x shown to the right. The coordinates of the points on the curve are

2.2. ASYMPTOTES OF FUNCTIONS 5

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

1 x

f(x) = 1xgraphed on Cartesian coordinates. The x and y-axes are the asymptotes.

of the form(x, 1

x

)where x is a number other than 0. For example, the graph contains the points (1, 1), (2, 0.5),

(5, 0.2), (10, 0.1), ... As the values of x become larger and larger, say 100, 1000, 10,000 ..., putting them far tothe right of the illustration, the corresponding values of y , .01, .001, .0001, ..., become infinitesimal relative to thescale shown. But no matter how large x becomes, its reciprocal 1

x is never 0, so the curve never actually touches thex-axis. Similarly, as the values of x become smaller and smaller, say .01, .001, .0001, ..., making them infinitesimalrelative to the scale shown, the corresponding values of y , 100, 1000, 10,000 ..., become larger and larger. So thecurve extends farther and farther upward as it comes closer and closer to the y-axis. Thus, both the x and y-axes areasymptotes of the curve. These ideas are part of the basis of concept of a limit in mathematics, and this connectionis explained more fully below.[7]

2.2 Asymptotes of functions

The asymptotes most commonly encountered in the study of calculus are of curves of the form y = ƒ(x). These canbe computed using limits and classified into horizontal, vertical and oblique asymptotes depending on its orientation.Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. As thename indicate they are parallel to the x-axis. Vertical asymptotes are vertical lines (perpendicular to the x-axis) nearwhich the function grows without bound. Oblique asymptotes are diagonal lines so that the difference between thecurve and the line approaches 0 as x tends to +∞ or −∞. More general type of asymptotes can be defined in this case.Only open curves that have some infinite branch, can have an asymptote. No closed curve can have an asymptote.

2.2.1 Vertical asymptotes

The line x = a is a vertical asymptote of the graph of the function y = ƒ(x) if at least one of the following statementsis true:

1. limx→a− f(x) = ±∞

6 CHAPTER 2. ASYMPTOTE

2. limx→a+ f(x) = ±∞.

The function ƒ(x) may or may not be defined at a, and its precise value at the point x = a does not affect the asymptote.For example, for the function

f(x) =

{1x if x > 0,

5 if x ≤ 0.

has a limit of +∞ as x→ 0+, ƒ(x) has the vertical asymptote x = 0, even though ƒ(0) = 5. The graph of this functiondoes intersect the vertical asymptote once, at (0,5). It is impossible for the graph of a function to intersect a verticalasymptote (or a vertical line in general) in more than one point. Moreover, if a function is continuous at each pointwhere it is defined, it is impossible that its graph does intersect any vertical asymptote.A common example of a vertical asymptote is the case of a rational function at a point x such that the denominatoris zero and the numerator is non-zero.

2.2.2 Horizontal asymptotes

The graph of a function can have two horizontal asymptotes. An example of such a function would be y = arctan(x).

Horizontal asymptotes are horizontal lines that the graph of the function approaches as x→ ±∞. The horizontal liney = c is a horizontal asymptote of the function y = ƒ(x) if

limx→−∞ f(x) = c or limx→+∞ f(x) = c .

In the first case, ƒ(x) has y = c as asymptote when x tends to −∞, and in the second that ƒ(x) has y = c as an asymptoteas x tends to +∞For example the arctangent function satisfies

limx→−∞ arctan(x) = −π2 and limx→+∞ arctan(x) = π

2 .

So the line y = −π/2 is a horizontal tangent for the arctangent when x tends to −∞, and y = π/2 is a horizontal tangentfor the arctangent when x tends to +∞.Functions may lack horizontal asymptotes on either or both sides, or may have one horizontal asymptote that is thesame in both directions. For example, the function ƒ(x) = 1/(x2+1) has a horizontal asymptote at y = 0 when x tendsboth to −∞ and +∞ because, respectively,

limx→−∞

1

x2 + 1= lim

x→+∞

1

x2 + 1= 0.

2.2. ASYMPTOTES OF FUNCTIONS 7

In the graph of f(x) = x+ 1x, the y-axis (x = 0) and the line y = x are both asymptotes.

2.2.3 Oblique asymptotes

When a linear asymptote is not parallel to the x- or y-axis, it is called an oblique asymptote or slant asymptote. Afunction f(x) is asymptotic to the straight line y = mx + n (m ≠ 0) iflimx→+∞ [f(x)− (mx+ n)] = 0 or limx→−∞ [f(x)− (mx+ n)] = 0.

In the first case the line y = mx + n is an oblique asymptote of ƒ(x) when x tends to +∞, and in the second case theline y = mx + n is an oblique asymptote of ƒ(x) when x tends to −∞An example is ƒ(x) = x + 1/x, which has the oblique asymptote y = x (that is m = 1, n = 0) as seen in the limits

limx→±∞

[f(x)− x]

= limx→±∞

[(x+

1

x

)− x

]

= limx→±∞

1

x= 0.

8 CHAPTER 2. ASYMPTOTE

2.3 Elementary methods for identifying asymptotes

The asymptotes of many elementary functions can be found without the explicit use of limits (although the derivationsof such methods typically use limits).

2.3.1 General computation of oblique asymptotes for functions

The oblique asymptote, for the function f(x), will be given by the equation y=mx+n. The value for m is computedfirst and is given by

mdef= lim

x→af(x)/x

where a is either−∞ or+∞ depending on the case being studied. It is good practice to treat the two cases separately.If this limit doesn't exist then there is no oblique asymptote in that direction.Having m then the value for n can be computed by

ndef= lim

x→a(f(x)−mx)

where a should be the same value used before. If this limit fails to exist then there is no oblique asymptote in thatdirection, even should the limit defining m exist. Otherwise y = mx + n is the oblique asymptote of ƒ(x) as x tends toa.For example, the function ƒ(x) = (2x2 + 3x + 1)/x has

m = limx→+∞

f(x)/x = limx→+∞

2x2 + 3x+ 1

x2= 2

n = limx→+∞

(f(x)−mx) = limx→+∞

(2x2 + 3x+ 1

x− 2x

)= 3

so that y = 2x + 3 is the asymptote of ƒ(x) when x tends to +∞.The function ƒ(x) = ln x has

m = limx→+∞

f(x)/x = limx→+∞

lnxx

= 0

n = limx→+∞

(f(x)−mx) = limx→+∞

lnx

So y = ln x does not have an asymptote when x tends to +∞.

2.3.2 Asymptotes for rational functions

A rational function has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many verticalasymptotes.The degree of the numerator and degree of the denominator determine whether or not there are any horizontalor oblique asymptotes. The cases are tabulated below, where deg(numerator) is the degree of the numerator, anddeg(denominator) is the degree of the denominator.The vertical asymptotes occur only when the denominator is zero (If both the numerator and denominator are zero,the multiplicities of the zero are compared). For example, the following function has vertical asymptotes at x = 0,and x = 1, but not at x = 2.

f(x) =x2 − 5x+ 6

x3 − 3x2 + 2x=

(x− 2)(x− 3)

x(x− 1)(x− 2)

2.3. ELEMENTARY METHODS FOR IDENTIFYING ASYMPTOTES 9

Oblique asymptotes of rational functions

-1

0

1

2

3

4

5

6

7

-1 0 1 2 3 4 5 6 7

y

x

f(x) = x + (x+1) -1

asymptote error

Black: the graph of f(x) = (x2 + x + 1)/(x + 1) . Red: the asymptote y = x . Green: difference between the graph and itsasymptote for x = 1, 2, 3, 4, 5, 6

When the numerator of a rational function has degree exactly one greater than the denominator, the function has anoblique (slant) asymptote. The asymptote is the polynomial term after dividing the numerator and denominator. Thisphenomenon occurs because when dividing the fraction, there will be a linear term, and a remainder. For example,consider the function

f(x) =x2 + x+ 1

x+ 1= x+

1

x+ 1

shown to the right. As the value of x increases, f approaches the asymptote y = x. This is because the other term, y= 1/(x+1) becomes smaller.If the degree of the numerator is more than 1 larger than the degree of the denominator, and the denominator doesnot divide the numerator, there will be a nonzero remainder that goes to zero as x increases, but the quotient will notbe linear, and the function does not have an oblique asymptote.

2.3.3 Transformations of known functions

If a known function has an asymptote (such as y=0 for f(x)=ex), then the translations of it also have an asymptote.

• If x=a is a vertical asymptote of f(x), then x=a+h is a vertical asymptote of f(x-h)

• If y=c is a horizontal asymptote of f(x), then y=c+k is a horizontal asymptote of f(x)+k

10 CHAPTER 2. ASYMPTOTE

If a known function has an asymptote, then the scaling of the function also have an asymptote.

• If y=ax+b is an asymptote of f(x), then y=cax+cb is an asymptote of cf(x)

For example, f(x)=ex−1+2 has horizontal asymptote y=0+2=2, and no vertical or oblique asymptotes.

2.4 General definition

x2 + y2 = x2 y2

asymptote

−3 −2 −1 0 1 2 3

−3

−2

−1

1

2

3

(sec(t), cosec(t)), or x2 + y2 = (xy)2, with 2 horizontal and 2 vertical asymptotes.

Let A : (a,b) → R2 be a parametric plane curve, in coordinates A(t) = (x(t),y(t)). Suppose that the curve tends toinfinity, that is:

limt→b

(x2(t) + y2(t)) = ∞.

A line ℓ is an asymptote of A if the distance from the point A(t) to ℓ tends to zero as t → b.[8]

For example, the upper right branch of the curve y = 1/x can be defined parametrically as x = t, y = 1/t (where t>0).First, x→∞ as t →∞ and the distance from the curve to the x-axis is 1/t which approaches 0 as t →∞. Therefore

2.5. CURVILINEAR ASYMPTOTES 11

the x-axis is an asymptote of the curve. Also, y→∞ as t → 0 from the right, and the distance between the curve andthe y-axis is t which approaches 0 as t → 0. So the y-axis is also an asymptote. A similar argument shows that thelower left branch of the curve also has the same two lines as asymptotes.Although the definition here uses a parameterization of the curve, the notion of asymptote does not depend on theparameterization. In fact, if the equation of the line is ax + by + c = 0 then the distance from the point A(t) =(x(t),y(t)) to the line is given by

|ax(t) + by(t) + c|√a2 + b2

if γ(t) is a change of parameterization then the distance becomes

|ax(γ(t)) + by(γ(t)) + c|√a2 + b2

which tends to zero simultaneously as the previous expression.An important case is when the curve is the graph of a real function (a function of one real variable and returningreal values). The graph of the function y = ƒ(x) is the set of points of the plane with coordinates (x,ƒ(x)). For this, aparameterization is

t 7→ (t, f(t)).

This parameterization is to be considered over the open intervals (a,b), where a can be −∞ and b can be +∞.An asymptote can be either vertical or non-vertical (oblique or horizontal). In the first case its equation is x = c, forsome real number c. The non-vertical case has equation y = mx + n, where m and n are real numbers. All three typesof asymptotes can be present at the same time in specific examples. Unlike asymptotes for curves that are graphs offunctions, a general curve may have more than two non-vertical asymptotes, and may cross its vertical asymptotesmore than once.

2.5 Curvilinear asymptotes

Let A : (a,b) → R2 be a parametric plane curve, in coordinates A(t) = (x(t),y(t)), and B be another (unparameterized)curve. Suppose, as before, that the curve A tends to infinity. The curve B is a curvilinear asymptote of A if theshortest of the distance from the point A(t) to a point on B tends to zero as t → b. Sometimes B is simply referred toas an asymptote of A, when there is no risk of confusion with linear asymptotes.[9]

For example, the function

y =x3 + 2x2 + 3x+ 4

x

has a curvilinear asymptote y = x2 + 2x + 3, which is known as a parabolic asymptote because it is a parabola ratherthan a straight line.[10]

2.6 Asymptotes and curve sketching

Asymptotes are used in procedures of curve sketching. An asymptote serves as a guide line to show the behaviorof the curve towards infinity.[11] In order to get better approximations of the curve, curvilinear asymptotes have alsobeen used [12] although the term asymptotic curve seems to be preferred.[13]

12 CHAPTER 2. ASYMPTOTE

- 4 - 2 2 4

- 20

20

40

x2+2x+3 is a parabolic asymptote to (x3+2x2+3x+4)/ x

2.7 Algebraic curves

The asymptotes of an algebraic curve in the affine plane are the lines that are tangent to the projectivized curve througha point at infinity.[14] For example, one may identify the asymptotes to the unit hyperbola in this manner. Asymptotesare often considered only for real curves,[15] although they also make sense when defined in this way for curves overan arbitrary field.[16]

A plane curve of degree n intersects its asymptote at most at n−2 other points, by Bézout’s theorem, as the intersectionat infinity is of multiplicity at least two. For a conic, there are a pair of lines that do not intersect the conic at anycomplex point: these are the two asymptotes of the conic.A plane algebraic curve is defined by an equation of the form P(x,y) = 0 where P is a polynomial of degree n

P (x, y) = Pn(x, y) + Pn−1(x, y) + · · ·+ P1(x, y) + P0

where Pk is homogeneous of degree k. Vanishing of the linear factors of the highest degree term Pn defines theasymptotes of the curve: setting Q = Pn, if Pn(x, y) = (ax − by) Qn₋₁(x, y), then the line

Q′x(b, a)x+Q′

y(b, a)y + Pn−1(b, a) = 0

is an asymptote if Q′x(b, a) and Q′

y(b, a) are not both zero. If Q′x(b, a) = Q′

y(b, a) = 0 and Pn−1(b, a) ̸= 0, there is no asymptote, but the curve has a branch that looks like a branch of parabola. Such a branch is called aparabolic branch, even when it does not have any parabola that is a curvilinear asymptote. IfQ′

x(b, a) = Q′y(b, a) =

Pn−1(b, a) = 0, the curve has a singular point at infinity which may have several asymptotes or parabolic branches.Over the complex numbers, Pn splits into linear factors, each of which defines an asymptote (or several for multiplefactors). 0ver the reals, Pn splits in factors that are linear or quadratic factors. Only the linear factors correspond toinfinite (real) branches of the curve, but if a linear factor has multiplicity greater than one, the curve may have severalasymptotes or parabolic branches. It may also occur that such a multiple linear factor corresponds to two complexconjugate branches, and does not corresponds to any infinite branch of the real curve. For example, the curve x4 +y2 - 1 = 0 has no real points outside the square |x| ≤ 1, |y| ≤ 1 , but its highest order term gives the linear factor xwith multiplicity 4, leading to the unique asymptote x=0.

2.8. ASYMPTOTIC CONE 13

1

0

−1

−2

x

2

−2 −1 0 1 2

y

A cubic curve, the folium of Descartes (solid) with a single real asymptote (dashed).

2.8 Asymptotic cone

The hyperbola

x2

a2− y2

b2= 1

has the two asymptotes

y = ± b

ax.

The equation for the union of these two lines is

x2

a2− y2

b2= 0.

Similarly, the hyperboloid

14 CHAPTER 2. ASYMPTOTE

Hyperbolas, obtained cutting the same right circular cone with a plane and their asymptotes.

x2

a2− y2

b2− z2

c2= 1

is said to have the asymptotic cone[17][18]

x2

a2− y2

b2− z2

c2= 0.

The distance between the hyperboloid and cone approaches 0 as the distance from the origin approaches infinity.More generally, let us consider a surface that has an implicit equation Pd(x, y, z) + Pd−2(x, y, z) + · · ·P0 = 0,where the Pi are homogeneous polynomials of degree i and Pd−1 = 0 . Then the equation Pd(x, y, z) = 0 defines acone which is centered at the origin. It is called an asymptotic cone, because the distance to the cone of a point ofthe surface tends to zero when the point on the surface tends to infinity.

2.9 See also• Asymptotic analysis

2.10. REFERENCES 15

• Asymptotic curve

• Big O notation

2.10 ReferencesGeneral references

• Kuptsov, L.P. (2001), “Asymptote”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

Specific references

[1] “Asymptotes” by Louis A. Talman

[2] Williamson, Benjamin (1899), “Asymptotes”, An elementary treatise on the differential calculus

[3] Nunemacher, Jeffrey (1999), “Asymptotes, Cubic Curves, and the Projective Plane”, Mathematics Magazine 72 (3): 183–192, doi:10.2307/2690881, JSTOR 2690881

[4] Oxford English Dictionary, second edition, 1989.

[5] D.E. Smith, History of Mathematics, vol 2 Dover (1958) p. 318

[6] Apostol, Tom M. (1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (2nd ed.), NewYork: John Wiley & Sons, ISBN 978-0-471-00005-1, §4.18.

[7] Reference for section: “Asymptote” The Penny Cyclopædia vol. 2, The Society for the Diffusion of Useful Knowledge(1841) Charles Knight and Co., London p. 541

[8] Pogorelov, A. V. (1959), Differential geometry, Translated from the first Russian ed. by L. F. Boron, Groningen: P.Noordhoff N. V., MR 0114163, §8.

[9] Fowler, R. H. (1920), The elementary differential geometry of plane curves, Cambridge, University Press, ISBN 0-486-44277-2, p. 89ff.

[10] William Nicholson, The British enciclopaedia, or dictionary of arts and sciences; comprising an accurate and popular viewof the present improved state of human knowledge, Vol. 5, 1809

[11] Frost, P. An elementary treatise on curve tracing (1918) online

[12] Fowler, R. H. The elementary differential geometry of plane curves Cambridge, University Press, 1920, pp 89ff.(online atarchive.org)

[13] Frost, P. An elementary treatise on curve tracing, 1918, page 5

[14] C.G. Gibson (1998) Elementary Geometry of Algebraic Curves, § 12.6 Asymptotes, Cambridge University Press ISBN0-521-64140-3,

[15] Coolidge, Julian Lowell (1959), A treatise on algebraic plane curves, New York: Dover Publications, ISBN 0-486-49576-0,MR 0120551, pp. 40–44.

[16] Kunz, Ernst (2005), Introduction to plane algebraic curves, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4381-2,MR 2156630, p. 121.

[17] L.P. Siceloff, G. Wentworth, D.E. Smith Analytic geometry (1922) p. 271

[18] P. Frost Solid geometry (1875) This has a more general treatment of asymptotic surfaces.

2.11 External links• Asymptote at PlanetMath.org.

• Hyperboloid and Asymptotic Cone, string surface model, 1872 from the Science Museum

Chapter 3

Asymptotic expansion

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is aformal series of functions which has the property that truncating the series after a finite number of terms provides anapproximation to a given function as the argument of the function tends towards a particular, often infinite, point.The most common type of asymptotic expansion is a power series in either positive or negative powers. Methodsof generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as theLaplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion.Since a convergent Taylor series fits the definition of asymptotic expansion as well, the phrase “asymptotic series”usually implies a non-convergent series. Despite non-convergence, the asymptotic expansion is useful when truncatedto a finite number of terms. Typically, the best approximation is given when the series is truncated at the smallestterm. This way of optimally truncating an asymptotic expansion is known as superasymptotics.[1] The error isthen typically of the form ∼ exp (−c/ϵ) where ε is the expansion parameter. The error is thus beyond all ordersin the expansion parameter. It is possible to improve on the superasymptotic error, e.g. by employing resummationmethods such as Borel resummation to the divergent tail. Such methods are often referred to as hyperasymptoticapproximations.See asymptotic analysis, big O notation, and little o notation for the notation used in this article.

3.1 Formal Definition

First we define an asymptotic scale, and then give the formal definition of an asymptotic expansion.If φn is a sequence of continuous functions on some domain, and if L is a limit point of the domain, then the sequenceconstitutes an asymptotic scale if for every n, φn+1(x) = o(φn(x)) (x → L) . (L may be taken to be infinity.) Inother words, a sequence of functions is an asymptotic scale if each function in the sequence grows strictly slower (inthe limit x → L ) than the preceding function.If f is a continuous function on the domain of the asymptotic scale, then f has an asymptotic expansion of order Nwith respect to the scale as a formal series

∑Nn=0 anφn(x) if

f(x)−N−1∑n=0

anφn(x) = O(φN (x)) (x → L)

or

f(x)−N−1∑n=0

anφn(x) = o(φN−1(x)) (x → L).

If one or the other holds for all N, then we write

16

3.2. EXAMPLES OF ASYMPTOTIC EXPANSIONS 17

f(x) ∼∞∑

n=0

anφn(x) (x → L).

In contrast to a convergent series for f , wherein the series converges for any fixed x in the limit N → ∞ , one canthink of the asymptotic series as converging for fixed N in the limit x → L (with L possibly infinite).

3.2 Examples of asymptotic expansions

• Gamma function

ex

xx√2πx

Γ(x+ 1) ∼ 1 +1

12x+

1

288x2− 139

51840x3− · · · (x → ∞)

• Exponential integral

xexE1(x) ∼∞∑

n=0

(−1)nn!

xn(x → ∞)

• Riemann zeta function

ζ(s) ∼∑N−1

n=1 n−s + N1−s

s−1 +N−s∑∞

m=1B2ms2m−1

(2m)!N2m−1

whereB2m are Bernoulli numbers and s2m−1 is a rising factorial. This expansion is valid forall complex s and is often used to compute the zeta function by using a large enough value ofN, for instance N > |s| .

• Error function

√πxex

2erfc(x) ∼ 1 +∞∑

n=1

(−1)n(2n)!

n!(2x)2n(x → ∞).

3.3 Detailed example

Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking ofvalues outside of its domain of convergence. Thus, for example, one may start with the ordinary series

1

1− w=

∞∑n=0

wn.

The expression on the left is valid on the entire complex plane w ̸= 1 , while the right hand side converges only for|w| < 1 . Multiplying by e−w/t and integrating both sides yields

∫ ∞

0

e−w/t

1− wdw =

∞∑n=0

tn+1

∫ ∞

0

e−uun du,

18 CHAPTER 3. ASYMPTOTIC EXPANSION

after the substitution u = w/t on the right hand side. The integral on the left hand side, understood as a Cauchyprincipal value, can be expressed in terms of the exponential integral. The integral on the right hand side may berecognized as the gamma function. Evaluating both, one obtains the asymptotic expansion

e−1/t Ei(1

t

)=

∞∑n=0

n! tn+1.

Here, the right hand side is clearly not convergent for any non-zero value of t. However, by truncating the series on theright to a finite number of terms, one may obtain a fairly good approximation to the value of Ei(1/t) for sufficientlysmall t. Substituting x = −1/t and noting that Ei(x) = −E1(−x) results in the asymptotic expansion given earlierin this article.

3.4 Notes[1] Boyd, John P. (1999). “The Devil’s Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series”. Acta Appli-

candae Mathematicae 56 (1): 1–98. doi:10.1023/A:1006145903624.

3.5 References• Bleistein, N. and Handelsman, R., Asymptotic Expansions of Integrals, Dover, New York, 1975.

• Copson, E. T., Asymptotic Expansions, Cambridge University Press, 1965.

• A. Erdélyi, Asymptotic Expansions, Dover, New York, 1955.

• Hardy, G. H., Divergent Series, Oxford University Press, 1949.

• Paris, R. B. and Kaminsky, D., Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, 2001.

• Whittaker, E. T. and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press,1963.

3.6 External links• Hazewinkel, Michiel, ed. (2001), “Asymptotic expansion”, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Wolfram Mathworld: Asymptotic Series

Chapter 4

Constructive analysis

In mathematics, constructive analysis is mathematical analysis done according to some principles of constructivemathematics. This contrasts with classical analysis, which (in this context) simply means analysis done according tothe (ordinary) principles of classical mathematics.Generally speaking, constructive analysis can reproduce theorems of classical analysis, but only in application toseparable spaces; also, some theorems may need to be approached by approximations. Furthermore, many classicaltheorems can be stated in ways that are logically equivalent according to classical logic, but not all of these forms willbe valid in constructive analysis, which uses intuitionistic logic.

4.1 Examples

4.1.1 The intermediate value theorem

For a simple example, consider the intermediate value theorem (IVT). In classical analysis, IVT says that, given anycontinuous function f from a closed interval [a,b] to the real line R, if f(a) is negative while f(b) is positive, thenthere exists a real number c in the interval such that f(c) is exactly zero. In constructive analysis, this does not hold,because the constructive interpretation of existential quantification (“there exists”) requires one to be able to constructthe real number c (in the sense that it can be approximated to any desired precision by a rational number). But if fhovers near zero during a stretch along its domain, then this cannot necessarily be done.However, constructive analysis provides several alternative formulations of IVT, all of which are equivalent to theusual form in classical analysis, but not in constructive analysis. For example, under the same conditions on f as inthe classical theorem, given any natural number n (no matter how large), there exists (that is, we can construct) a realnumber cn in the interval such that the absolute value of f(cn) is less than 1/n. That is, we can get as close to zero aswe like, even if we can't construct a c that gives us exactly zero.Alternatively, we can keep the same conclusion as in the classical IVT — a single c such that f(c) is exactly zero —while strengthening the conditions on f. We require that f be locally non-zero, meaning that given any point x in theinterval [a,b] and any natural number m, there exists (we can construct) a real number y in the interval such that |y -x| < 1/m and |f(y)| > 0. In this case, the desired number c can be constructed. This is a complicated condition, butthere are several other conditions which imply it and which are commonly met; for example, every analytic functionis locally non-zero (assuming that it already satisfies f(a) < 0 and f(b) > 0).For another way to view this example, notice that according to classical logic, if the locally non-zero condition fails,then it must fail at some specific point x; and then f(x) will equal 0, so that IVT is valid automatically. Thus inclassical analysis, which uses classical logic, in order to prove the full IVT, it is sufficient to prove the constructiveversion. From this perspective, the full IVT fails in constructive analysis simply because constructive analysis does notaccept classical logic. Conversely, one may argue that the true meaning of IVT, even in classical mathematics, is theconstructive version involving the locally non-zero condition, with the full IVT following by “pure logic” afterwards.Some logicians, while accepting that classical mathematics is correct, still believe that the constructive approach givesa better insight into the true meaning of theorems, in much this way.

19

20 CHAPTER 4. CONSTRUCTIVE ANALYSIS

4.1.2 The least upper bound principle and compact sets

Another difference between classical and constructive analysis is that constructive analysis does not accept the leastupper bound principle, that any subset of the real line R has a least upper bound (or supremum), possibly infinite.However, as with the intermediate value theorem, an alternative version survives; in constructive analysis, any locatedsubset of the real line has a supremum. (Here a subset S of R is located if, whenever x < y are real numbers, eitherthere exists an element s of S such that x < s, or y is an upper bound of S.) Again, this is classically equivalent to thefull least upper bound principle, since every set is located in classical mathematics. And again, while the definitionof located set is complicated, nevertheless it is satisfied by several commonly studied sets, including all intervals andcompact sets.Closely related to this, in constructive mathematics, fewer characterisations of compact spaces are constructivelyvalid—or from another point of view, there are several different concepts which are classically equivalent but notconstructively equivalent. Indeed, if the interval [a,b] were sequentially compact in constructive analysis, then theclassical IVT would follow from the first constructive version in the example; one could find c as a cluster point ofthe infinite sequence (cn)n.

4.1.3 Uncountability of the real numbers

A constructive version of “the famous theorem of Cantor, that the real numbers are uncountable” is: “Let {an} bea sequence of real numbers. Let x0 and y0 be real numbers, x0 < y0. Then there exists a real number x with x0 ≤x ≤ y0 and x ≠ an (n ∈ Z+) . . . The proof is essentially Cantor’s 'diagonal' proof.” (Theorem 1 in Errett Bishop,Foundations of Constructive Analysis, 1967, page 25.) It should be stressed that the constructive component of thediagonal argument already appeared in Cantor’s work.[1] According to Kanamori, a historical misrepresentation hasbeen perpetuated that associates diagonalization with non-constructivity.

4.2 References[1] Akihiro Kanamori, The Mathematical Development of Set Theory from Cantor to Cohen, Bulletin of Symbolic Logic /

Volume 2 / Issue 01 / March 1996, pp 1-71

4.3 See also• Computable analysis

• Indecomposability

4.4 Further reading• Bridger, Mark (2007). Real Analysis: A Constructive Approach. Hoboken: Wiley. ISBN 0-471-79230-6.

Chapter 5

Mathematical analysis

A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis withmany applications to science and engineering.

Mathematical analysis is a branch of mathematics that studies continuous change and includes the theories ofdifferentiation, integration, measure, limits, infinite series, and analytic functions.[1][2]

These theories are usually studied in the context of real and complex numbers and functions. Analysis evolvedfrom calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguishedfrom geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (atopological space) or specific distances between objects (a metric space).

21

22 CHAPTER 5. MATHEMATICAL ANALYSIS

5.1 History

Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more andmore sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.

Mathematical analysis formally developed in the 17th century during the Scientific Revolution,[3] but many of its ideascan be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days ofancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno’s paradox of the dichotomy.[4]Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the conceptsof limits and convergence when they used the method of exhaustion to compute the area and volume of regions andsolids.[5] The explicit use of infinitesimals appears in Archimedes’ The Method of Mechanical Theorems, a workrediscovered in the 20th century.[6] In Asia, the Chinese mathematician Liu Hui used the method of exhaustionin the 3rd century AD to find the area of a circle.[7] Zu Chongzhi established a method that would later be calledCavalieri’s principle to find the volume of a sphere in the 5th century.[8] The Indian mathematician Bhāskara II gaveexamples of the derivative and used what is now known as Rolle’s theorem in the 12th century.[9]

In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and theTaylor series, of functions such as sine, cosine, tangent and arctangent.[10] Alongside his development of the Taylorseries of the trigonometric functions, he also estimated the magnitude of the error terms created by truncating theseseries and gave a rational approximation of an infinite series. His followers at the Kerala school of astronomy andmathematics further expanded his works, up to the 16th century.The modern foundations of mathematical analysis were established in 17th century Europe.[3] Descartes and Fermatindependently developed analytic geometry, and a few decades later Newton and Leibniz independently developedinfinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, intoanalysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, andgenerating functions. During this period, calculus techniques were applied to approximate discrete problems bycontinuous ones.In the 18th century, Euler introduced the notion of mathematical function.[11] Real analysis began to emerge as anindependent subject when Bernard Bolzano introduced the modern definition of continuity in 1816,[12] but Bolzano’swork did not becomewidely known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundationby rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler. Instead,Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity requiredan infinitesimal change in x to correspond to an infinitesimal change in y. He also introduced the concept of theCauchy sequence, and started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studiedpartial differential equations and harmonic analysis. The contributions of these mathematicians and others, suchas Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematicalanalysis.In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century sawthe arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, andintroduced the “epsilon-delta” definition of limit. Then, mathematicians started worrying that they were assuming theexistence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekindcuts, in which irrational numbers are formally defined, which serve to fill the “gaps” between rational numbers, therebycreating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in termsof decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study

5.2. IMPORTANT CONCEPTS 23

of the “size” of the set of discontinuities of real functions.Also, "monsters" (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves)began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is nowcalled naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formal-ized using an axiomatic set theory. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces tosolve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functionalanalysis.

5.2 Important concepts

5.2.1 Metric spaces

Main article: Metric space

In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set isdefined.Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane,Euclidean space, other vector spaces, and the integers. Examples of analysis without a metric include measure theory(which describes size rather than distance) and functional analysis (which studies topological vector spaces that neednot have any sense of distance).Formally, Ametric space is an ordered pair (M,d) whereM is a set and d is a metric onM , i.e., a function

d : M ×M → R

such that for any x, y, z ∈ M , the following holds:

1. d(x, y) = 0 if and only if x = y (identity of indiscernibles),

2. d(x, y) = d(y, x) (symmetry) and

3. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality) .

By taking the third property and letting z = x , it can be shown that d(x, y) ≥ 0 (non-negative).

5.2.2 Sequences and limits

Main article: Sequence

A sequence is an ordered list. Like a set, it contains members (also called elements, or terms). Unlike a set, ordermatters, and exactly the same elements can appear multiple times at different positions in the sequence. Most pre-cisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the naturalnumbers.One of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit.Continuing informally, a (singly-infinite) sequence has a limit if it approaches some point x, called the limit, as nbecomes very large. That is, for an abstract sequence (an) (with n running from 1 to infinity understood) the distancebetween an and x approaches 0 as n→∞, denoted

limn→∞

an = x.

24 CHAPTER 5. MATHEMATICAL ANALYSIS

5.3 Main branches

5.3.1 Real analysis

Main article: Real analysis

Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealingwith the real numbers and real-valued functions of a real variable.[13][14] In particular, it deals with the analyticproperties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculusof the real numbers, and continuity, smoothness and related properties of real-valued functions.

5.3.2 Complex analysis

Main article: Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of math-ematical analysis that investigates functions of complex numbers.[15] It is useful in many branches of mathematics,including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics,thermodynamics, mechanical engineering, electrical engineering, and particularly, quantum field theory.Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally,meromorphic functions). Because the separate real and imaginary parts of any analytic functionmust satisfy Laplace’sequation, complex analysis is widely applicable to two-dimensional problems in physics.

5.3.3 Functional analysis

Main article: Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spacesendowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operatorsacting upon these spaces and respecting these structures in a suitable sense.[16][17] The historical roots of functionalanalysis lie in the study of spaces of functions and the formulation of properties of transformations of functions suchas the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. Thispoint of view turned out to be particularly useful for the study of differential and integral equations.

5.3.4 Differential equations

Main article: Differential equations

A differential equation is a mathematical equation for an unknown function of one or several variables that relatesthe values of the function itself and its derivatives of various orders.[18][19][20] Differential equations play a prominentrole in engineering, physics, economics, biology, and other disciplines.Differential equations arise in many areas of science and technology, specifically whenever a deterministic relationinvolving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time(expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of abody is described by its position and velocity as the time value varies. Newton’s laws allow one (given the position,velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differentialequation for the unknown position of the body as a function of time. In some cases, this differential equation (calledan equation of motion) may be solved explicitly.

5.4. OTHER TOPICS IN MATHEMATICAL ANALYSIS 25

5.3.5 Measure theory

Main article: Measure (mathematics)

A measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpretedas its size.[21] In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularlyimportant example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, andvolume of Euclidean geometry to suitable subsets of the n -dimensional Euclidean space Rn . For instance, theLebesguemeasure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically,1.Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a setX . Itmust assign 0 to the empty set and be (countably) additive: the measure of a 'large' subset that can be decomposedinto a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the “smaller” subsets.In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms ofa measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measureonly on a sub-collection of all subsets; the so-calledmeasurable subsets, which are required to form a σ -algebra. Thismeans that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarilycomplicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivialconsequence of the axiom of choice.

5.3.6 Numerical analysis

Main article: Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolicmanipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).[22]

Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain inpractice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintainingreasonable bounds on errors.Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21stcentury, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differentialequations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for dataanalysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine andbiology.

5.4 Other topics in mathematical analysis

• Calculus of variations deals with extremizing functionals, as opposed to ordinary calculus which deals withfunctions.

• Harmonic analysis deals with Fourier series and their abstractions.

• Geometric analysis involves the use of geometrical methods in the study of partial differential equations andthe application of the theory of partial differential equations to geometry.

• Clifford analysis, the study of Clifford valued functions that are annihilated by Dirac or Dirac-like operators,termed in general as monogenic or Clifford analytic functions.

• p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interestingand surprising ways from its real and complex counterparts.

• Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treat-ment of infinitesimals and infinitely large numbers.

• Computable analysis, the study of which parts of analysis can be carried out in a computable manner.

26 CHAPTER 5. MATHEMATICAL ANALYSIS

• Stochastic calculus – analytical notions developed for stochastic processes.

• Set-valued analysis – applies ideas from analysis and topology to set-valued functions.

• Convex analysis, the study of convex sets and functions.

• Tropical analysis (or idempotent analysis) – analysis in the context of the semiring of the max-plus algebrawhere the lack of an additive inverse is compensated somewhat by the idempotent rule A + A = A. Whentransferred to the tropical setting, many nonlinear problems become linear.[23]

5.5 Applications

Techniques from analysis are also found in other areas such as:

5.5.1 Physical sciences

The vastmajority of classicalmechanics, relativity, and quantummechanics is based on applied analysis, and differentialequations in particular. Examples of important differential equations include Newton’s second law, the Schrödingerequation, and the Einstein field equations.Functional analysis is also a major factor in quantum mechanics.

5.5.2 Signal processing

When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysiscan isolate individual components of a compound waveform, concentrating them for easier detection and/or removal.A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.[24]

5.5.3 Other areas of mathematics

Techniques from analysis are used in many areas of mathematics, including:

• Analytic number theory

• Analytic combinatorics

• Continuous probability

• Differential entropy in information theory

• Differential games

• Differential geometry, the application of calculus to specific mathematical spaces known as manifolds thatpossess a complicated internal structure but behave in a simple manner locally.

• Differential topology

• Mathematical finance

5.6 See also

• Constructive analysis

• History of calculus

• Non-classical analysis

5.7. NOTES 27

• Paraconsistent mathematics

• Smooth infinitesimal analysis

• Timeline of calculus and mathematical analysis

5.7 Notes[1] Edwin Hewitt and Karl Stromberg, “Real and Abstract Analysis”, Springer-Verlag, 1965

[2] Stillwell, John Colin. “analysis | mathematics”. Encyclopedia Britannica. Retrieved 2015-07-31.

[3] Jahnke, Hans Niels (2003). A History of Analysis. American Mathematical Society. p. 7. ISBN 978-0-8218-2623-2.

[4] Stillwell (2004). “Infinite Series”. Mathematics and its History (2nd ed.). Springer Science + Business Media Inc. p. 170.ISBN 0-387-95336-1. Infinite series were present in Greek mathematics, [...] There is no question that Zeno’s paradox ofthe dichotomy (Section 4.1), for example, concerns the decomposition of the number 1 into the infinite series 1 ⁄2 + 1 ⁄22+ 1 ⁄23 + 1 ⁄24 + ... and that Archimedes found the area of the parabolic segment (Section 4.4) essentially by summing theinfinite series 1 + 1 ⁄4 + 1 ⁄42 + 1 ⁄43 + ... = 4 ⁄3. Both these examples are special cases of the result we express as summationof a geometric series

[5] (Smith, 1958)

[6] Pinto, J. Sousa (2004). Infinitesimal Methods of Mathematical Analysis. Horwood Publishing. p. 8. ISBN 978-1-898563-99-0.

[7] Dun, Liu; Fan, Dainian; Cohen, Robert Sonné (1966). “A comparison of Archimedes’ and Liu Hui’s studies of circles”.Chinese studies in the history and philosophy of science and technology 130. Springer. p. 279. ISBN 0-7923-3463-9.,Chapter , p. 279

[8] Zill, Dennis G.; Wright, Scott; Wright, Warren S. (2009). Calculus: Early Transcendentals (3 ed.). Jones & BartlettLearning. p. xxvii. ISBN 0-7637-5995-3., Extract of page 27

[9] Seal, Sir Brajendranath (1915), The positive sciences of the ancient Hindus, Longmans, Green and co.

[10] C. T. Rajagopal and M. S. Rangachari (June 1978). “On an untapped source of medieval Keralese Mathematics”. Archivefor History of Exact Sciences 18 (2): 89–102. doi:10.1007/BF00348142.

[11] Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America. p. 17.

[12] • Cooke, Roger (1997). “Beyond the Calculus”. The History of Mathematics: A Brief Course. Wiley-Interscience. p.379. ISBN 0-471-18082-3. Real analysis began its growth as an independent subject with the introduction of themodern definition of continuity in 1816 by the Czech mathematician Bernard Bolzano (1781–1848)

[13] Rudin, Walter. Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.).McGraw–Hill. ISBN 978-0-07-054235-8.

[14] Abbott, Stephen (2001). Understanding Analysis. Undergraduate Texts in Mathematics. New York: Springer-Verlag.ISBN 0-387-95060-5.

[15] Ahlfors.,Complex Analysis (McGraw-Hill)

[16] Rudin, W.: Functional Analysis, McGraw-Hill Science, 1991

[17] Conway, J. B.: A Course in Functional Analysis, 2nd edition, Springer-Verlag, 1994, ISBN 0-387-97245-5

[18] E. L. Ince, Ordinary Differential Equations, Dover Publications, 1958, ISBN 0-486-60349-0

[19] Witold Hurewicz, Lectures on Ordinary Differential Equations, Dover Publications, ISBN 0-486-49510-8

[20] Evans, L. C. (1998), Partial Differential Equations, Providence: American Mathematical Society, ISBN 0-8218-0772-2

[21] Terence Tao, 2011. An Introduction to Measure Theory. American Mathematical Society.

[22] Hildebrand, F. B. (1974). Introduction to Numerical Analysis (2nd ed.). McGraw-Hill. ISBN 0-07-028761-9.

[23] THE MASLOV DEQUANTIZATION, IDEMPOTENT AND TROPICAL MATHEMATICS: A BRIEF INTRODUC-TION

[24] Theory and application of digital signal processing Rabiner, L. R.; Gold, B. Englewood Cliffs, N.J., Prentice-Hall, Inc.,1975.

28 CHAPTER 5. MATHEMATICAL ANALYSIS

5.8 References• Aleksandrov, A. D., Kolmogorov, A. N., Lavrent'ev, M. A. (eds.). 1984. Mathematics, its Content, Methods,and Meaning. 2nd ed. Translated by S. H. Gould, K. A. Hirsch and T. Bartha; translation edited by S. H.Gould. MIT Press; published in cooperation with the American Mathematical Society.

• Apostol, Tom M. 1974. Mathematical Analysis. 2nd ed. Addison–Wesley. ISBN 978-0-201-00288-1.

• Binmore, K.G. 1980–1981. The foundations of analysis: a straightforward introduction. 2 volumes. Cam-bridge University Press.

• Johnsonbaugh, Richard, & W. E. Pfaffenberger. 1981. Foundations of mathematical analysis. New York: M.Dekker.

• Nikol’skii, S. M. 2002. “Mathematical analysis”. In Encyclopaedia of Mathematics, Michiel Hazewinkel (edi-tor). Springer-Verlag. ISBN 1-4020-0609-8.

• Rombaldi, Jean-Étienne. 2004. Éléments d'analyse réelle : CAPES et agrégation interne de mathématiques.EDP Sciences. ISBN 2-86883-681-X.

• Rudin, Walter. 1976. Principles of Mathematical Analysis. McGraw–Hill Publishing Co.; 3rd revised edition(September 1, 1976), ISBN 978-0-07-085613-4.

• Smith, David E. 1958. History of Mathematics. Dover Publications. ISBN 0-486-20430-8.

• Whittaker, E. T. and Watson, G. N.. 1927. A Course of Modern Analysis. 4th edition. Cambridge UniversityPress. ISBN 0-521-58807-3.

• Real Analysis - Course Notes

5.9 External links• Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis

• Basic Analysis: Introduction to Real Analysis by Jiri Lebl (Creative Commons BY-NC-SA)

• Mathematical Analysis-Encyclopedia Britannica

• Calculus and Analysis

5.10. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 29

5.10 Text and image sources, contributors, and licenses

5.10.1 Text• Arithmetization of analysis Source: https://en.wikipedia.org/wiki/Arithmetization_of_analysis?oldid=641605102Contributors: Miguel~enwiki,

Chinju, Ahoerstemeier, WhisperToMe, Robbot, MathMartin, Wile E. Heresiarch, Chameleon, Oleg Alexandrov, BradBeattie, Trovatore,SmackBot, Bluebot, Colonies Chris, Akriasas, Harryboyles, Mets501, Gregbard, Thijs!bot, Gentlemath, Addbot, Erik9bot, FrescoBot,Tkuvho, ZéroBot, Bomazi, Wcherowi, Brirush and Anonymous: 1

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• Asymptotic expansion Source: https://en.wikipedia.org/wiki/Asymptotic_expansion?oldid=668065694 Contributors: Michael Hardy,Charles Matthews, Zero0000, Hadal, Giftlite, Gene Ward Smith, Alberto da Calvairate~enwiki, EmilJ, Ddlamb, Count Iblis, OlegAlexandrov, Linas, Reedbeta, Mathbot, Scythe33, Bgwhite, YurikBot, KSmrq, Jess Riedel, Nakon, Alex Selby, WISo, HappyInGeneral,Mintz l, Policron, Fylwind, Adam Zivner, TXiKiBoT, Exp(-Pi), DrTLesterThomas, JohnWStockwell, Qwfp, Crowsnest, Addbot, Yobot,Ht686rg90, Unara, Sznagy, Ptrf, Pinethicket, DixonDBot, ZéroBot, ChuispastonBot, Vinícius Machado Vogt, BG19bot, Bilingsley, Kas-parBot and Anonymous: 12

• Constructive analysis Source: https://en.wikipedia.org/wiki/Constructive_analysis?oldid=663217929Contributors: TobyBartels, B4hand,Timwi, Wik, Hyacinth, Cleduc, Gdr, Bender235, Bookandcoffee, Oleg Alexandrov, Marudubshinki, Jshadias, Quuxplusone, Hairy Dude,SmackBot, CRGreathouse, CBM, Gregbard, Cydebot, JohnBlackburne, Classicalecon, Mbridger, Addbot, Unzerlegbarkeit, Ptbotgourou,Obscuranym, Unara, Erik9bot, Paolo Lipparini, Brad7777, KasparBot and Anonymous: 4

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30 CHAPTER 5. MATHEMATICAL ANALYSIS

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