# Asymptotic analysis

**asymptoticasymptoticallyasymptoticsasymptotic behaviorasymptotic theoryasymptotic formulaasymptotic behaviourasymptotic boundsasymptotic equivalenceasymptotic expansions**

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.wikipedia

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### Prime number theorem

**distribution of primesdistribution of prime numbersprime number theorem for arithmetic progressions**

An example of an important asymptotic result is the prime number theorem.

In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers.

### Prime-counting function

**prime counting functioncounting prime numbersdenoting the number of prime numbers**

denote the prime-counting function (which is not directly related to the constant Pi), i.e.

Of great interest in number theory is the growth rate of the prime-counting function.

### Prime number

**primeprime factorprime numbers**

is the number of prime numbers that are less than or equal to x.

At the start of the 19th century, Legendre and Gauss conjectured that as x tends to infinity, the number of primes up to x is asymptotic to x/\log x, where \log x is the natural logarithm of x. Ideas of Bernhard Riemann in his 1859 paper on the zeta-function sketched an outline for proving this.

### Big O notation

**Obig-O notationlittle-o notation**

The alternative definition, in little-o notation, is that

Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity.

### Stirling's approximation

**Stirling's formulaStirling approximationStirling series**

where the sign ~ means that the two quantities are asymptotic: their ratio tends to 1 as n tends to infinity.

### Tilde

**~ŨĨ**

is the tilde.

It can be used to denote the asymptotic equality of two functions.

### Partition (number theory)

**partitionpartitionsinteger partition**

No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly.

### Asymptotic expansion

**asymptotic seriesasymptotic expansionsasymptotic**

An asymptotic expansion of a function

See asymptotic analysis and big O notation for the notation used in this article.

### Asymptote

**asymptoticasymptoticallyasymptotes**

This is based on the notion of an asymptotic function which cleanly approaches a constant value (the asymptote) as the independent variable goes to infinity; "clean" in this sense meaning that for any desired closeness epsilon there is some value of the independent variable after which the function never differs from the constant by more than epsilon.

The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of asymptotic analysis.

### Bessel function

**Bessel functionsmodified Bessel functionBessel function of the first kind**

The Bessel functions have the following asymptotic forms.

### Limit (mathematics)

**limitlimitsconverge**

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.

### Airy function

**Airy equationAiryAiry differential equation**

For |arg(z)| < π we have the following asymptotic formula for Ai(z):

### Applied mathematics

**applied mathematicianappliedapplications of mathematics**

Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis); and applied probability.

### Asymptotic computational complexity

**asymptotic time complexityasymptoticallyasymptotic**

In computational complexity theory, asymptotic computational complexity is the usage of asymptotic analysis for the estimation of computational complexity of algorithms and computational problems, commonly associated with the usage of the big O notation.

### Asymptotic distribution

**asymptotically normalasymptotic normalitylimiting distribution**

In mathematical statistics, an asymptotic distribution is a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions.

### Analysis of algorithms

**computational complexitycomplexity analysiscomputationally expensive**

Usually asymptotic estimates are used because different implementations of the same algorithm may differ in efficiency.

### Asymptotology

The field of asymptotics is normally first encountered in school geometry with the introduction of the asymptote, a line to which a curve tends at infinity.

### Watson's lemma

In mathematics, Watson's lemma, proved by G. N. Watson (1918, p. 133), has significant application within the theory on the asymptotic behavior of integrals.

### Asymptotic theory (statistics)

**asymptotic theoryasymptoticallyasymptotic**

### Method of matched asymptotic expansions

**a mathematical approachasymptotic methodsmatched asymptotic expansions**

### Method of dominant balance

**dominant balance**

*Asymptotic analysis

### Mathematical analysis

**analysisclassical analysisanalytic**

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.

### Pi

**ππ\pi**

denote the prime-counting function (which is not directly related to the constant Pi), i.e.

### Equivalence relation

**equivalenceequivalentmodulo**

The relation is an equivalence relation on the set of functions of x; the functions f and g are said to be asymptotically equivalent.

### Domain of a function

**domaindomainsdomain of definition**

The domain of f and g can be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers.