Chebyshev nodes

30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí. In mathematics, an asymptotic formula for a quantity (function or expression) depending on natural numbers, or on a variable taking real numbers as values, is a function of natural numbers, or of a real variable, whose values are nearly equal to the values of the former when both are evaluated for the same large values of the variable. An asymptotic formula for a quantity is a function which is asymptotically equivalent to the former.

More generally, an asymptotic formula is "a statement of equality between two functions which is not a true equality but which means the ratio of the two functions approaches 1 as the variable approaches some value, usually infinity".

Definition

Let P(n) be a quantity or function depending on n which is a natural number. A function F(n) of n is an asymptotic formula for P(n) if P(n) is asymptotically equivalent toF(n), that is, if

$\lim _{n\rightarrow \infty }{\frac {P(n)}{F(n)}}=1.$

This is symbolically denoted by

$P(n)\sim F(n)\,$

Examples

Prime number theorem

For a real number x, let π (x) denote the number of prime numbers less than or equal to x. The classical prime number theorem gives an asymptotic formula for π (x):

$\pi (x)\sim {\frac {x}{\log(x)}}.$

Stirling's formula

Stirling's approximation approaches the factorial function as n increases.

Stirling's approximation is a well-known asymptotic formula for the factorial function:

$n!=1\times 2\times \ldots \times n$.

The asymptotic formula is

$n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}.$

Asymptotic formula for the partition function

For a positive integer n, the partition function P(n), sometimes also denoted p(n), gives the number of ways of writing the integer n as a sum of positive integers, where the order of addends is not considered significant. Thus, for example, P(4) = 5. G.H. Hardy and Srinivasa Ramanujan in 1918 obtained the following asymptotic formula for P(n):

$P(n)\sim {\frac {1}{4n{\sqrt {3}}}}e^{\pi {\sqrt {2n/3}}}.$

Asymptotic formula for Airy function

The Airy function Ai(x) which is a solution of the differential equation

$y''-xy=0\,$

and which has many applications in physics, has the following asymptotic formula:

$\mathrm {Ai} (x)\sim {\frac {e^{-{\frac {2}{3}}x^{3/2}}}{2{\sqrt {\pi }}x^{1/4}}}.$

See also

• Asymptotic analysis

References

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