Asymptotic analysis

In mathematical analysis, asymptotic analysis, also known as asymptotics, is the development and application of methods that generate an approximate analytical solution to a mathematical problem when a variable or parameter assumes a value that is large, small or near a specified value.

An example of asymptotic analysis is function approximation. For example, the function ${\textstyle {\widetilde {y}}(x)=x}$ accurately approximates the function ${\textstyle y(x)=x+e^{-x}}$ for large positive ${\textstyle x}$ values (Figure 1). For any desired accuracy, there is a corresponding range of ${\textstyle x}$ values where this accuracy occurs. In this case, a chosen accuracy with a relative error of less than 1% occurs when the ${\textstyle x}$ values are greater than 3.4.

The prime number theorem is an example of an important asymptotic result. For any real number ${\textstyle x}$, the prime counting function, denoted as ${\textstyle \operatorname {\pi } (x)}$, is the number of prime numbers less than or equal to ${\textstyle x}$. The function, ${\textstyle x/\ln(x)}$, approximates the prime counting function for large numbers ${\textstyle x}$. As in the preceding example, as the number ${\textstyle x}$ becomes increasingly larger, the approximating function becomes increasingly more accurate, leading to an increasingly smaller relative error. This asymptotic relationship is expressed using this notation: $${\displaystyle \pi (x)\sim {\frac {x}{\ln(x)}}\quad (x\to \infty )}$$

Asymptotic analysis impacts on many areas of mathematics. Mathematicians use asymptotic analysis for computing function approximations, implicit functions, integrals, iterated functions, series summation, partial sums, solutions of difference equations. solutions of differential equations and properties of computer algorithms.