Asymptotics and Integral Transforms

Example of asymptotics

• f(x) = 1/x sinx - (sin ε)/ε ∼ 1 − ε²/6 + ε⁴/120

• Stirling’s formula: n! ∼ √(2πn)nⁿe⁻ⁿ

• Prime Number Theorem. If π(n) is the number of primes less than or equal to n, then π(n) ∼ n/ log n for large n

• Much of applied maths research

Asymptotics

____ is the study of mathematical objects (e.g. roots of equations, solutions to PDEs, distribution of primes, etc) as a parameter ε gets small. Or, as a parameter gets large: if X → ∞ then ε = 1/X → 0.

Asymptotics can

• Give often accurate solutions with very little computational effort.

• Often make sense of why things happen, what is important, and the mechanism behind them.

• Give analytic approximate solutions where no exact analytic solution exists

• Works when numerical solutions don’t.

Big O Notation

For functions f(x) and g(x), we say that f(x) = O(g(x)) as x → a

⇔ |f(x)/g(x)| is bounded as x → a

⇔ ∃ M, δ s.t. |f(x)/g(x)| < M ∀ |x − a| < δ

⇔ lim sup_x→a |f(x)/g(x)| < ∞