Asymptotic analysis is just a scheme by Big Oh to sell more functions

#math #CS #ComputerScience #AsymptoticBehaviour

I love big-O notation when doing theory to focus on big ideas,

but constants can matter in practice, and often do!

(Leading constants moreso. To focus on leading constant but still ignore lower order terms you can use asymptotic equality ~)

(Inspired by @ccanonne on the birdiste.)

#complexity #AsymptoticBehaviour #algorithms

GAUTSCHI'S INEQUALITY:

\(\bullet\) Let \(x\in\mathbb{R}^+\) and \(s\in(0,1)\). Then, an inequality for ratios of gamma functions known as Gautschi's inequality:
\[x^{1-s}<\dfrac{\Gamma(x+1)}{\Gamma(x+s)}<(x+1)^{1-s}\]
\(\bullet\) Asymptotic behaviour of the ratios of gamma functions:
\[\displaystyle\lim_{x\rightarrow\infty}\dfrac{\Gamma(x+1)}{\Gamma(x+s)x^{1-s}}=1\]
#GammaFunction #GautaschiInequality #Inequality #RealAnalysis #AsymptoticBehaviour #mathematics #analysis #lowerbound #upperbound