Lemmas

This document introduces definitions, lemmas, and tricks that are universally used in mathematics and are difficult to categorize into specific categories.

• 🔒(25/06/19)Professor Choi Byung-sun’s Freely Distributed Textbooks on Mathematics, Statistics, and Economics

Definitions

• Pi $\pi$
• Interval $[a,b]$
• Difference between root and solution
• Fixed point $x_{0} = f \left( x_{0} \right)$
• Meaning of weak and strong in mathematics
• Closed form
• Hamming distance $H$
• Recurrence relation
• Continued fraction
• Pseudo Code

Summaries

• Multiplication formula table
• Nested radical expansion formula $\sqrt{a + b \pm 2\sqrt{a b}} = \sqrt{a} \pm \sqrt{b}$
• Gabi’s Ri

Means

• Geometric mean
• Harmonic mean
• Cumulative average formula 🔵
• Inequality among arithmetic, geometric, and harmonic means

Tricks

• Quickly finding the extremum of a quadratic function
• Quick rationalization of fractions containing roots
• Easily computing powers of two-digit numbers ending in 5
• Moving Big-O notation from denominator to numerator
• Product of indicator functions 🔵

Inequalities

• Cauchy–Schwarz inequality
  • Cauchy–Schwarz inequality for random variables 🔵
• Inequality of zero
• Bernoulli’s inequality
• Finite form of Jensen’s inequality
• Integral form of Jensen’s inequality
• Expectation form of Jensen’s inequality 🔵
• Markov’s inequality 🔵
• Chebyshev’s inequality 🔵
• Grönwall’s inequality
• Unified inequality summarizing various inequalities 🔵
• Inequality for logarithm $1 - \dfrac{1}{x} \le \ln x \le x - 1$

Integration Formulas

• $e^{-x^{2}}$ Gaussian integral, Euler–Poisson integral
  • Generalization of the Gaussian integral
• Indefinite integral of $e^{x^{2}}$
• Table of trigonometric integrals
• Integrability of $\dfrac{1}{x^p}$
• $\displaystyle \int_{-\infty}^{\infty} \dfrac{1}{1+x^{2}}dx = \pi$, $\displaystyle \int \dfrac{1}{1+x^{2}}dx = \tan^{-1} x + C$
• Common definite integral of a quadratic $\int_{\alpha}^{\beta}(x-\alpha)(x-\beta),dx = -\frac{(\beta-\alpha)^{3}}{6}$
• Integration by parts for powers of natural logarithm $\int (\ln x)^{n},dx = x(\ln x)^{n} - \int n(\ln x)^{n-1},dx$

Limits

• $\lim\limits_{x \to 0} \dfrac{\sin x}{x} = 1$
• $\lim\limits_{x \to 0} \dfrac{1 - \cos x}{x} = 0$
• $\lim\limits_{n \to \infty} \sqrt[n]{n} = 1$, $\lim\limits_{n \to \infty} \sqrt[n]{\dfrac{1}{n}} = 1$

Discrete Mathematics

Arithmetic

• Sum of an arithmetic sequence
  • Partial sums of an arithmetic sequence are arithmetic
• Sum of a geometric sequence
  • Partial sums of a geometric sequence are geometric
• Sum of squares $\sum_{k=1}^{n} k^{2} = n(n+1)(2n+1)/6$
• General term of the Fibonacci sequence

Combinatorics

• 🔒(25/08/01)Permutations
  • 🔒(25/08/09)The principle of partition in combinatorics
• Definition of combinations and proof of the binomial theorem
  • 🔒(25/08/13)Derivation of Pascal’s identity $\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}$
  • Sum of binomial coefficients $2^{n} = \sum \binom{n}{k}$
  • 🔒(25/08/05)Subtraction formula for binomial coefficients $\binom{n}{k}(n/k)^{-1} = \binom{n-1}{k-1}$
  • 🔒(25/08/17)Square-sum formula for binomial coefficients $\binom{2n}{n} = \sum \binom{n}{k}^{2}$
• 🔒(25/08/21)Pigeonhole principle
• Birthday Problem: Probability of Sharing the Same Birthday