Lemmas
This document introduces definitions, lemmas, and tricks that are universally used in mathematics and are difficult to categorize into specific categories.
Mark | Subcategory |
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🔵 | Mathematical Statistics |
Definitions
- Pi $\pi$
- Interval $[a,b]$
- 🔒(25/09/10)Hypercube
- Fixed point $x_{0} = f \left( x_{0} \right)$
- Hamming distance $H$
- Recurrence relation
- Continued fraction
- Pseudo Code
- 🔒(25/11/20)Formal Sum
용어
- Difference between root and solution
- Meaning of weak and strong in mathematics
- Closed form
- 🔒(25/10/04)Maintaining Generality in Mathematics
- 🔒(25/11/16)The Expression of Trivial
- 🔒(25/11/18)The Expression Well-Defined
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
- 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
- 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
- Sum of a geometric sequence
- Sum of squares $\sum_{k=1}^{n} k^{2} = n(n+1)(2n+1)/6$
- General term of the Fibonacci sequence
Combinatorics
- Permutations
- Definition of combinations and proof of the binomial theorem
- Derivation of Pascal’s identity $\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}$
- Sum of binomial coefficients $2^{n} = \sum \binom{n}{k}$
- 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
All posts
- Arithmetic, Geometric, and Harmonic Means Inequality
- Frequently Used Definite Integrals of Quadratic Functions
- Gabi's Proof of Li
- The Number of Subsets of a Finite Set with n Elements
- Finding the Extremum of a Quadratic Function Quickly
- Integration of the Natural Logarithm Raised to a Power
- Cauchy-Schwarz Inequality Proof
- Multiplication Formula Table
- Finding the Sum of a Geometric Sequence
- Finding the Sum of an Arithmetic Sequence
- Finding the Sum of Squares
- Proof that the Partial Sums of a Geometric Sequence are Also Geometric
- Proof that the Partial Sums of an Arithmetic Sequence Also Form an Arithmetic Sequence
- Binomial Theorem Proof
- Definite Integration of the form e^-x^2, Gaussian Integral, Euler-Poisson Integral
- Proof of the Integral Form of Jensen's Inequality
- Indefinite Integral of the Form e^{x^2}
- Simplifying the Exponentiation of Two-digit Numbers Ending in 5
- Derivation of the General Term of the Fibonacci Sequence
- Proof of the Finite Form of Jensen's Inequality
- Proof of Young's Inequality
- How to Move the Big O Notation from Denominator to Numerator
- Bounded Linear Operators Squared Norm
- Rationalizing Fractions Containing Roots Quickly
- Proof of Bernoulli's Inequality
- Markov Inequality Proof
- Chebyshev's Inequality Proof
- Proof of the Expected Form of Jensen's Inequality
- Generalization of Gaussian Integrals
- Integrals of Trigonometric Functions Table
- Proof of Gronwall's Inequality
- Summarizing Inequalities in the Form of an Inequality
- Product of Indicator Functions
- Proof of the Cauchy-Schwarz Inequality in Mathematical Statistics
- Inequalities for the Logarithmic Function 1-1/x < log x < x-1
- Definition of Pi
- Definition of Intervals in Mathematics
- Cumulative Average Formula Derivation
- Meaning of Weak and Strong in Mathematics
- Difference Between Root and Solution
- Fixed Points in Mathematics
- Square Root Expansion Formula
- Harmonic Mean
- Integrability of 1/x^p
- The limit of sin(x)/x
- The Limit of 1-cos(x)/x
- Integration of 1/(1+x^2)
- What is a Closed Form in Mathematics?
- Hamming Distance
- Continued Fraction
- Recurrence Relation
- The limit of n^(1/n)
- Birthday Problem: Probability of Sharing the Same Birthday
- Professor Choi Byung-sun's Freely Distributed Textbooks on Mathematics, Statistics, and Economics
- Geometric Mean