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
Mark | Subcategory |
---|---|
🔵 | Mathematical Statistics |
Definitions
- Pi
- Interval
- Difference between root and solution
- Fixed point
- Meaning of weak and strong in mathematics
- Closed form
- Hamming distance
- Recurrence relation
- Continued fraction
- Pseudo Code
Summaries
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
Integration Formulas
- Gaussian integral, Euler–Poisson integral
- Indefinite integral of
- Table of trigonometric integrals
- Integrability of
- ,
- Common definite integral of a quadratic
- Integration by parts for powers of natural logarithm
Limits
Discrete Mathematics
Arithmetic
- Sum of an arithmetic sequence
- Sum of a geometric sequence
- Sum of squares
- 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
- 🔒(25/08/21)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?
- Geometric Mean
- Hamming Distance
- Continued Fraction
- Recurrence Relation
- The limit of n^(1/n)