Lemmas
This document introduces definitions, lemmas, and tricks that are universally used in mathematics and are difficult to categorize into specific categories.
Mark | Subcategory |
---|---|
🔵 | Mathematical Statistics |
Definitions
- Pi $\pi$
- Interval $[a,b]$
- Difference between a root and a solution
- Fixed point $x_{0} = f \left( x_{0} \right)$
- Harmonic mean
- Meaning of weak and strong in mathematics
- Closed form
- 🔒(24/10/15) 해밍 거리 $H$
- Recurrence relation
- Continued fraction
Theorems
- Multiplication Formula Table
- Quickly Rationalizing Fractions Containing Roots
- Easily Squaring Two-Digit Numbers Ending in 5
- Expansion Formula for Double Radicals $\sqrt{a + b \pm 2\sqrt{ab}} = \sqrt{a} \pm \sqrt{b}$
- Quickly Finding the Extremes of a Quadratic Function
- Calculating the Sum of Square Numbers
- Calculating the Sum of an Arithmetic Series
- Calculating the Sum of a Geometric Series
- General Term of the Fibonacci Sequence
- Binomial Theorem
- Gabi’s Theorem
- Number of Subsets of a Finite Set with n Elements
Tricks
- How to move Big O notation from the denominator to the numerator
- Product of indicator functions 🔵
- Cumulative average formula 🔵
Inequalities
- Inequality between arithmetic, geometric, and harmonic means
- Cauchy-Schwarz inequality
- Young’s inequality
- Bernoulli’s inequality
- Jensen’s inequality in finite form
- Jensen’s inequality in integral form
- Jensen’s inequality in expectation form 🔵
- Markov’s inequality 🔵
- Chebyshev’s inequality 🔵
- Grönwall’s inequality
- Inequality summarizing several inequalities 🔵
- Inequality for the logarithm function $1 - \dfrac{1}{x} \le \ln x \le x - 1$
Integral Formulas
- Definite integral of $e^{-x^{2}}$, Gaussian integral, Euler-Poisson integral
- Indefinite integral of $e^{x^{2}}$
- Table of integrals of trigonometric functions
- Integrability of $\dfrac{1}{x^p}$
- $\displaystyle \int_{-\infty}^{\infty} \dfrac{1}{1+x^{2}}dx = \pi$, $\displaystyle \int \dfrac{1}{1+x^{2}}dx = \tan^{-1} x + C$
- Frequently Used Definite Integrals of Quadratic Functions $\int_{\alpha}^{\beta}{(x-\alpha)(x-\beta)dx}=-\frac{{(\beta-\alpha)}^3}{6}$
- Integration of Natural Logarithms Raised to a Power $\int {(\ln x)^n} dx=x{(\ln x)^n}-\int n{(\ln x)^{n-1}}dx$
극한
- $\lim \limits_{x \to 0} \dfrac{\sin x}{x} = 1$
- $\lim \limits_{x \to 0} \dfrac{1 - \cos x}{x} = 0$
- $\lim\limits_{n \to \infty} \sqrt[n]{n} = 1$, $\lim\limits_{n \to \infty} \sqrt[n]{\dfrac{1}{n}} = 1$
All posts
- Continued Fraction
- Recurrence Relation
- The Limit of n^(1/n)
- 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
- 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