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 $[a,b]$
- Difference between a root and a solution
- Fixed point $x_{0} = f \left( x_{0} \right)$
- 🔒(24/02/28) Harmonic mean
- Meaning of weak and strong in mathematics
- 🔒(24/05/30) Closed form
Theorems
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 $\dfrac{1}{x^p}$
- $\displaystyle \int_{-\infty}^{\infty} \dfrac{1}{1+x^{2}}dx = \pi$, $\displaystyle \int \dfrac{1}{1+x^{2}}dx = \tan^{-1} x + C$
극한
All posts
- Arithmetic, Geometric, and Harmonic Means Inequality
- Cauchy-Schwarz Inequality 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}
- 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
- 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
- Harmonic Mean
- Integrability of 1/x^p
- The Limit of 1-cos(x)/x
- Integration of 1/(1+x^2)