Formula Library

LaTeX Formula Library

Browse, search, and copy commonly used LaTeX formulas. Click any formula to copy it to your clipboard.

Showing 65 of 65 mathematical formulas

Quadratic FormulaBeginner

Solution to quadratic equations ax² + bx + c = 0

Algebra9th centuryMuhammad ibn Musa al-Khwarizmi

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

View

Binomial TheoremIntermediate

Expansion of (x+y)ⁿ

Algebra1665Isaac Newton

(x + y)^{n} = k = 0 \sum n (k n) x^{n - k} y^{k}

(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

View

Difference of SquaresBeginner

Factorization of difference of two squares

AlgebraAncientAncient mathematicians

a^{2} - b^{2} = (a + b) (a - b)

a^2 - b^2 = (a+b)(a-b)

View

Derivative DefinitionIntermediate

Definition of derivative as a limit

Calculus17th centuryIsaac Newton and Gottfried Leibniz

f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

View

Chain RuleIntermediate

Derivative of composite functions

Calculus17th centuryGottfried Leibniz

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

View

Fundamental Theorem of CalculusAdvanced

Connection between derivatives and integrals

Calculus17th centuryIsaac Newton and Gottfried Leibniz

\int_{a}^{b} f (x) d x = F (b) - F (a), F^{'} (x) = f (x)

\int_a^b f(x) \, dx = F(b) - F(a), \quad F'(x) = f(x)

View

Normal DistributionIntermediate

Gaussian distribution probability density

Statistics18th centuryCarl Friedrich Gauss

f (x) = \frac{1}{σ 2 π} e^{- \frac{1}{2} (\frac{x - μ}{σ})^{2}}

f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

View

Bayes' TheoremIntermediate

Update probability based on new evidence

Statistics18th centuryThomas Bayes

P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{P ( B )}

P(A|B) = \frac{P(B|A)P(A)}{P(B)}

View

Eigenvalue EquationAdvanced

Matrix eigenvalue and eigenvector

Linear Algebra18th centuryLeonhard Euler

A v = λ v

A\vec{v} = \lambda\vec{v}

View

Dot ProductBeginner

Inner product of two vectors

Linear Algebra19th centuryWilliam Rowan Hamilton

a \cdot b = i = 1 \sum n a_{i} b_{i} = ∣ a ∣∣ b ∣ cos θ

\vec{a} \cdot \vec{b} = \sum_{i=1}^{n} a_i b_i = |\vec{a}||\vec{b}|\cos\theta

View

Pythagorean IdentityBeginner

Fundamental trigonometric identity

TrigonometryAncientAncient Greek mathematicians

sin^{2} θ + cos^{2} θ = 1

\sin^2\theta + \cos^2\theta = 1

View

Euler's FormulaAdvanced

Complex exponential and trigonometric functions

Trigonometry18th centuryLeonhard Euler

e^{i θ} = cos θ + i sin θ

e^{i\theta} = \cos\theta + i\sin\theta

View

Newton's Second LawBeginner

Force equals mass times acceleration

Physics1687Isaac Newton

F = ma

F = ma

View

Mass-Energy EquivalenceAdvanced

Einstein's famous equation

Physics1905Albert Einstein

E = m c^{2}

E = mc^2

View

Schrödinger EquationAdvanced

Fundamental equation of quantum mechanics

Physics1926Erwin Schrödinger

i ℏ \frac{\partial}{\partial t} Ψ = \hat{H} Ψ

i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi

View

Area of CircleBeginner

Area of circle with radius r

GeometryAncientAncient mathematicians

A = π r^{2}

A = \pi r^2

View

Volume of SphereIntermediate

Volume of sphere with radius r

GeometryAncientArchimedes

V = \frac{4}{3} π r^{3}

V = \frac{4}{3}\pi r^3

View

Cauchy-Riemann EquationsAdvanced

Necessary conditions for complex differentiability

Complex Analysis19th centuryAugustin-Louis Cauchy and Bernhard Riemann

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

View

Laplace TransformAdvanced

Integral transform for solving ODEs

Differential Equations18th centuryPierre-Simon Laplace

L {f (t)} = F (s) = \int_{0}^{\infty} e^{- s t} f (t) d t

\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st}f(t) \, dt

View

Taylor SeriesAdvanced

Taylor series expansion around point a

Calculus18th centuryBrook Taylor

f (x) = n = 0 \sum \infty \frac{f ^{(n)} ( a )}{n !} (x - a)^{n}

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

View

Fourier SeriesAdvanced

Representation of periodic functions as sum of sines and cosines

Calculus19th centuryJoseph Fourier

f (x) = \frac{a _{0}}{2} + n = 1 \sum \infty (a_{n} cos (\frac{nπ x}{L}) + b_{n} sin (\frac{nπ x}{L}))

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n\cos\left(\frac{n\pi x}{L}\right) + b_n\sin\left(\frac{n\pi x}{L}\right)\right)

View

Stokes' TheoremAdvanced

Connection between line integrals and surface integrals

Calculus19th centuryGeorge Gabriel Stokes

\oint_{C} F \cdot d r = \iint_{S} (\nabla \times F) \cdot d S

\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{S}

View

Central Limit TheoremAdvanced

Sample means approach normal distribution as sample size increases

Statistics18th centuryAbraham de Moivre, Pierre-Simon Laplace

n \to \infty lim P (\frac{X ˉ _{n} - μ}{σ / n} \leq z) = Φ (z)

\lim_{n \to \infty} P\left(\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \leq z\right) = \Phi(z)

View

3D Pythagorean TheoremBeginner

Distance from origin in 3D space

GeometryAncientAncient mathematicians

d = x^{2} + y^{2} + z^{2}

d = \sqrt{x^2 + y^2 + z^2}

View

L'Hôpital's RuleIntermediate

Evaluate limits of indeterminate forms

Calculus17th centuryGuillaume de l'Hôpital

x \to c lim \frac{f ( x )}{g ( x )} = x \to c lim \frac{f ^{'} ( x )}{g ^{'} ( x )}

\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}

View

Integration by PartsIntermediate

Product rule for integration

Calculus17th centuryBrook Taylor

\int u d v = uv - \int v d u

\int u \, dv = uv - \int v \, du

View

Matrix Inverse (2×2)Intermediate

Inverse of 2×2 matrix

Linear Algebra19th centuryArthur Cayley

A^{- 1} = \frac{1}{det ( A )} (d - c - b a)

A^{-1} = \frac{1}{\det(A)}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

View

Confidence IntervalIntermediate

Interval estimate for population mean

Statistics20th centuryJerzy Neyman

\overset{x}{ˉ} \pm z_{α /2} \frac{σ}{n}

\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}

View

Maxwell's EquationsAdvanced

Fundamental equations of electromagnetism

Physics19th centuryJames Clerk Maxwell

\nabla \cdot E = \frac{ρ}{ϵ _{0}}, \nabla \times E = - \frac{\partial B}{\partial t}

\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}, \quad \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}

View

Completing the SquareIntermediate

Method to convert quadratic to vertex form

AlgebraAncientAncient mathematicians

a x^{2} + b x + c = a (x + \frac{b}{2 a})^{2} + c - \frac{b ^{2}}{4 a}

ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}

View

Geometric SeriesIntermediate

Sum of finite geometric series

AlgebraAncientAncient mathematicians

k = 0 \sum n a r^{k} = a \frac{1 - r ^{n + 1}}{1 - r}, r \neq = 1

\sum_{k=0}^{n} ar^k = a\frac{1-r^{n+1}}{1-r}, \quad r \neq 1

View

Arithmetic SeriesBeginner

Sum of arithmetic sequence

AlgebraAncientAncient mathematicians

k = 1 \sum n (a + (k - 1) d) = \frac{n}{2} (2 a + (n - 1) d)

\sum_{k=1}^{n} (a + (k-1)d) = \frac{n}{2}(2a + (n-1)d)

View

Definite IntegralIntermediate

Fundamental theorem of calculus

Calculus17th centuryIsaac Newton and Gottfried Leibniz

\int_{a}^{b} f (x) d x = F (b) - F (a)

\int_{a}^{b} f(x) \, dx = F(b) - F(a)

View

Product RuleBeginner

Derivative of product of functions

Calculus17th centuryGottfried Leibniz

\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)

\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

View

Quotient RuleIntermediate

Derivative of quotient of functions

Calculus17th centuryGottfried Leibniz

\frac{d}{d x} [\frac{f ( x )}{g ( x )}] = \frac{f ^{'} ( x ) g ( x ) - f ( x ) g ^{'} ( x )}{[ g ( x ) ] ^{2}}

\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

View

Power RuleBeginner

Derivative of power function

Calculus17th centuryIsaac Newton and Gottfried Leibniz

\frac{d}{d x} [x^{n}] = n x^{n - 1}

\frac{d}{dx}[x^n] = nx^{n-1}

View

Mean (Average)Beginner

Arithmetic mean of a dataset

StatisticsAncientAncient mathematicians

\overset{x}{ˉ} = \frac{1}{n} i = 1 \sum n x_{i}

\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i

View

VarianceIntermediate

Measure of data spread

Statistics19th centuryRonald Fisher

σ^{2} = \frac{1}{n} i = 1 \sum n (x_{i} - μ)^{2}

\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2

View

Standard DeviationIntermediate

Square root of variance

Statistics19th centuryKarl Pearson

σ = \frac{1}{n} i = 1 \sum n (x_{i} - μ)^{2}

\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2}

View

CovarianceIntermediate

Measure of joint variability

Statistics19th centuryFrancis Galton

Cov (X, Y) = \frac{1}{n} i = 1 \sum n (x_{i} - \overset{x}{ˉ}) (y_{i} - \overset{y}{ˉ})

\text{Cov}(X,Y) = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})

View

Correlation CoefficientIntermediate

Pearson correlation coefficient

Statistics19th centuryKarl Pearson

r = \frac{Cov ( X , Y )}{σ _{X} σ _{Y}}

r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}

View

Binomial ProbabilityIntermediate

Probability of k successes in n trials

Statistics17th centuryJacob Bernoulli

P (X = k) = (k n) p^{k} (1 - p)^{n - k}

P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}

View

Matrix MultiplicationIntermediate

Definition of matrix product

Linear Algebra19th centuryArthur Cayley

(A B)_{ij} = k = 1 \sum n A_{ik} B_{kj}

(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}

View

2×2 DeterminantBeginner

Determinant of 2×2 matrix

Linear Algebra19th centuryArthur Cayley

det (a c b d) = a d - b c

\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

View

Cross ProductIntermediate

Vector product in 3D space

Linear Algebra19th centuryWilliam Rowan Hamilton

a \times b = i a_{1} b_{1} j a_{2} b_{2} k a_{3} b_{3}

\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

View

Matrix TraceIntermediate

Sum of diagonal elements

Linear Algebra19th centuryArthur Cayley

tr (A) = i = 1 \sum n a_{ii}

\text{tr}(A) = \sum_{i=1}^{n} a_{ii}

View

Law of SinesIntermediate

Relationship between sides and angles in triangles