Formula gallery

Mathematical examples

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Basic algebra

Quadratic formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Binomial theorem

(a+b)n=∑k=0n(nk)an−kbk(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

Difference of squares

a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)

Geometry

Pythagorean theorem

a2+b2=c2a^2 + b^2 = c^2

Area of a circle

A=πr2A = \pi r^2

Volume of a sphere

V=43πr3V = \frac{4}{3}\pi r^3

Calculus

Definition of derivative

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Fundamental theorem

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)

Gaussian integral

∫−∞∞e−x2 dx=π\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}

Famous identities

Euler's identity

eiπ+1=0e^{i\pi} + 1 = 0

Golden ratio

φ=1+52≈1.618\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618

Fibonacci closed form (Binet)

Fn=φn−ψn5,ψ=1−52F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}, \quad \psi = \frac{1 - \sqrt{5}}{2}

Mandelbrot set

Iteration map

zn+1=zn2+cz_{n+1} = z_n^2 + c

Membership condition

Escape: ∣zn∣>2 for some n∈N\text{Escape: } |z_n| > 2 \text{ for some } n \in \mathbb{N}

Boundary in the complex plane

M={c∈C:lim sup⁡n→∞∣zn∣≤2}M = \{ c \in \mathbb{C} : \limsup_{n\to\infty} |z_n| \le 2 \}

Polar curves

Rose curve

r=acos⁡(kθ)r = a \cos(k\theta)

Cardioid

r=a(1+cos⁡θ)r = a(1 + \cos\theta)

Lemniscate of Bernoulli

r2=a2cos⁡(2θ)r^2 = a^2 \cos(2\theta)