List of some fundamental formulas that are commonly used in various mathematical disciplines:

- Pythagorean theorem: a² + b² = c² (for right triangles)
- Quadratic formula: x = (-b ± √(b² – 4ac)) / 2a (for quadratic equations)
- Distance formula: d = √((x₂ – x₁)² + (y₂ – y₁)²) (for finding the distance between two points in a plane)
- Slope formula: m = (y₂ – y₁) / (x₂ – x₁) (for finding the slope of a line)
- Sum of an arithmetic series: S = (n/2)(a₁ + aₙ) (for finding the sum of a series with a constant difference between terms)
- Sum of a geometric series: S = (a₁(1 – rⁿ)) / (1 – r) (for finding the sum of a series with a constant ratio between terms)
- Fundamental theorem of calculus: ∫(a to b) f(x) dx = F(b) – F(a) (for finding the definite integral of a function)
- Law of cosines: c² = a² + b² – 2abcos(θ) (for finding the length of a side of a triangle)
- Law of sines: sin(α) / a = sin(β) / b = sin(γ) / c (for finding the sides or angles of a triangle)
- Euler’s formula: e^(iθ) = cos(θ) + i sin(θ) (relating the exponential function to trigonometry)
- Binomial theorem: (a + b)^n = Σ(n choose k) a^(n-k) b^k (for expanding a binomial expression)
- Fundamental theorem of algebra: every non-constant polynomial equation has at least one complex root

## More Formulas

- Vieta’s formulas: the sum and product of the roots of a polynomial equation
- Bayes’ theorem: P(A|B) = P(B|A) P(A) / P(B) (relating conditional probabilities)
- Fermat’s little theorem: a^(p-1) ≡ 1 mod p (for modular arithmetic)
- Taylor series: f(x) = Σ(n=0 to infinity) f^(n)(a) (x-a)^n / n! (for approximating functions)
- Integration by parts: ∫u dv = uv – ∫v du (for integrating products of functions)
- Cauchy-Riemann equations: ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x (defining complex differentiability)
- Stokes’ theorem: ∫∫(∇ x F) · dA = ∫(∂F/∂n) ds (relating surface and line integrals)
- Laplace transform: F(s) = ∫(0 to infinity) f(t) e^(-st) dt (for solving differential equations)
- Bernoulli’s inequality: (1+x)^n ≥ 1 + nx (for positive integers n and real x > -1)
- Ratio test: if lim(n→∞) |a(n+1) / a(n)| < 1, then Σ a(n) converges absolutely (for testing series convergence)
- Pick’s theorem: A = i + b/2 – 1 (relating the area of a polygon to its lattice points and boundary points)
- Schrödinger equation: iħ ∂ψ/∂t = H ψ (for describing quantum mechanics)
- Law of large numbers: as the sample size n grows, the sample mean approaches the population mean
- Black-Scholes formula: the price of a European call option on a non-dividend-paying stock
- Pythagorean trigonometric identity: sin²θ + cos²θ = 1 (for relating trigonometric functions)
- Gaussian elimination: a method for solving systems of linear equations
- Newton’s law of gravitation: F = G (m₁m₂/r²) (for describing the force between two masses)

The above fundamental formulas can be very helpful in solving complex problems