A Lexicon of Core Mathematical and Scientific Principles
This codex catalogs fundamental equations across various scientific and mathematical domains. Each entry is a foundational principle, representing a coherent and verifiable relationship within its respective system.
1. Geometry
Area of a Circle
A = \pi r^2 Definition: Describes the area (A) enclosed by a circle with a given radius (r).
2. Algebra & Binomial Expansion
Binomial Theorem
(x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^k a^{n-k} Definition: Expands a binomial raised to a power into a sum of terms involving binomial coefficients.
Newton’s Binomial Series (Special Case)
(1 + x)^n = 1 + \frac{nx}{1!} + \frac{n(n-1)x^2}{2!} + \cdots Definition: A power series expansion for latex^n[/latex] that is useful in calculus and for approximations, particularly when n is not an integer.
3. Fourier Series
General Form of a Fourier Series
f(x) = a_0 + \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) Definition: Represents a periodic function, f(x), as an infinite sum of sine and cosine functions, decomposing it into its constituent frequencies.
4. Trigonometry
Pythagorean Theorem
a^2 + b^2 = c^2 Definition: Relates the lengths of the sides (a, b) and the hypotenuse (c) of a right-angled triangle.
Sum and Difference of Sines
\sin\alpha \pm \sin\beta = 2\sin\left(\frac{\alpha \pm \beta}{2}\right)\cos\left(\frac{\alpha \mp \beta}{2}\right) Definition: Expresses the sum or difference of two sine values as a product of sine and cosine functions.
Sum of Cosines
\cos\alpha + \cos\beta = 2\cos\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha - \beta}{2}\right) Definition: Expresses the sum of two cosine values as a product of two cosine functions.
5. Quadratic Formula
Solution to a Quadratic Equation
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} Definition: Provides the solution(s) for x in a quadratic equation of the form ax^2 + bx + c = 0.
6. Calculus & Series
Maclaurin Series Expansion for an Exponential Function
e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots, \quad -\infty < x < \infty Definition: A power series that represents the exponential function e^x for all real numbers x, providing a way to approximate its value.