Mathematical Examples

Showcase of classic and important mathematical formulas that have significant meaning in the development and application of mathematics.

Algebra

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

A general formula used to solve quadratic equations in one variable, where a, b, c are equation coefficients.

Example： For example: Solve x² - 5x + 6 = 0, where a=1, b=-5, c=6

Complex Analysis

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

An important formula that connects exponential functions with trigonometric functions, holding a central position in complex analysis.

Example： When θ = π, we get the famous Euler's identity: e^{iπ} + 1 = 0

Geometry

a^2 + b^2 = c^2

In a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse, a fundamental theorem in geometry.

Example： If the two legs of a right triangle are 3 and 4 respectively, then the hypotenuse length is 5

Calculus

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

Expanding a function as a power series around a point, an important tool in calculus.

Example： Taylor expansion of sin(x) at x=0: x - x³/3! + x⁵/5! - x⁷/7! + ...

Integration

\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}

The integral of the Gaussian function over the entire real axis, very important in probability theory and statistics.

Example： This integral plays a key role in the standardization process of normal distribution

Analysis

\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)e^{-2\pi ix\xi} dx

Converting time-domain signals to frequency-domain representation, widely used in signal processing and image processing.

Example： Used in fields such as audio compression, image processing, and communication systems

Showcasing the powerful functionality of LaTeX in mathematical formula rendering

\\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix}

\\lim_{x \\to \\infty} \\frac{1}{x} = 0

\\sum_{i=1}^{n} i = \\frac{n(n+1)}{2}

\\frac{dy}{dx} + P(x)y = Q(x)