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Basic Mathematics

Examples of basic mathematical expressions rendered with MathJax.

Quadratic Formula

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

Pythagorean Theorem

\[ a^2 + b^2 = c^2 \]

Binomial Theorem

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k \]

Geometric Series

\[ \sum_{k=0}^{\infty} ar^k = \frac{a}{1 - r}, \quad |r| < 1 \]

Advanced Mathematics

More complex mathematical expressions and notations.

Euler's Identity

\[ e^{i\pi} + 1 = 0 \]

Gamma Function

\[ \Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt \]

Riemann Zeta Function

\[ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}} \]

Matrix Operations

Matrix notation and operations with MathJax.

Matrix Multiplication

\[ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{pmatrix} \]

Determinant

\[ \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \]

Eigenvalue Equation

\[ A\mathbf{v} = \lambda\mathbf{v} \]

3D Rotation Matrix

\[ R_x(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix} \]

Calculus

Differential and integral calculus examples.

Derivative Definition

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

Integration by Parts

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

Taylor Series

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

Gaussian Integral

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

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