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# 🧮 Welcome to CollabMath > *A collaborative space where ideas meet rigor — write, discuss, and explore mathematics together.* --- ## What is CollabMath? **CollabMath** is a platform built for mathematicians, students, and curious minds. Whether you're working through a proof, sharing an elegant identity, or debugging a derivation — this is your space. We support full **Markdown** and **KaTeX** rendering, so your math looks exactly as it should. --- ## ✍️ Writing Math You can write inline math like $e^{i\pi} + 1 = 0$ seamlessly within sentences, or display it as a full block: $$ \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi} $$ Display equations are great for derivations. Here's the **Cauchy integral formula**: $$ f(z_0) = \frac{1}{2\pi i} \oint_{\gamma} \frac{f(z)}{z - z_0} \, dz $$ --- ## 📐 A Sample Proof > **Theorem** *(Pythagorean Theorem)*: For a right triangle with legs $a$, $b$ and hypotenuse $c$, > $$a^2 + b^2 = c^2$$ **Proof sketch.** Consider a square of side $(a + b)$. Arranged inside it are four congruent right triangles, each of area $\frac{1}{2}ab$, surrounding a central square of side $c$. $$ (a + b)^2 = 4 \cdot \frac{1}{2}ab + c^2 \implies a^2 + 2ab + b^2 = 2ab + c^2 \implies a^2 + b^2 = c^2 \qquad \blacksquare $$ --- ## 🔢 Formatting Features ### Text Formatting You can write **bold**, *italic*, ~~strikethrough~~, and `inline code` for variable names like `f(x)`. ### Lists Ordered and unordered lists work great for proof steps: 1. Assume $\epsilon > 0$ 2. Choose $\delta = \epsilon / 2$ 3. Show $|f(x) - L| < \epsilon$ whenever $0 < |x - a| < \delta$ Some common function families: - Polynomials: $p(x) = a_n x^n + \cdots + a_1 x + a_0$ - Exponentials: $f(x) = e^x$ - Trigonometric: $\sin^2 x + \cos^2 x = 1$ ### Tables | Identity | Formula | |---|---| | Euler's formula | $e^{ix} = \cos x + i \sin x$ | | Binomial theorem | $(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}$ | | Geometric series | $\sum_{k=0}^{\infty} r^k = \frac{1}{1-r}, \quad \|r\| < 1$ | | Stirling's approx. | $n! \approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$ | ### Code Blocks Algorithms and computations live naturally alongside math: ```python def newton_raphson(f, df, x0, tol=1e-9): """Solve f(x) = 0 near x0 using Newton's method.""" x = x0 while abs(f(x)) > tol: x -= f(x) / df(x) return x ``` The iteration step corresponds to the update rule: $$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$ --- ## 🧵 CollabMath Thread Conventions Use **blockquotes** to reply to or highlight another user's statement: > *"Is there a closed form for $\sum_{n=1}^{\infty} \frac{1}{n^2}$?"* Yes! This is the **Basel problem**, solved by Euler in 1734: $$ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} $$ More generally, the **Riemann zeta function** is defined for $\text{Re}(s) > 1$ as: $$ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} $$ --- ## 🔗 Horizontal Rules & Section Dividers Use `---` to cleanly divide a long thread into sections, keeping discussions readable even as they grow. --- *Happy proving. May your limits converge and your series be absolutely convergent.* 🎓