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    <h1 class="sp">Complex Analysis: A Short Documentation</h1>
    <h2 class="sp" align=center>Made by SJR with Mistral Vibe for Mathematical Physics</h1>
    <div class="toc">
        <h2>Table of Contents</h2>
        <ul>
            <li><a href="#brief-revision">Brief Revision of Complex Numbers</a></li>
            <li><a href="#graphical-representation">Graphical Representation of Complex Numbers</a></li>
            <li><a href="#eulers-formula">Euler's Formula</a></li>
            <li><a href="#de-moivres-theorem">De Moivre's Theorem</a></li>
            <li><a href="#roots">Roots of Complex Numbers</a></li>
            <li><a href="#functions">Functions of Complex Variables</a></li>
            <li><a href="#analyticity">Analyticity and Cauchy-Riemann Conditions</a></li>
            <li><a href="#examples">Examples of Analytic Functions</a></li>
            <li><a href="#singular-functions">Singular Functions: Poles and Branch Points</a></li>
            <li><a href="#order">Order of Singularity</a></li>
            <li><a href="#branch-cuts">Branch Cuts</a></li>
            <li><a href="#integration">Integration of a Function of a Complex Variable</a></li>
            <li><a href="#cauchys-inequality">Cauchy's Inequality</a></li>
            <li><a href="#cauchys-integral">Cauchy’s Integral Formula</a></li>
        </ul>
    </div>

    <div class="section" id="brief-revision">
        <h2>Brief Revision of Complex Numbers</h2>
        <p>A <strong>complex number</strong> <span class="math">\( z \)</span> is defined as:</p>
        <p class="math">\[ z = a + bi \]</p>
        <p>where:</p>
        <ul>
            <li><span class="math">\( a, b \in \mathbb{R} \)</span> are the <strong>real</strong> and <strong>imaginary parts</strong>, respectively.</li>
            <li><span class="math">\( i \)</span> is the <strong>imaginary unit</strong>, satisfying <span class="math">\( i^2 = -1 \)</span>.</li>
        </ul>

        <h3>Properties</h3>
        <ul>
            <li><strong>Real Part</strong>: <span class="math">\( \Re(z) = a \)</span></li>
            <li><strong>Imaginary Part</strong>: <span class="math">\( \Im(z) = b \)</span></li>
            <li><strong>Complex Conjugate</strong>: <span class="math">\( \overline{z} = a - bi \)</span></li>
            <li><strong>Modulus</strong>: <span class="math">\( |z| = \sqrt{a^2 + b^2} \)</span></li>
            <li><strong>Argument</strong>: <span class="math">\( \arg(z) = \theta = \arctan\left(\frac{b}{a}\right) \)</span></li>
        </ul>

        <h3>Algebraic Operations</h3>
        <ul>
            <li><strong>Addition</strong>: <span class="math">\( (a + bi) + (c + di) = (a + c) + (b + d)i \)</span></li>
            <li><strong>Subtraction</strong>: <span class="math">\( (a + bi) - (c + di) = (a - c) + (b - d)i \)</span></li>
            <li><strong>Multiplication</strong>: <span class="math">\( (a + bi)(c + di) = (ac - bd) + (ad + bc)i \)</span></li>
            <li><strong>Division</strong>: <span class="math">\( \frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} \)</span></li>
        </ul>
    </div>

    <div class="section" id="graphical-representation">
        <h2>Graphical Representation of Complex Numbers</h2>
        <p>Complex numbers are represented in the <strong>complex plane</strong> (Argand diagram), where:</p>
        <ul>
            <li>The <strong>x-axis</strong> represents the <strong>real part</strong> <span class="math">\( \Re(z) \)</span>.</li>
            <li>The <strong>y-axis</strong> represents the <strong>imaginary part</strong> <span class="math">\( \Im(z) \)</span>.</li>
        </ul>

        <div class="svg-container">
            <svg viewBox="0 0 200 250" xmlns="http://www.w3.org/2000/svg">
                <line x1="0" y1="110" x2="200" y2="110" stroke="black" stroke-width="1" />
                <line x1="100" y1="10" x2="100" y2="240" stroke="black" stroke-width="1" />
                <text x="200" y="105" text-anchor="end">Re(z) (3 Units)</text>
                <text x="105" y="20" text-anchor="middle">Im(z) (4 Units)</text>
                <circle cx="130" cy="60" r="3" fill="red" />
                <text x="130" y="60" text-anchor="middle" font-size="12">3+4i</text>
                <line x1="100" y1="110" x2="130" y2="60" stroke="blue" stroke-width="1" />
            </svg>
        </div>
    </div>

    <div class="section" id="eulers-formula">
        <h2>Euler's Formula</h2>
        <p>Euler's formula establishes a deep connection between exponential and trigonometric functions:</p>
        <p class="math">\[ e^{i\theta} = \cos(\theta) + i\sin(\theta) \]</p>

        <h3>Polar Form</h3>
        <p>A complex number <span class="math">\( z = a + bi \)</span> can be expressed in <strong>polar form</strong> as:</p>
        <p class="math">\[ z = r(\cos\theta + i\sin\theta) = re^{i\theta} \]</p>
        <p>where:</p>
        <ul>
            <li><span class="math">\( r = |z| = \sqrt{a^2 + b^2} \)</span></li>
            <li><span class="math">\( \theta = \arg(z) = \arctan\left(\frac{b}{a}\right) \)</span></li>
        </ul>
    </div>

    <div class="section" id="de-moivres-theorem">
        <h2>De Moivre's Theorem</h2>
        <p>De Moivre's theorem states:</p>
        <p class="math">\[ (\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta) \]</p>
        <p>for any integer <span class="math">\( n \)</span> and real <span class="math">\( \theta \)</span>.</p>

        <h3>Applications</h3>
        <ul>
            <li><strong>Powers of Complex Numbers</strong>: Simplifies <span class="math">\( (z)^n \)</span> for <span class="math">\( z = r(\cos\theta + i\sin\theta) \)</span>.</li>
            <li><strong>Roots of Complex Numbers</strong>: Used to find the <span class="math">\( n \)</span>-th roots of <span class="math">\( z \)</span>.</li>
        </ul>
    </div>

    <div class="section" id="roots">
        <h2>Roots of Complex Numbers</h2>
        <p>The <strong><span class="math">\( n \)</span>-th roots</strong> of a complex number <span class="math">\( z = r(\cos\theta + i\sin\theta) \)</span> are given by:</p>
        <p class="math">\[ z_k = r^{1/n} \left[ \cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right) \right] \]</p>
        <p>for <span class="math">\( k = 0, 1, 2, \dots, n-1 \)</span>.</p>

        <h3>Example</h3>
        <p>Find the cube roots of <span class="math">\( z = 8(\cos(\pi/3) + i\sin(\pi/3)) \)</span>:</p>
        <ul>
            <li><span class="math">\( z_0 = 2\left[ \cos\left(\frac{\pi}{9}\right) + i\sin\left(\frac{\pi}{9}\right) \right] \)</span></li>
            <li><span class="math">\( z_1 = 2\left[ \cos\left(\frac{7\pi}{9}\right) + i\sin\left(\frac{7\pi}{9}\right) \right] \)</span></li>
            <li><span class="math">\( z_2 = 2\left[ \cos\left(\frac{13\pi}{9}\right) + i\sin\left(\frac{13\pi}{9}\right) \right] \)</span></li>
        </ul>
    </div>

    <div class="section" id="functions">
        <h2>Functions of Complex Variables</h2>
        <p>A <strong>function of a complex variable</strong> is a rule that assigns a complex number <span class="math">\( w \)</span> to each complex number <span class="math">\( z \)</span> in a domain <span class="math">\( D \subseteq \mathbb{C} \)</span>.</p>

        <h3>Common Functions</h3>
        <ol>
            <li><strong>Polynomials</strong>:
                <p class="math">\[ f(z) = a_n z^n + a_{n-1} z^{n-1} + \dots + a_0 \]</p>
            </li>
            <li><strong>Rational Functions</strong>:
                <p class="math">\[ f(z) = \frac{P(z)}{Q(z)} \]</p>
                where <span class="math">\( P \)</span> and <span class="math">\( Q \)</span> are polynomials.
            </li>
            <li><strong>Exponential Function</strong>:
                <p class="math">\[ f(z) = e^z = e^x (\cos y + i\sin y) \]</p>
                where <span class="math">\( z = x + iy \)</span>.
            </li>
            <li><strong>Trigonometric Functions</strong>:
                <ul>
                    <li><span class="math">\( \sin(z) = \frac{e^{iz} - e^{-iz}}{2i} \)</span></li>
                    <li><span class="math">\( \cos(z) = \frac{e^{iz} + e^{-iz}}{2} \)</span></li>
                </ul>
            </li>
            <li><strong>Logarithmic Function</strong>:
                <p class="math">\[ f(z) = \ln|z| + i\arg(z) \]</p>
            </li>
        </ol>
    </div>

    <div class="section" id="analyticity">
        <h2>Analyticity and Cauchy-Riemann Conditions</h2>
        <p>A function <span class="math">\( f(z) \)</span> is <strong>analytic</strong> (holomorphic) at a point <span class="math">\( z_0 \)</span> if it is differentiable in some neighborhood of <span class="math">\( z_0 \)</span>.</p>

        <h3>Cauchy-Riemann Conditions</h3>
        <p>For <span class="math">\( f(z) = u(x, y) + iv(x, y) \)</span> to be analytic, it must satisfy:</p>
        <ol>
            <li><span class="math">\( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \)</span></li>
            <li><span class="math">\( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)</span></li>
        </ol>
        <p>Additionally, <span class="math">\( u \)</span> and <span class="math">\( v \)</span> must satisfy the <strong>Laplace equation</strong>:</p>
        <ul>
            <li><span class="math">\( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \)</span></li>
            <li><span class="math">\( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0 \)</span></li>
        </ul>
    </div>

    <div class="section" id="examples">
        <h2>Examples of Analytic Functions</h2>
        <ol>
            <li><span class="math">\( f(z) = z^2 \)</span>: Satisfies Cauchy-Riemann conditions everywhere.</li>
            <li><span class="math">\( f(z) = e^z \)</span>: Analytic everywhere (entire function).</li>
            <li><span class="math">\( f(z) = \sin(z) \)</span>: Analytic everywhere.</li>
            <li><span class="math">\( f(z) = \frac{1}{z} \)</span>: Analytic everywhere except at <span class="math">\( z = 0 \)</span> (singularity).</li>
        </ol>
    </div>

    <div class="section" id="singular-functions">
        <h2>Singular Functions: Poles and Branch Points</h2>

        <h3>Singularities</h3>
        <p>A point <span class="math">\( z_0 \)</span> is a <strong>singularity</strong> of <span class="math">\( f(z) \)</span> if <span class="math">\( f(z) \)</span> is not analytic at <span class="math">\( z_0 \)</span> but is analytic in some punctured neighborhood of <span class="math">\( z_0 \)</span>.</p>

        <h4>Types of Singularities</h4>
        <ol>
            <li><strong>Removable Singularity</strong>: If <span class="math">\( \lim_{z\to z_0} f(z) \)</span> exists and is finite.</li>
            <li><strong>Pole</strong>: If <span class="math">\( \lim_{z\to z_0} f(z) = \infty \)</span>. The <strong>order</strong> of the pole is the smallest integer <span class="math">\( n \)</span> such that <span class="math">\( (z - z_0)^n f(z) \)</span> is bounded near <span class="math">\( z_0 \)</span>.</li>
            <li><strong>Essential Singularity</strong>: If <span class="math">\( \lim_{z\to z_0} f(z) \)</span> does not exist in any form (finite or infinite).</li>
        </ol>

        <h3>Branch Points</h3>
        <p>A <strong>branch point</strong> is a point <span class="math">\( z_0 \)</span> such that a function <span class="math">\( f(z) \)</span> is not single-valued in any neighborhood of <span class="math">\( z_0 \)</span>. Common in <strong>multi-valued functions</strong> like <span class="math">\( \log(z) \)</span> and <span class="math">\( z^{1/n} \)</span>.</p>
    </div>

    <div class="section" id="order">
        <h2>Order of Singularity</h2>
        <p>The <strong>order</strong> of a singularity at <span class="math">\( z_0 \)</span> is determined by the <strong>Laurent series expansion</strong> of <span class="math">\( f(z) \)</span> around <span class="math">\( z_0 \)</span>:</p>
        <ul>
            <li>If the <strong>principal part</strong> has a finite number of terms, the singularity is a <strong>pole</strong> of order equal to the highest power of <span class="math">\( (z - z_0) \)</span> in the principal part.</li>
            <li>If the principal part has infinitely many terms, the singularity is <strong>essential</strong>.</li>
        </ul>
    </div>

    <div class="section" id="branch-cuts">
        <h2>Branch Cuts</h2>
        <p>A <strong>branch cut</strong> is a curve in the complex plane that restricts the domain of a multi-valued function to make it single-valued and continuous.</p>

        <h3>Example</h3>
        <p>For the function <span class="math">\( f(z) = \log(z) \)</span>, a branch cut is typically made along the <strong>negative real axis</strong> (from <span class="math">\( 0 \)</span> to <span class="math">\( -\infty \)</span>).</p>

        <div class="svg-container">
            <svg viewBox="0 0 200 200" xmlns="http://www.w3.org/2000/svg">
                <line x1="0" y1="100" x2="200" y2="100" stroke="black" stroke-width="1" />
                <line x1="100" y1="0" x2="100" y2="200" stroke="black" stroke-width="1" />
                <path d="M100,100 L50,100" stroke="red" stroke-width="2" stroke-dasharray="5,5" />
                <text x="25" y="95" text-anchor="middle" font-size="12">Branch Cut</text>
            </svg>
        </div>
    </div>

    <div class="section" id="integration">
        <h2>Integration of a Function of a Complex Variable</h2>
        <p>The integral of a complex function <span class="math">\( f(z) \)</span> along a contour <span class="math">\( C \)</span> is defined as:</p>
        <p class="math">\[ \int_C f(z) \, dz = \int_a^b f(\gamma(t)) \, \gamma'(t) \, dt \]</p>
        <p>where <span class="math">\( \gamma(t) \)</span> is a parameterization of <span class="math">\( C \)</span> for <span class="math">\( t \in [a, b] \)</span>.</p>

        <h3>Key Theorems</h3>
        <ol>
            <li><strong>Cauchy's Theorem</strong>: If <span class="math">\( f(z) \)</span> is analytic in a simply connected domain <span class="math">\( D \)</span>, then:
                <p class="math">\[ \int_C f(z) \, dz = 0 \]</p>
                for any closed contour <span class="math">\( C \)</span> in <span class="math">\( D \)</span>.
            </li>
            <li><strong>Cauchy-Goursat Theorem</strong>: A generalization of Cauchy's theorem for domains that are not simply connected.</li>
        </ol>
    </div>

    <div class="section" id="cauchys-inequality">
        <h2>Cauchy's Inequality</h2>
        <p>Cauchy's inequality provides a bound on the coefficients of a <strong>Taylor series</strong> for an analytic function. If <span class="math">\( f(z) \)</span> is analytic in a disk <span class="math">\( |z - z_0| < R \)</span>, then:</p>
        <p class="math">\[ |a_n| \leq \frac{M}{R^n} \]</p>
        <p>where:</p>
        <ul>
            <li><span class="math">\( M \)</span> is the maximum value of <span class="math">\( |f(z)| \)</span> on the circle <span class="math">\( |z - z_0| = R \)</span>.</li>
            <li><span class="math">\( a_n \)</span> is the coefficient of <span class="math">\( (z - z_0)^n \)</span> in the Taylor series expansion of <span class="math">\( f(z) \)</span>.</li>
        </ul>
    </div>

    <div class="section" id="cauchys-integral">
        <h2>Cauchy’s Integral Formula</h2>
        <p>If <span class="math">\( f(z) \)</span> is analytic in a simply connected domain <span class="math">\( D \)</span>, and <span class="math">\( C \)</span> is a simple closed contour in <span class="math">\( D \)</span> that encloses <span class="math">\( z_0 \)</span>, then:</p>
        <p class="math">\[ f(z_0) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z - z_0} \, dz \]</p>

        <h3>Generalized Cauchy’s Integral Formula</h3>
        <p>For the <strong><span class="math">\( n \)</span>-th derivative</strong> of <span class="math">\( f(z) \)</span> at <span class="math">\( z_0 \)</span>:</p>
        <p class="math">\[ f^{(n)}(z_0) = \frac{n!}{2\pi i} \int_C \frac{f(z)}{(z - z_0)^{n+1}} \, dz \]</p>
    </div>

    <div class="section">
        <h2>Conclusion</h2>
        <p>This documentation provides a overview of <strong>Complex Analysis</strong>, with <strong>LaTeX-formatted equations</strong> and <strong>SVG graphs</strong> for clarity. Each section can be expanded with additional examples, proofs, or visualizations as needed.</p>
    </div>
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