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Commit dc9cc39

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authored
Merge pull request #40 from coolbluealan/master
Fix typos.
2 parents 29807d5 + fcbdc00 commit dc9cc39

24 files changed

+99
-99
lines changed

‎README.md‎

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@@ -37,7 +37,7 @@ Some of the topics covered here are basic arithmetic of complex numbers, complex
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Riemann surfaces, limits, derivatives, domain coloring, analytic landscapes and
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some applications of conformal mappings.
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What distinguishes this online book from other traditional texts in the first instance is the use of interactive applets that allow you to explore properties of complex numbers geometrically and analyze complex functions by using different techniques to visualize them. For the design of applets I used the following open-source softwares: [GeoGebra](https://geogebra.org/), [p5.js](https://p5js.org/), [Cindy.js](https://cindyjs.org/) and [MathCell](http://mathcell.org/).
40+
What distinguishes this online book from other traditional texts in the first instance is the use of interactive applets that allow you to explore properties of complex numbers geometrically and analyze complex functions by using different techniques to visualize them. For the design of applets I used the following open-source software: [GeoGebra](https://geogebra.org/), [p5.js](https://p5js.org/), [Cindy.js](https://cindyjs.org/) and [MathCell](http://mathcell.org/).
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Although I advocate for the use of computers as an aid to geometric reasoning,
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I highly encourage you to practice your problem solving skills by solving
@@ -48,7 +48,7 @@ Think of the computer as a physicist would his laboratory. It may be used
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to check existing ideas about our world, or as a tool to discover new phenomena
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which then poses new ideas or challenges for their explanation.
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Throughout the sections I have provided detailed instructions (in some cases)
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to explore concepts and relationships about complex numbers using specific softwares,
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to explore concepts and relationships about complex numbers using specific software,
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nevertheless you must still keep in mind that computer hardware and software
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are ephemeral things in comparison with mathematical ideas, which are timeless.
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‎content/analytic_landscapes.html‎

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if(newN!=N,
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N = newN;
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//The following line is DIRTY, but it makes the application run smooth for high degrees. :-)
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//Nethertheless, it might cause render errors for high degree surfaces. In fact, only a subset of the surface is rendered.
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//Nevertheless, it might cause render errors for high degree surfaces. In fact, only a subset of the surface is rendered.
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//Adapt limit according to hardware.
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//values of kind 4*n-1 are good values, as it means to use vectors of length 4*n.
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N = min(N,11);
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if(!intersect & id>0,
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s = gets(id); //s = floor(log_2(id))
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//the intervals [a,b] are chossen such that (id in binary notation)
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//the intervals [a,b] are chosen such that (id in binary notation)
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//id = 1 => [a,b]=[l,u]
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//id = 10 => [a,b]=[l,(u+l)/2]
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//id = 101 => [a,b]=[l,(u+3*l)/4]
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<script async src="js/prism.js"></script>
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</body>
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</html>
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‎content/cauchy_goursat_theorem.html‎

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\end{eqnarray*}
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</div>
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<p>
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Both intergrals, on the right side, are zero by the <a href="complex_differentiation.html">Cauchy-Riemann equations</a>. $\hspace{5pt} \blacksquare$
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Both integrals, on the right side, are zero by the <a href="complex_differentiation.html">Cauchy-Riemann equations</a>. $\hspace{5pt} \blacksquare$
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</p>
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<p>Observe that once it has been established that the value
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<p><strong>Example 1:</strong>
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Consider the function $f(z)= \exp\left(z^3\right).$ If $C$ is
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any simple closed countour, in either direction, then
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any simple closed contour, in either direction, then
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\[
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\int_C \exp\left(z^3\right),円dz =0.
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\]
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<p>
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If $f$ is analytic in a multiply connected domain $D$ then we cannot conclude
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that $\int_C f(z),円dz=0$ for every simple closed contour $C$ in $D.$
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Suppose that $D$ is muliply connected with two "holes".
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Suppose that $D$ is multiply connected with two "holes".
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Let $C,$ $C_1$ and $C_2$ be simple
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closed contours such that each $C_k$ surrounds only one "hole" in the
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domain and are inside $C.$ See Figure 8.
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sides by line segments in the same way as shown in Figure 14
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and proceed to the equivalent of (\ref{sumcontours}) and (\ref{triangle01}).
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The integral of $f$ along one of these new triangular contous, let's call it
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$\Delta_2$ then satisifies
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$\Delta_2$ then satisfies
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\[
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\left|\int_{\Delta_1} f(z),円dz\right| \leq 4 \left|\int_{\Delta_2} f(z),円dz\right| .
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\]
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respectively. Then, if we keep in mind how the triangle $\Delta_1$ was constructed,
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it is a straightforward problem in similar triangles to show that
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$L_1$ is related to $L$ by $L_1=\dfrac{1}{2}L.$ Likewise,
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if $L_2$ is the legth of $\Delta_2,$ then $L_2=\dfrac{1}{2}L_1 = \dfrac{1}{2^2}L.$
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if $L_2$ is the length of $\Delta_2,$ then $L_2=\dfrac{1}{2}L_1 = \dfrac{1}{2^2}L.$
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In general we have that if $L_n$ is the length of $\Delta_n,$ then
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$L_n= \dfrac{1}{2^n}L.$
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</p>
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<figure>
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<img src="../images/chp04/cauchy-theorem-final-proof.gif" alt="Approximation by polygonal path" title="Approximation by polygonal path" style="width:550px;">
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<figcaption>
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The countour $C$ is approximated by a polygonal contour $P.$
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The contour $C$ is approximated by a polygonal contour $P.$
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</figcaption>
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</figure>
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<p>
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<em>Proof of Cauchy-Goursat Theorem.</em>
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Consider a simple closed contour $C$ and $n$ points $z_1, z_2, \ldots, z_n$
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on $C$ through wich a polygonal path $P$ has been constructed.
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on $C$ through which a polygonal path $P$ has been constructed.
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Then it can be shown that the difference
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\[
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\abs{\int_C f(z),円dz - \int_P f(z),円dz}
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</article>
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</main>
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<a href="cauchy_integral_formula.html" style="text-decoration: none;">
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<p class="nextPage">Cauchy Integral Formula <i class="fa-solid fa-angles-right"></i></p>
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‎content/cauchy_integral_formula.html‎

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<p>
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Choose the radius $r$ of the circle $C_r$ smaller thatn the number
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Choose the radius $r$ of the circle $C_r$ smaller than the number
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$\delta$ in the second of these inequalities. Since
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$\abs{z-z_0}=r\lt \delta$ when $z $ is on $C_r,$ it follows that
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the first of inequalities in (\ref{formula-03}) holds
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<div class="practice">
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<p>
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<strong>Exercise 2:</strong>
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Use the formal defition of derivative to verify that
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Use the formal definition of derivative to verify that
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$f'(z)$ exists and the expression (\ref{integral-derivative})
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is in fact valid.
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</p>
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<p>
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As a consequence, when a function $ f (z) = u(x, y) + iv(x, y)$
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is analytic at a point $z = (x, y),$ the differentiability of
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$f'$ ensures the continuity od $f'$ there. Then, since
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$f'$ ensures the continuity of $f'$ there. Then, since
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\[
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f'(z) = u_x + iv_x = v_y - i u_y,
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\]
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</article>
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</main>
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<a href="fundamental_theorem_of_algebra.html" style="text-decoration: none;">
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<p class="nextPage">The Fundamental Theorem of Algebra<i class="fa-solid fa-angles-right"></i></p>
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</html>
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‎content/complex_functions.html‎

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</article>
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</main>
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<script src="https://www.geogebra.org/apps/deployggb.js"></script>
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<a href="limits.html" style="text-decoration: none;">
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</html>
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‎content/complex_integration.html‎

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</p>
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<div class="scroll-wrapper">
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\begin{eqnarray}
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\int_a^b k ,円 w(t),円 dt &amp;=&amp; k \int_a^b w(t),円dt, \quad k\text{ is a complex contant,}\\
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\int_a^b k ,円 w(t),円 dt &amp;=&amp; k \int_a^b w(t),円dt, \quad k\text{ is a complex constant,}\\
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\int_a^b \big[ w(t) + s(t)\big] ,円dt &amp;=&amp; \int_a^b w(t),円dt + \int_a^b s(t) ,円dt,\\
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\int_a^b w(t) ,円dt &amp;=&amp; \int_a^c w(t),円dt + \int_c^b w(t) ,円dt \quad c\in[a,b], \\
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\int_a^b w(t) ,円dt &amp;=&amp; -\int_b^a w(t),円dt.
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385385
<p>
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From definition (\ref{contour-integral}), and the properties just mentioned above,
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it also follows immediatly that
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it also follows immediately that
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</p>
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<div class="scroll-wrapper">
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\begin{eqnarray}
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\int_{C} z_0 ,円 f(z) ,円 dz &amp;=&amp; z_0 \int_{C} f(z) ,円 dz, \quad z_0 \text{ is a complex contant,}\\
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\int_{C} z_0 ,円 f(z) ,円 dz &amp;=&amp; z_0 \int_{C} f(z) ,円 dz, \quad z_0 \text{ is a complex constant,}\\
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\int_{C} \big[ f(z) + g(z) \big] ,円 dz &amp;=&amp; \int_{C} ,円 f(z) ,円 dz + \int_{C} g(z) ,円 dz \\
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\end{eqnarray}
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</div>
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\[
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z'(t) = x'(t) + iy'(t)
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\]
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are continuos on the entire interval $[a,b].$
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are continuous on the entire interval $[a,b].$
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Then the real-valued function
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\[
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</p>
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<p>
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If $C$ is a contour of lenght $L$ and $f$ is a piecewise continuous
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function on $C,$ and $M$ is a nonnegative contant such that
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If $C$ is a contour of length $L$ and $f$ is a piecewise continuous
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function on $C,$ and $M$ is a nonnegative constant such that
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$\left|f(z)\right|\leq M$ for all points $z$ on $C$ at which $f(z)$
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\begin{eqnarray}\label{ML-inequality}
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<div class="theorem" id="FTC">
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with initial fixed point $z_1$ and
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final fixed point $z_2,$ we have that
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Now, since $f$ is continuous, for any $\epsilon\gt 0 $ there
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exists a $\delta\gt 0$ such that $\abs{f(s)-f(z)}\lt \epsilon$
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whenever $\abs{s-z}\lt \delta.$
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it follows from the ML-inequality that
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‎content/conformal_mapping.html‎

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<h2>Analytics functions</h2>
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<h2>Analytic functions</h2>
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<p>
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A remarkable geometrical property enjoyed by all complex
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<div class="theorem">
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If $f$ is analytic in a domain $D$ containing $z_0,$ and if
331-
$f'(z_0)\neq 0,$ then $w=f(z)$ is a conformal mapping at $z_0.$
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$f'(z_0)\neq 0,$ then $w=f(z)$ is conformal at $z_0.$
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<div class="proof">
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<div class="scroll-wrapper">
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\begin{eqnarray*}
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\text{arg}\left(f'\left(z_0\right)\cdot z_2'\right)-\text{arg}\left(f'\left(z_0\right)\cdot z_1'\right) &amp;= &amp;
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\text{arg}\left(f'\left(z_0\right) z_2'\right) +
386+
\text{arg}\left(f'\left(z_0\right)\right) +
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\text{arg}\left(z_2'\right)-\left[\text{arg}\left(f'\left(z_0\right)\right)+ \text{arg}\left(z_1'\right)
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\right]\\
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&amp;=&amp; \text{arg}\left(z_2'\right) - \text{arg}\left(z_1'\right).
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<p>
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throughout $V.$
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Since the Cauchy-Riemann equations hold for $u$ and $v,$ they also
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hold for $x$ and $y,$ that is
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\[
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x_u = y_v \quad \text{and}\quad x_v = - y_u.
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‎content/continuity.html‎

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<div class="theorem" id="bounded-function">
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both bounded and closed, there there exists
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$M\gt 0$ such that $\abs{f(z)}\leq M$ for all points in $R,$
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<div class="theorem" id="contuity-properties">
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Suppose that $f$ and $g$ are continous at $z_0,$ then the following
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<ol class="roman-list">
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