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Borishaase (talk | contribs) (Fundamental theorems of calculus and approximation theorem) |
Borishaase (talk | contribs) (Cauchy's integral theorem, fundamental theorem of algebra and Newton’s method) |
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= Welcome to MWiki = | = Welcome to MWiki = | ||
== Theorems of the month == | == Theorems of the month == | ||
− | === | + | === Cauchy's integral theorem === |
− | + | Given the <abbr title="neighbourhood relation">NR</abbr>s <math>B \subseteq {D}^{2}</math> and <math>A \subseteq [a, b]</math> for some <math>h</math>-domain <math>D \subseteq {}^{\omega}\mathbb{C}</math>, infinitesimal <math>h</math>, <math>f \in \mathcal{O}(D)</math> and a <abbr title="closed path">CP</abbr> <math>\gamma: [a, b[\rightarrow \partial D</math>, choosing <math>{}^\curvearrowright \gamma(t) = \gamma({}^\curvearrowright t)</math> for <math>t \in [a, b[</math> gives | |
+ | <div style="text-align:center;"><math>{\uparrow}_{\gamma }{f(z){\downarrow}z}=0.</math></div> | ||
+ | '''Proof:''' By the Cauchy-Riemann differential equations and Green's theorem, with <math>x := \text{Re} \, z, y := \text{Im} \, z, u := \text{Re} \, f, v := \text{Im} \, f</math> and <math>{D}^{-} := \{z \in D : z + h + \underline{h} \in D\}</math>, it holds that | ||
+ | <div style="text-align:center;"><math>{\uparrow}_{\gamma }{f(z){\downarrow}z}={\uparrow}_{\gamma }{\left( u+\underline{v} \right)\left( {\downarrow}x+{\downarrow}\underline{y} \right)}={\uparrow}_{z\in {{D}^{-}}}{\left( \left( \tfrac{{\downarrow} \underline{u}}{{\downarrow} x}-\tfrac{{\downarrow} \underline{v}}{{\downarrow} y} \right)-\left( \tfrac{{\downarrow} v}{{\downarrow} x}+\tfrac{{\downarrow} u}{{\downarrow} y} \right) \right){\downarrow}(x,y)}=0.\square</math></div> | ||
+ | === Fundamental theorem of algebra === | ||
+ | Every non-constant polynomial <math>p \in {}^{(\omega)}\mathbb{C}</math> has at least one complex root. | ||
− | + | '''Indirect proof:''' By performing an affine substitution of variables, reduce to the case <math>\widetilde{p(0)} \ne \mathcal{O}(\iota)</math>. Suppose that <math>p(z) \ne 0</math> for all <math>z \in {}^{(\omega)}\mathbb{C}</math>. | |
− | + | Since <math>f(z) := \widetilde{p(z)}</math> is holomorphic, it holds that <math>f(\tilde{\iota}) = \mathcal{O}(\iota)</math>. By the mean value inequality <math>|f(0)| \le {|f|}_{\gamma}</math> for <math>\gamma = \partial\mathbb{B}_{r}(0)</math> and arbitrary <math>r \in {}^{(\omega)}\mathbb{R}_{>0}</math>, and hence <math>f(0) = \mathcal{O}(\iota)</math>, which is a contradiction (and hence exactly <math>z(m) = m</math> holds).<math>\square</math> | |
− | <math>{ | ||
− | === | + | === Newton’s method === |
− | + | Demanding above <math>f(\curvearrowright z)=f(z)+f^\prime(z){\downarrow}z=0</math> implies <math>z_{\grave{n}} := z_n-{f^\prime(z_n)}^{-1}f(z_n)</math> if <math>{f^\prime(z_n)}^{-1}</math> is invertible resulting in quadratic convergence close to a zero.<math>\square</math> | |
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== Recommended readings == | == Recommended readings == |
Revision as of 17:11, 31 January 2024
Welcome to MWiki
Theorems of the month
Cauchy's integral theorem
Given the NRs [math]\displaystyle{ B \subseteq {D}^{2} }[/math] and [math]\displaystyle{ A \subseteq [a, b] }[/math] for some [math]\displaystyle{ h }[/math]-domain [math]\displaystyle{ D \subseteq {}^{\omega}\mathbb{C} }[/math], infinitesimal [math]\displaystyle{ h }[/math], [math]\displaystyle{ f \in \mathcal{O}(D) }[/math] and a CP [math]\displaystyle{ \gamma: [a, b[\rightarrow \partial D }[/math], choosing [math]\displaystyle{ {}^\curvearrowright \gamma(t) = \gamma({}^\curvearrowright t) }[/math] for [math]\displaystyle{ t \in [a, b[ }[/math] gives
Proof: By the Cauchy-Riemann differential equations and Green's theorem, with [math]\displaystyle{ x := \text{Re} \, z, y := \text{Im} \, z, u := \text{Re} \, f, v := \text{Im} \, f }[/math] and [math]\displaystyle{ {D}^{-} := \{z \in D : z + h + \underline{h} \in D\} }[/math], it holds that
Fundamental theorem of algebra
Every non-constant polynomial [math]\displaystyle{ p \in {}^{(\omega)}\mathbb{C} }[/math] has at least one complex root.
Indirect proof: By performing an affine substitution of variables, reduce to the case [math]\displaystyle{ \widetilde{p(0)} \ne \mathcal{O}(\iota) }[/math]. Suppose that [math]\displaystyle{ p(z) \ne 0 }[/math] for all [math]\displaystyle{ z \in {}^{(\omega)}\mathbb{C} }[/math].
Since [math]\displaystyle{ f(z) := \widetilde{p(z)} }[/math] is holomorphic, it holds that [math]\displaystyle{ f(\tilde{\iota}) = \mathcal{O}(\iota) }[/math]. By the mean value inequality [math]\displaystyle{ |f(0)| \le {|f|}_{\gamma} }[/math] for [math]\displaystyle{ \gamma = \partial\mathbb{B}_{r}(0) }[/math] and arbitrary [math]\displaystyle{ r \in {}^{(\omega)}\mathbb{R}_{>0} }[/math], and hence [math]\displaystyle{ f(0) = \mathcal{O}(\iota) }[/math], which is a contradiction (and hence exactly [math]\displaystyle{ z(m) = m }[/math] holds).[math]\displaystyle{ \square }[/math]
Newton’s method
Demanding above [math]\displaystyle{ f(\curvearrowright z)=f(z)+f^\prime(z){\downarrow}z=0 }[/math] implies [math]\displaystyle{ z_{\grave{n}} := z_n-{f^\prime(z_n)}^{-1}f(z_n) }[/math] if [math]\displaystyle{ {f^\prime(z_n)}^{-1} }[/math] is invertible resulting in quadratic convergence close to a zero.[math]\displaystyle{ \square }[/math]