From ac66c13835c264bd637742eddeacdf9aa81c56ef Mon Sep 17 00:00:00 2001 From: Chewico Date: Wed, 17 Sep 2025 20:17:41 +0200 Subject: [PATCH] MATH_HOME_17_9_25 --- .../obsidian-excalidraw-plugin/data.json | 2 +- .obsidian/workspace.json | 16 +- ...簃eros complejos 09-09-25_10_1.excalidraw.md | 4711 +++++++---------- Ilustraci贸nes/img_02-09-25_20.excalidraw.md | 1053 +--- Matematicas/N煤meros complejos.md | 45 +- 5 files changed, 2127 insertions(+), 3700 deletions(-) diff --git a/.obsidian/plugins/obsidian-excalidraw-plugin/data.json b/.obsidian/plugins/obsidian-excalidraw-plugin/data.json index 7f5cbb2..07a08d2 100644 --- a/.obsidian/plugins/obsidian-excalidraw-plugin/data.json +++ b/.obsidian/plugins/obsidian-excalidraw-plugin/data.json @@ -439,7 +439,7 @@ } } ], - "numberOfCustomPens": 0, + "numberOfCustomPens": 1, "pdfScale": 4, "pdfBorderBox": true, "pdfFrame": false, diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index f26b539..9a88f84 100644 --- 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complejos.md +++ b/Matematicas/N煤meros complejos.md @@ -41,7 +41,17 @@ $$ $$ \huge \frac{z}{z'} = \frac{r}{r'} \big(\cos(\alpha - \alpha') + i \sin(\alpha - \alpha')\big) $$ -## Potencia de un n煤mero complejo (F贸rmula de De Moivre) +# Producto polar +La formula para calcular el producto en forma polar +$$ +\huge r_{\alpha} \cdot r'_{\alpha'}=(r\cdot r')_{\alpha + \alpha '} +$$ +# Divisi贸n polar +La formula para dividir en forma polar +$$ +\huge \frac {z_1} {z_2} = (\frac {r_1} {r_2})_{\alpha_1-\alpha_2} +$$ +## Potencia de un n煤mero complejo Si $z = r(\cos \alpha + i \sin \alpha)$ y $\mu \in \mathbb{Z}$, entonces: @@ -66,31 +76,20 @@ $$ # Ra铆z n-茅sima de un n煤mero complejo -# Ra铆z n-茅sima - -# Ra铆z n-茅sima de un n煤mero complejo - -# Ra铆z n-茅sima de un n煤mero complejo - Para calcular la ra铆z $n$-茅sima de un n煤mero complejo $z = r(\cos \alpha + i \sin \alpha)$, se utiliza la siguiente f贸rmula: +Si tenemos $$ -z^{1/n} = r^{1/n} \left[ \cos\left(\frac{\alpha + 2k\pi}{n}\right) + i \sin\left(\frac{\alpha + 2k\pi}{n}\right) \right], \quad k = 0, 1, \dots, n-1 +\huge ^{n}\sqrt r \space (cos \space \alpha + i \space sin \space \alpha) $$ - -- Aqu铆 $r = |z|$ es el m贸dulo y $\alpha = \arg(z)$ es el argumento de $z$. -- Cada valor de $k$ da una **ra铆z diferente**, por eso hay $n$ ra铆ces distintas. -- Notaci贸n equivalente: si $z' = r'(\cos \alpha' + i \sin \alpha')$ es una ra铆z, entonces: - $$ - r'^n = r, \quad n \alpha' = \alpha + 2k\pi - $$ - - +Podemos obtener n soluciones, que pueden ser repetidas, cada soluci贸n sera: $$ -<<<<<<< HEAD -\huge^{\mu}\sqrt{r_\alpha}=r'_{\alpha'} +\large R = \space^{n} \sqrt r +$$ +y su parte $\alpha$ sera +$$ +\large \alpha = \frac {\theta + 2\pi\cdot K} n,\quad donde \space K \{0, 1, 2...n-1\} $$ -tal que $r'^{\mu}=r$ $\mu\alpha'=\alpha(+k2\pi)$ # Exponencial de un numero complejo $$ @@ -103,7 +102,5 @@ Si $|z| = r \quad argz = \alpha$ $$ \huge z=re^{\alpha i} $$ -======= - r'^n = r, \quad n \alpha' = \alpha + 2k\pi - $$ ->>>>>>> 3f6ee7f (Fixed math) + +