vault backup: 2025-11-11 17:01:13

.obsidian/workspace.json

@@ -62,9 +62,23 @@
               "icon": "lucide-file",
               "title": "Trigonometri"
             }
+          },
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+                "file": "Komplexa tal.md",
+                "mode": "source",
+                "source": false
+              },
+              "icon": "lucide-file",
+              "title": "Komplexa tal"
+            }
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+        "currentTab": 4
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@@ -224,10 +238,13 @@
       "obsidian-git:Open Git source control": false
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+  "active": "be47d5ede3a9176b",
   "lastOpenFiles": [
-    "Funktioner Forts.md",
+    "k2.png",
+    "k1.png",
     "Trigonometri.md",
+    "Komplexa tal.md",
+    "Funktioner Forts.md",
     "Grafer.md",
     "Funktioner.md",
     "f_inverse.png",

Komplexa tal.md

@@ -0,0 +1,24 @@
+- Komplexa tal
+	- **Def**: $x^2+1=0$ saknar reell lösning. Vi antar talet $i\notin\mathbb{R}$ löser ekvationen, d.v.s $i^2=-1$
+	- Mängd av komplexa talen: $\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}$
+	- Om $z=a+bi,a=Re(z)$ och $b=Im(z)$
+	- **Konjugat**: $z=a+bi\Rightarrow\bar{z}=a-bi$
+	- **Regler**:
+		- $\bar{\bar{z}}=z$
+		- $\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}$
+		- $\overline{z_1\times{z_2}}=\overline{z_1}\times{z_2}$
+	- **Absolut belopp**: $$\mid{z}\mid=\mid\overline{z}\mid=\sqrt{z\overline{z}}=\sqrt{a^2+b^2}\text{ om }z=a+bi$$
+	- **Triangelsformeln**: $\mid{z_1+z_2}\mid\leq\mid{z_1}\mid+\mid{z_2}\mid$
+	- **Ex**: $$\begin{align}z_1=2+3i\\z_2=2-i\\\\z_1+z_2=(2+3i)+(2-1)\\=4+2i\\\overline{z_1+z_2}=4-2i\\\overline{z_1}=2-3i,\;\overline{z_2}=2+i\\\overline{z_1}+\overline{z_2}=2-3i+2+i\\=3-2i\\\\z_1\times{z_2}=(2+3i)(2-i)\\=4-2i+6i-3i^2\\=4+4i+3\\=7+4i\\\overline{z_1\times{z_2}}=7-4i\\\overline{z_1}=2-3i,\;\overline{z_2}=2+i\\\overline{z_1}\times\overline{z_2}=(2-3i(2+i)\\=4+2i-6i-3i^2\\=4-2i+3\\=7-4i\end{align}$$
+	- **Ex 2**: $$\begin{align}z=a+bi\\\overline{z}=a-bi\\z\times\overline{z}=(a+bi)(a-bi)\\=a^2-\left(bi\right)^2\\=a^2-b^2i^2\\=a^2+b^2\end{align}$$
+	- **Ex 3**: $$\begin{align}\mid{z_1+z_2}\mid=\mid4+2i\mid\\=\sqrt{4^2+2^2}\\=\sqrt{16+4}=2\sqrt{5}\\\mid{z_1}\mid=\mid2+3i=\sqrt{2^2+3^2}\\=\sqrt{13}\\\mid{z_2}\mid=\mid2-i\mid=\sqrt{2^2+(-i)^2}=\sqrt{5}\end{align}$$
+	- **Ex 4**: $$\begin{align}\frac{z_1}{z_2}=\frac{2+3i}{2-i}\\=\frac{2+3i}{2-i}\times\frac{2+i}{2+i}\\=\frac{4+2i+6i+3i^2}{2^2-i^2}\\=\frac{1+8i}{5}\\=\frac{1}{5}+\frac{8}{5}i\end{align}$$
+- Grafer
+	- ![[k1.png]]
+	- ![[k2.png]]
+- Polär form
+	- **Eulers formel**: $e^{i\theta}=\cos\theta+i\sin\theta$
+	- Varje komplex tal $z=x+yi$ kan skrivas på pol'r form som $$z=re^{i\theta}$$ där $$r=\sqrt{x^2+y^2}$$ och $arg(z)=\theta$ är så att $$\cos\theta=\frac{x}{\sqrt{x^2+y^2}}\text{ och }\sin\theta=\frac{y}{x^2+y^2}$$
+	- **de Moivre**: $z=re^{i\theta}\Rightarrow z^n=r^ne^{in\theta}=r^n(\cos(n\theta)+i\sin(n\theta))$
+	- **Ex**: Lös $z^3=1+i\sqrt3$ $$\begin{align}r=\sqrt{1^2+\sqrt3^2}\\=\sqrt{1+3}\\=\sqrt{4}\\=2\\\\\end{align}$$
+	- **Ex 2**: $$\begin{align}z=-\frac{\sqrt3}{2}+\frac{1}{2}i\\z=ne^{i\theta}\\n=\sqrt{\left(-\frac{\sqrt{3}}{2}\right)^2+\left(\frac{1}{2}\right)^2}=1\\\theta\text{ är så att }\cos\theta=\frac{\sqrt{3}}{2},\sin\theta=\frac{1}{2}\\\text{En lösning}:\end{align}$$

Trigonometri.md

@@ -22,4 +22,13 @@
 	- $\sin\theta=\sin{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\\pi-a+2n\pi,n\in\mathbb{Z}\end{align}\right.$
 	- $\cos\theta=\cos{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\-a+2n\pi,n\in{Z}\end{align}\right.$
 	- $\tan\theta=\tan{a}\Leftrightarrow\theta=a+n\pi,n\in\mathbb{Z}$
-	- Ex: Solve $\sin(x+\frac{\pi}{6})=\frac{\sqrt{3}}{2}$ 
+	- Ex: Solve $\sin(x+\frac{\pi}{6})=\frac{\sqrt{3}}{2}$ 
+- Inverse trigonometric function
+	- **Def**: *$f(x)=\sin(x),x\in\left[\frac{-\pi}{2},\frac{\pi}{2}\right]$. Then $f$ is strictly increasing on $D_f$ and hence inverible. The fuction $\arcsin$ is defined as $$\arcsin(x)=f^{-1}(x)\text{ on }D_{arcsin}=R_f=[-1,1]$$*
+	- **Similarly**: *For $g(x)=\cos(x),x\in\left[0,\pi\right]$ (which is strictly decreasing) and $h(x)=\tan(x),x\in\left[\frac{-\pi}{2},\frac{\pi}{2}\right]$ (which is strictly increasing), the function $\arccos$ and $\arctan$ are defined as $$\begin{align}\arccos(x)=g^{-1}(x)\text{ on }D_{\arccos}=\left[-1,1\right]\\\arctan(x)=h^{-1}(x)\text{ on }D_{\arctan}=\mathbb{R}\end{align}$$*
+	- **Note**: *That the tanges $R_{\arcsin}=R_{\arctan}=\left[\frac{-\pi}{2},\frac{\pi}{2}\right]$ whereas $R_{\arccos}=\left[0,\pi\right]$*
+- Properties
+	- **Def**: $$\begin{align}\sin(\arcsin(x))=x\forall{x}\in\left[-1,1\right]\text{ | }\arcsin(sin(x))=x\text{ if }x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\\\cos(\arccos(x))=x\forall{x}\in\left[-1,1\right]\text{ | }\arccos(\cos(x))=x\text{ if }x\in\left[0,\pi\right]\\\tan(\arctan(x))=x\forall{x}\in\mathbb{R}\text{ | }\arctan(\tan(x))=x\text{ if }x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\end{align}$$
+	- **Complementary angles**: $$\arcsin(x)+\arccos(x)=\frac{pi}{2},\;\arctan(x)+\arccos(x)=\frac{\pi}{2}$$
+	- **Negatives**: *$\arcsin$ and $\arctan$ are odd functions. $$\begin{align}\arcsin(-x)=-\arcsin(x)\\\arccos(-x)=\pi-\arccos(x)\\\arctan(-x)=-\arctan(x)\end{align}$$*
+-