From 672968bdfbfc0983390aad9cdcffb3c39a408295 Mon Sep 17 00:00:00 2001
From: raj_mathe <raj.dahya@math.uni-leipzig.de>
Date: Thu, 15 Dec 2022 07:35:09 +0100
Subject: [PATCH] master > master:  notebook - wk10 - cleanup

---
 notebooks/week-10.ipynb | 56 +++++++++++++++++++++++++----------------
 1 file changed, 34 insertions(+), 22 deletions(-)

diff --git a/notebooks/week-10.ipynb b/notebooks/week-10.ipynb
index 825f8c5..604ff1a 100644
--- a/notebooks/week-10.ipynb
+++ b/notebooks/week-10.ipynb
@@ -101,26 +101,28 @@
     "Für $i\\in\\{0,1,\\ldots,n-1\\}$ ist die $i$-te von $\\mathcal{F}_{n}\\,u_{n,k}$\n",
     "\n",
     "$$\n",
+    "    \\begin{array}{rcl}\n",
     "    (\\mathcal{F}_{n}\\,u_{n,k})_{i}\n",
-    "    = \\sum_{j=0}^{n-1}\n",
-    "        (\\mathcal{F}_{n})_{ij}\\,(u_{n,k})_{j}\n",
-    "    = \\sum_{j=0}^{n-1}\n",
+    "    &= &\\displaystyle\\sum_{j=0}^{n-1}\n",
+    "        (\\mathcal{F}_{n})_{ij}\\,(u_{n,k})_{j}\\\\\n",
+    "    &= &\\displaystyle\\sum_{j=0}^{n-1}\n",
     "        \\tfrac{1}{\\sqrt{n}}\n",
     "        \\exp(-\\imath\\tfrac{2\\pi\\,ij}{n})\n",
     "        \\cdot\n",
     "        \\tfrac{1}{\\sqrt{n}}\n",
-    "        \\exp(\\imath\\tfrac{2\\pi\\,kj}{n})\n",
-    "    = \\frac{1}{n}\\sum_{j=0}^{n-1}\n",
+    "        \\exp(\\imath\\tfrac{2\\pi\\,kj}{n})\\\\\n",
+    "    &= &\\frac{1}{n}\\displaystyle\\sum_{j=0}^{n-1}\n",
     "        \\underbrace{\n",
     "            \\exp(\\imath\\tfrac{2\\pi\\,(k-i)j}{n})\n",
-    "        }_{= r^{j}}\n",
+    "        }_{= r^{j}}\\\\\n",
+    "    \\end{array}\n",
     "$$\n",
     "\n",
-    "wobei $r \\colonequals \\exp(\\imath\\tfrac{2\\pi\\,(k-i)}{n})$.\n",
+    "wobei $r := \\exp(\\imath\\tfrac{2\\pi\\,(k-i)}{n})$.\n",
     "Wir haben, dass\n",
     "    $r = 1$\n",
     "    $\\Leftrightarrow$\n",
-    "        $\\tfrac{k-i}{n} \\in \\Z$\n",
+    "        $\\tfrac{k-i}{n} \\in \\mathbb{Z}$\n",
     "    $\\Leftrightarrow$\n",
     "        $i = k$,\n",
     "da $0\\leq i,k \\leq n-1$.\n",
@@ -130,23 +132,34 @@
     "Aus der o.s. Berechnung erhält man also\n",
     "\n",
     "$$\n",
+    "    \\begin{array}{rcl}\n",
     "    (\\mathcal{F}_{n}\\,u_{n,k})_{i}\n",
-    "    = \\tfrac{1}{n}\\displaystyle\\sum_{j=0}^{n-1}r^{j}\n",
-    "    = \\tfrac{1}{n}\n",
-    "        \\begin{cases}\n",
+    "    &= &\\tfrac{1}{n}\\displaystyle\\sum_{j=0}^{n-1}r^{j}\\\\\n",
+    "    &= &\\tfrac{1}{n}\n",
+    "        \\left\\{\n",
+    "        \\begin{array}{lcl}\n",
     "            \\displaystyle\\sum_{j=0}^{n-1}1 &: &i = k\\\\\n",
-    "            \\frac{1 - r^{n}}{1 - r} &: &\\text{sonst}\\\\\n",
-    "        \\end{cases}\n",
-    "    = \\tfrac{1}{n}\n",
-    "        \\begin{cases}\n",
+    "            \\dfrac{1 - r^{n}}{1 - r} &: &\\text{sonst}\\\\\n",
+    "        \\end{array}\n",
+    "        \\right.\\\\\n",
+    "    &= &\\tfrac{1}{n}\n",
+    "        \\left\\{\n",
+    "        \\begin{array}{lcl}\n",
     "            n &: &i = k\\\\\n",
-    "            \\frac{1 - 1}{1 - r} &: &\\text{sonst}\\\\\n",
-    "        \\end{cases}\n",
-    "    = \\delta_{ik}.\n",
+    "            \\dfrac{1 - 1}{1 - r} &: &\\text{sonst}\\\\\n",
+    "        \\end{array}\n",
+    "        \\right.\\\\\n",
+    "    &= &\\left\\{\n",
+    "        \\begin{array}{lcl}\n",
+    "            1 &: &i = k\\\\\n",
+    "            0 &: &\\text{sonst}\\\\\n",
+    "        \\end{array}\n",
+    "        \\right.\\\\\n",
+    "    &= &\\delta_{ik}.\\\\\n",
+    "    \\end{array}\n",
     "$$\n",
     "\n",
-    "Darum $\n",
-    "    \\mathcal{F}_{n}\\,u_{n,k}\n",
+    "Darum $\\mathcal{F}_{n}\\,u_{n,k}\n",
     "    = \\left(\\begin{matrix}\n",
     "        0\\\\\n",
     "        0\\\\\n",
@@ -163,8 +176,7 @@
     "        \\\\\n",
     "        \\\\\n",
     "    \\end{matrix}\n",
-    "    = \\mathbf{e}_{k}\n",
-    "$.\n"
+    "    = \\mathbf{e}_{k}$.\n"
    ]
   },
   {