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src/Analysis1.typ

@@ -202,8 +202,6 @@
 
       1. $e^x = sum_(n=0)^infinity (x^n)/(n!)$
       2. $ln(x) = sum_(n=0)^infinity (-1)^n x^(n+1)$
-      3. $sin(x) = sum_(n=0)^infinity $
-      4. $cos(x) = sum_(n=0)^infinity $
   ])
 )
 

src/Analysis_rewrite.typ

@@ -87,6 +87,47 @@
     )
   ]
 
+  #bgBlock(fill: colorAllgemein)[
+    #subHeading(fill: colorAllgemein)[Trigonometrie]
+  ]
+
+    #bgBlock(fill: colorAllgemein)[
+    #subHeading(fill: colorAllgemein)[Sinus-Tabel]
+
+    #table(
+      inset: 1.5mm,
+      stroke: (thickness: 0.2mm),
+      columns: 4,
+      table.header(
+        [x], [deg], [cos(x)], [sin(x)]
+      ),
+      [$0$], [$0°$], [$1$], [$0$],
+      [$pi/6$], [$30°$], [$sqrt(3)/2$], [$1/2$],
+      [$pi/4$], [$45°$], [$sqrt(2)/2$], [$sqrt(2)/2$],
+      [$pi/3$], [$60°$], [$1/2$], [$sqrt(3)/2$],
+      [$pi/2$], [$90°$], [$0$], [$1$],
+      [$2/3pi$], [$120°$], [$-1/2$], [$sqrt(3)/2$],
+      [$3/4pi$], [$135°$], [$-sqrt(2)/2$], [$sqrt(2)/2$],
+      [$5/6pi$], [$150°$], [$-sqrt(3)/2$], [$1/2$],
+      [$pi$], [$180°$], [$-1$], [$0$],
+      [$3/2pi$], [$270°$], [$0$], [$-1$],
+      [$2pi$], [$360°$], [$1$], [$0$]
+    )
+  ]
+
+  #bgBlock(fill: colorAllgemein)[
+    #subHeading(fill: colorAllgemein)[Complexe Zahlen]
+    $z = r dot e^(phi i) = r (cos(phi) + i sin(phi))$
+
+    $z^n = r^n dot e^(phi i dot n) = r^n (cos(n phi) + i sin(n phi))$
+
+    #grid(
+      columns: (1fr, 1fr),
+      [$ sin(x) = (e^(i x) - e^(-i x))/(2i) $],
+      [$ cos(x) = (e^(i x) + e^(-i x))/(2) $]
+    )
+  ]
+
   #bgBlock(fill: colorFolgen)[
     #subHeading(fill: colorFolgen)[Folgen]
     $ lim_(x -> infinity) a_n $
@@ -170,11 +211,16 @@
       row-gutter: 2mm,
       align: bottom,
       MathAlignLeft($ lim_(n->infinity) 1/n = 0 $),
-      MathAlignLeft($ lim_(n->infinity) q^n = 0 $),
+      [],
-      MathAlignLeft($ lim_(n->infinity) q^n = 0 $),
+      MathAlignLeft($ lim_(n->infinity) k = k, k in RR $),
-      grid.cell(colspan: 2, MathAlignLeft($ lim_(n->infinity) sqrt(n) = + infinity $)), [],
+      grid.cell(colspan: 2, MathAlignLeft($ exp(x) =  e^x = lim_(n->infinity) (1 + x/n)^n $)), 
-      grid.cell(colspan: 2, MathAlignLeft($ lim_(n->infinity) k = k, k in RR $)), [],
+      MathAlignLeft($ lim_(n->infinity) sqrt(n) = + infinity $),
-      grid.cell(colspan: 2, MathAlignLeft($ exp(x) =  e^x = lim_(n->infinity) (1 + x/n)^n $))
+      grid.cell(colspan: 2, MathAlignLeft($ lim_(n->infinity) q^n = cases(
+        0 &abs(q),
+        1 &q = 1,
+        plus.minus infinity &q < -1,
+        plus infinity #h(5mm) &q > 1
+      ) $)), []
     )
   ]
 
@@ -195,6 +241,19 @@
     #subHeading(fill: colorReihen)[Potenzreihen]
   ]
 
+  #bgBlock(fill: colorReihen)[
+    #subHeading(fill: colorReihen)[Bekannte Reihen]
+    *Geometrische Reihe:* $sum_(n=0)^infinity q^n$
+    - konvergent $abs(q) < 1$, divergent  $abs(q) >= 1$
+    - Grenzwert: (Muss $n=0$) $=1/(1-q)$
+
+    *Harmonische Reihe:* $sum_(n=0)^infinity 1/n = +infinity$
+    
+    *Andere*
+    - $e^x = limits(sum)_(n=0)^infinity (x^n)/(n!)$
+    - $ln(x) = limits(sum)_(n=0)^infinity (-1)^n x^(n+1)$
+  ]
+
   #colbreak()
 
   #bgBlock(fill: colorAbleitung)[
@@ -279,3 +338,51 @@
 
   #colbreak()
 ]
+
+#pagebreak()
+
+== Folgen in $CC$
+
+$z_n in C: lim z_n  <=> lim abs(z_n -> infinity) = 0$
+
+Alle folgen regelen gelten
+
+Complexe Folge kann man in Realteil und Imag zerlegen
+
+z.B.
+
+$z_n = z^n z in CC$
+
+$z = abs(z) dot e^(i phi) = abs(z)^n$
+
+== Reihen in $CC$
+
+Fast alles gilt auch.
+
+Bis auf Leibnitzkriterium weil es keine Monotonie gibt
+
+Geometrische Reihe gilt.
+
+Exponential funktion
+
+#MathAlignLeft($ e^z = lim_(n -> infinity) (1 + z/n)^n = sum_(n=0)^infinity (z^n)/(n!) space z in CC $)
+
+Vorsicht: $(b^a)^n = b^(a dot c)$
+
+Potenzreihen: Eine Fn der form:
+
+#MathAlignLeft($ P(z) = sum^(infinity)_(n=0) a_n dot (z - z_0)^n space z, z_0 in CC $)
+
+=== Satz
+
+Konvergenz Radius $R = [0, infinity)$$$
+
+1. $R = 0$ Konvergiet nur bei $z = 0$
+
+2. $R in R : cases(
+  z in CC &abs(z - z_0) < R &: "abs Konvergent",
+  z in CC &abs(z - z_0) = R &: "keine Ahnung",
+  z in CC &abs(z - z_0) > R &: "Divergent"
+)$
+
+$ R = limsup_(n -> infinity) $