diff --git a/src/Analysis1.typ b/src/Analysis1.typ index 25e3631..6c5067b 100644 --- a/src/Analysis1.typ +++ b/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 $ ]) ) diff --git a/src/Analysis_rewrite.typ b/src/Analysis_rewrite.typ index b1bee0b..52f673b 100644 --- a/src/Analysis_rewrite.typ +++ b/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 $), - grid.cell(colspan: 2, MathAlignLeft($ lim_(n->infinity) sqrt(n) = + infinity $)), [], - grid.cell(colspan: 2, MathAlignLeft($ lim_(n->infinity) k = k, k in RR $)), [], - grid.cell(colspan: 2, MathAlignLeft($ exp(x) = e^x = lim_(n->infinity) (1 + x/n)^n $)) + [], + MathAlignLeft($ lim_(n->infinity) k = k, k in RR $), + grid.cell(colspan: 2, MathAlignLeft($ exp(x) = e^x = lim_(n->infinity) (1 + x/n)^n $)), + MathAlignLeft($ lim_(n->infinity) sqrt(n) = + infinity $), + grid.cell(colspan: 2, MathAlignLeft($ 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)[ @@ -278,4 +337,52 @@ ] #colbreak() -] \ No newline at end of file +] + +#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) $ \ No newline at end of file