moved more around
All checks were successful
Build Typst PDFs (Docker) / build-typst (push) Successful in 11s
All checks were successful
Build Typst PDFs (Docker) / build-typst (push) Successful in 11s
This commit is contained in:
alexander
546
src/cheatsheets/Analysis1.typ
Normal file
546
src/cheatsheets/Analysis1.typ
Normal file
@@ -0,0 +1,546 @@
|
||||
#import "../lib/common_rewrite.typ" : *
|
||||
#import "@preview/mannot:0.3.1"
|
||||
|
||||
#set page(
|
||||
paper: "a4",
|
||||
margin: (
|
||||
bottom: 10mm,
|
||||
top: 5mm,
|
||||
left: 5mm,
|
||||
right: 5mm
|
||||
),
|
||||
flipped:true,
|
||||
footer: context [
|
||||
#grid(
|
||||
align: center,
|
||||
columns: (1fr, 1fr, 1fr),
|
||||
[#align(left, datetime.today().display("[day].[month].[year]"))],
|
||||
[#align(center, counter(page).display("- 1 -"))],
|
||||
[#align(right, image("../images/cc0.png", height: 5mm,))]
|
||||
)
|
||||
],
|
||||
)
|
||||
|
||||
#place(top+center, scope: "parent", float: true, heading(
|
||||
[Analysis 1 (IE)]
|
||||
))
|
||||
|
||||
#let SeperatorLine = line(length: 100%, stroke: (paint: black, thickness: 0.3mm))
|
||||
#let MathAlignLeft(e) = {
|
||||
align(left, block(e))
|
||||
}
|
||||
|
||||
#let colorAllgemein = color.hsl(105.13deg, 92.13%, 75.1%)
|
||||
#let colorFolgen = color.hsl(202.05deg, 92.13%, 75.1%)
|
||||
#let colorReihen = color.hsl(280deg, 92.13%, 75.1%)
|
||||
#let colorAbleitung = color.hsl(356.92deg, 92.13%, 75.1%)
|
||||
#let colorIntegral = color.hsl(34.87deg, 92.13%, 75.1%)
|
||||
|
||||
|
||||
#columns(4, gutter: 2mm)[
|
||||
#bgBlock(fill: colorAllgemein)[
|
||||
#subHeading(fill: colorAllgemein)[Allgemeins]
|
||||
#grid(
|
||||
columns: (auto, auto),
|
||||
row-gutter: 2mm,
|
||||
column-gutter: 3mm,
|
||||
[Dreiecksungleichung], [
|
||||
$abs(x + y) <= abs(x) + abs(y)$ \
|
||||
$abs(abs(x) - abs(y)) <= abs(x - y)$
|
||||
],
|
||||
[Cauchy-Schwarz-Ungleichung], [
|
||||
$abs(x dot y) <= abs(abs(x) dot abs(y))$
|
||||
],
|
||||
[Geometrische Summenformel], [
|
||||
#MathAlignLeft($ limits(sum)_(k=1)^(n) k = (n(n+1))/2 $)
|
||||
],
|
||||
[Bernoulli-Ungleichung ], [
|
||||
$(1 + a)^n x in RR >= 1 + n a$
|
||||
],
|
||||
[Binomialkoeffizient], [
|
||||
$binom(n, k) = (n!)/(k!(n-k)!)$
|
||||
],
|
||||
[Binomische Formel], [
|
||||
#MathAlignLeft($ (a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $)
|
||||
],
|
||||
[Fakultäten], [$ 0! = 1! = 1 $],
|
||||
|
||||
[Gausklammer], [
|
||||
$floor(x) = text("floor")(x)$ \
|
||||
$ceil(x) = text("ceil")(x)$
|
||||
],
|
||||
[Bekannte Werte], [
|
||||
$e approx 2.71828$ ($2 < e < 3$) \
|
||||
$pi approx 3.14159$ ($3 < pi < 4$)
|
||||
]
|
||||
)
|
||||
]
|
||||
|
||||
#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) $]
|
||||
)
|
||||
#subHeading(fill: colorAllgemein)[Trigonmetrie]
|
||||
*Additionstheorem* \
|
||||
$sin(x+y) = cos(x)sin(y) + sin(x)cos(y)$ \
|
||||
$cos(x+y) = cos(x)cos(y) - sin(x)sin(y)$ \
|
||||
$tan(x) + tan(y) = (tan(a) + tan(b))/(1 - tan(a) tan(b))$ \
|
||||
$arctan(x) + arctan(y) = arctan((x+y)/(1 - x y))$ \
|
||||
|
||||
*Doppelwinkel Formel* \
|
||||
$cos(2x) = cos^2(x) - sin^2(x)$ \
|
||||
$sin(2x) = 2sin(x)cos(x)$
|
||||
|
||||
#grid(
|
||||
gutter: 5mm,
|
||||
columns: (auto, auto),
|
||||
[$cos^2(x) = (1 + cos(2x))/2$],
|
||||
[$sin^2(x) = (1 - cos(2x))/2$]
|
||||
)
|
||||
|
||||
$cos^2(x) + sin^2(x) = 1$
|
||||
git config pull.rebase falsegit config pull.rebase false
|
||||
#grid(
|
||||
gutter: 5mm,
|
||||
columns: (auto, auto),
|
||||
[$cos(-x) = cos(x)$],
|
||||
[$sin(-x) = -sin(x)$],
|
||||
)
|
||||
|
||||
Subsitution mit Hilfsvariable
|
||||
|
||||
#grid(
|
||||
gutter: 5mm,
|
||||
row-gutter: 3mm,
|
||||
columns: (auto, auto),
|
||||
[$tan(x)=sin(x)/cos(x)$],
|
||||
[$cot(x)=cos(x)/sin(x)$],
|
||||
[$tan(x)=-cot(x + pi/2)$],
|
||||
[$cot(x)=-tan(x + pi/2)$],
|
||||
[$cos(x - pi/2) = sin(x)$],
|
||||
[$sin(x + pi/2) = cos(x)$],
|
||||
)
|
||||
$sin(x)cos(y) = 1/2sin(x - y) + 1/2sin(x + y)$
|
||||
|
||||
Für $x in [-1, 1]$ \
|
||||
$arcsin(x) = -arccos(x) - pi/2 in [-pi/2, pi/2]$ \
|
||||
$arccos(x) = -arcsin(x) + pi/2 in [0, pi]$
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorFolgen)[
|
||||
#subHeading(fill: colorFolgen)[Folgen]
|
||||
$ lim_(x -> infinity) a_n $
|
||||
|
||||
*Beschränkt:* $exists k in RR$ sodass $abs(a_n) <= k$
|
||||
- Beweiße: durch Induktion
|
||||
- Beweiße: Hat min. ein konvergent Teilefolge
|
||||
- (Beweiße: Ungleichung $abs(a_n) <= k$)
|
||||
|
||||
*Monoton fallend/steigended*
|
||||
- Beweise: Induktion
|
||||
#grid(columns: (1fr, 1fr),
|
||||
gutter: 1mm,
|
||||
row-gutter: 2mm,
|
||||
align(top+center, [*Fallend*]), align(top+center, [*Steigend*]),
|
||||
[$ a_(n+1) <= a_(n) $],
|
||||
[$ a_(n+1) >= a_(n) $],
|
||||
[$ a_(n+1)/a_(n) < 1 $],
|
||||
[$ a_(n+1)/a_(n) > 1 $],
|
||||
)
|
||||
|
||||
*Konvergentz Allgemein*
|
||||
$ lim_(n -> infinity) a_n = a $
|
||||
|
||||
$forall epsilon > 0 space exists n_epsilon in NN$ sodass \
|
||||
- Konvergent $-> a$: $a_n in [a - epsilon, a + epsilon] $
|
||||
- Divergent $-> infinity$: $a_n in [epsilon, infinity) $
|
||||
- Divergent $-> infinity$: $a_n in (-infinity, epsilon) $
|
||||
|
||||
$space forall n > n_epsilon$
|
||||
|
||||
*Konvergentz Häufungspunkte*
|
||||
- $a_n -> a <=>$ Alle Teilfolgen $-> a$
|
||||
|
||||
*Konvergenz Beweißen*
|
||||
- Monoton UND Beschränkt $=>$ Konvergenz
|
||||
NICHT Umgekehert
|
||||
- (Cauchyfolge \
|
||||
$forall epsilon > 0 space exists n_epsilon in NN space$ sodass \
|
||||
$forall m,n >= n_epsilon : abs(a_n - a_m) < epsilon$ \
|
||||
Cauchyfolge $=>$ Konvergenz)
|
||||
- $a_n$ unbeschränkt $=>$ divergenz
|
||||
|
||||
*Konvergent Grenzwert finden*
|
||||
- Von Bekannten Ausdrücken aufbauen
|
||||
- Fixpunk Gleichung: $a = f(a)$ \
|
||||
für rekusive $a_(n+1) = f(a_n)$ (Zu erst machen!)
|
||||
- Bernoulli-Ungleichung Folgen der Art $(a_n)^n$: \
|
||||
$(1 + a)^n >= 1 + n a$
|
||||
- Sandwitchtheorem:\
|
||||
$b_n -> x$: $a_n <= b_n <= c_n$, wenn $a_n -> x$ und $c_n -> x$ \
|
||||
$b_n -> -infinity$: $b_n <= c_n$, wenn $c_n -> -infinity$ \
|
||||
$b_n -> +infinity$: $c_n <= b_n $, wenn $a_n -> +infinity$
|
||||
- Zwerlegen in Konvergente Teil folgen \
|
||||
(Vorallem bei $(-1)^n dot a_n$)
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorFolgen)[
|
||||
#subHeading(fill: colorFolgen)[Konvergent Folge Regeln]
|
||||
#grid(
|
||||
columns: (auto, auto),
|
||||
align: bottom,
|
||||
gutter: 2mm,
|
||||
[$ lim_(n->infinity) (a_n + b_n) = a + b $],
|
||||
grid.cell(
|
||||
rowspan: 2,
|
||||
[$ lim_(n->infinity) (a_n / b_n) = a / b $ für ($b != 0$)],
|
||||
),
|
||||
MathAlignLeft($ lim_(n->infinity) (a_n dot b_n) = a dot b $),
|
||||
MathAlignLeft($ lim_(n->infinity) sqrt(a_n) = sqrt(a) $),
|
||||
MathAlignLeft($ lim_(n->infinity) abs(a_n) = abs(a) $),
|
||||
MathAlignLeft($ lim_(n->infinity) c dot a_n = c dot lim_(n->infinity) a_n $),
|
||||
)
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorFolgen)[
|
||||
#subHeading(fill: colorFolgen)[Bekannte Folgen]
|
||||
#grid(
|
||||
columns: (auto, auto, auto),
|
||||
column-gutter: 4mm,
|
||||
row-gutter: 2mm,
|
||||
align: bottom,
|
||||
MathAlignLeft($ lim_(n->infinity) 1/n = 0 $),
|
||||
[],
|
||||
MathAlignLeft($ 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
|
||||
) $)), []
|
||||
)
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorFolgen)[
|
||||
#subHeading(fill: colorFolgen)[Teilfolgen]
|
||||
$ a_k subset a_n space (text("z.B") k= 2n + 1) $
|
||||
- Index muss streng monoton steigen!
|
||||
- Beschränkte $a_n => text("min eine konvergente") a_k$
|
||||
- Konvergenz-Werte von $a_k$ sind Häufungspunkte
|
||||
- Wenn alle $a_k$ gegen #underline([genau eine]) Häufungspunk konverigiert $<=> a_n$ konvergent
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorReihen)[
|
||||
#subHeading(fill: colorReihen)[Reihen]
|
||||
$limits(lim)_(n->infinity) a_n != 0 => limits(sum)_(n=1)^infinity a_n$ konverigiert NICHT \
|
||||
|
||||
- *Absolute Konvergenz* \
|
||||
$limits(sum)_(n=1)^infinity abs(a_n) = a => limits(sum)_(n=1)^infinity a_n$ konvergent
|
||||
|
||||
|
||||
|
||||
- *Partialsummen* \
|
||||
ALLE Partialsummen von $limits(sum)_(k=1)^infinity abs(a)$ beschränkt\
|
||||
$=>$ _Absolute Konvergent_
|
||||
|
||||
- *(Cauchy-Kriterium)*\
|
||||
konvergent wenn $forall epsilon > 0 space exists n_epsilon in NN$ \
|
||||
sodass $abs(s_n - s_m) = abs(limits(sum)_(k=m+1)^(n)) < epsilon space$ \
|
||||
$forall n_epsilon < m < n $
|
||||
|
||||
- *Leibnitzkriterium* \
|
||||
Alternierend + Nullfolge \
|
||||
$=> limits(sum)_(n=1)^infinity (-1)^n dot a_n$ konvergent
|
||||
|
||||
- *Vergleichskriterium* \
|
||||
$a_n, b_n : abs(a_n) <= b_n space forall n in NN > N_0, N_0 in NN$
|
||||
1. $limits(sum)_(n=0)^infinity b_n$ konvergent $=> limits(sum)_(n=0)^infinity abs(a_n)$ konvergent \
|
||||
Suche $b_n$ für Konvergenz
|
||||
2. $limits(sum)_(n=0)^infinity abs(a_n)$ divergent $=> limits(sum)_(n=0)^infinity b_n$ divergent \
|
||||
Suche $abs(a_n)$ für Divergenz
|
||||
|
||||
Nützlich:
|
||||
- Dreiecksungleichung
|
||||
- $forall space n > N_0 in NN space exists k,q in RR$ \
|
||||
sodass $q > 1$: $n^k <= q^n$ (Potenz stärker Polynom)
|
||||
|
||||
- *Quotientenkriterium und Wurzelkriterium*
|
||||
1. $rho = lim_(n -> infinity) abs((a_(n+1))/(a_n)) $
|
||||
2. $rho = lim_(n -> infinity) root(n, abs(a_(n+1))) $ \
|
||||
|
||||
divergent: $rho > 1$, keine Aussage $rho = 1$, konvergent $rho < 1$
|
||||
|
||||
- *Geometrische Reihe*
|
||||
$limits(sum)_(n=0)^infinity q^n$
|
||||
- konvergent $abs(q) < 1$, divergent $abs(q) >= 1$
|
||||
- Grenzwert: (Muss $n=0$) $=1/(1-q)$
|
||||
- *Harmonische Reihe* $limits(sum)_(n=0)^infinity 1/n = +infinity$
|
||||
|
||||
- *Reihendarstellungen*
|
||||
1. $e^x = limits(sum)_(n=0)^infinity (x^n)/(n!)$
|
||||
2. $ln(x) = limits(sum)_(n=0)^infinity (-1)^n x^(n+1)$
|
||||
3. $sin(x) = limits(sum)_(n=0)^infinity $
|
||||
4. $cos(x) = limits(sum)_(n=0)^infinity $
|
||||
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorReihen)[
|
||||
#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)[
|
||||
#subHeading(fill: colorAbleitung)[Funktionen]
|
||||
Sei $f : [a,b] -> RR$, stetig auf $x in [a,b]$
|
||||
- *Zwischenwertsatz* \
|
||||
$=> forall y in [f(a), f(b)] exists text("min. ein") x in [a,b] : f(x) = y$ \
|
||||
_Beweiß für mindest. n Nst_
|
||||
- *Satze von Rolle* \
|
||||
diffbar $x in (a,b)$\
|
||||
$f(a) = f(b) => exists text("min. ein") x_0 in (a,b) : f'(x_0) = 0$
|
||||
_Beweiß für max. n Nst, durchWiederspruchsbweiß mit $f(a)=f(b)=0$ und Wiederholte Ableitung_
|
||||
|
||||
- *Mittelwertsatz*
|
||||
diffbar $x in (a,b)$ \
|
||||
$=> exists x_0 : f'(x_0)=(f(b) - f(a))/(a-b)$
|
||||
|
||||
- *Monotonie* \
|
||||
$x in I : f'(x) < 0$: Streng monoton steigended \
|
||||
$x_0,x_1 in I, x_0 < x_1 => f(x_0) < f(x_1)$ \
|
||||
(Analog bei (streng ) steigned/fallended)
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorAbleitung)[
|
||||
#subHeading(fill: colorAbleitung)[Stetigkeit]
|
||||
*Allgemein*
|
||||
|
||||
$f(x)$ ist stetig wenn: \
|
||||
$ limits(lim)_(x->x_0-) f(x) = limits(lim)_(x->x_0+) f(x) = f(x_0) $ \
|
||||
$x in DD$ Beachten! Definitionslücken $!=$ unstätig \
|
||||
Definition gilt auch für $I subset RR$
|
||||
|
||||
*Regeln*
|
||||
|
||||
$f(x),g(x)$ seinen stetig dann sind auch Stetig:
|
||||
|
||||
#grid(columns: (auto, auto, auto, auto, auto),
|
||||
column-gutter: 4mm,
|
||||
row-gutter: 2mm,
|
||||
$f(x) + g(x)$, $f circle.small g$, $alpha dot f(x)$,
|
||||
$f(x)/g(x)$, $f(x) dot g(x)$
|
||||
)
|
||||
|
||||
*Bekannte Funktion*
|
||||
#table(
|
||||
columns: (1fr, 1fr),
|
||||
table.header(
|
||||
[*Stetig*], [*Nicht Stetig*]
|
||||
),
|
||||
stroke: (x, y) => (x: 0mm, y: 0.2mm),
|
||||
[
|
||||
- Polynome, gebrochen Rationale Fn
|
||||
- $floor(x),ceil(x)$ für $x in RR without ZZ$
|
||||
- Betrags Funktion
|
||||
- $sin, cos, tan$
|
||||
],
|
||||
[
|
||||
- Stufenfunktion
|
||||
- Fall Unterscheidungen
|
||||
- $floor(x),ceil(x)$ für $x in RR$
|
||||
]
|
||||
)
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorAbleitung)[
|
||||
#subHeading(fill: colorAbleitung)[Ableitung]
|
||||
*Differenzierbarkeit*
|
||||
- $f(x)$ ist an der Stelle $x_0 in DD$ diffbar wenn \
|
||||
#MathAlignLeft($ f'(x_0) = lim_(x->x_0 plus.minus) (f(x_0 + h - f(x_0))/h) $)
|
||||
- $f(x)$ diffbar $=>$ $f(x)$ stetig
|
||||
- Tangente an $x_0$: $f(x_0) + f'(x_0)(x - x_0)$
|
||||
- Beste #underline([linear]) Annäherung
|
||||
- Tangente $t(x)$ von $f(x)$ an der Stelle $x_0$: $ lim_(x->0) (f(x) - f(x_0))/(x-x_0) -f'(x_0) =0 $
|
||||
|
||||
*Ableitung Regeln*
|
||||
|
||||
#grid(
|
||||
row-gutter: 3mm,
|
||||
columns: (1fr, 1fr),
|
||||
grid.cell(
|
||||
colspan: 2,
|
||||
[$f(x) + g(x) : f'(x) + g'(x) $]
|
||||
),
|
||||
grid.cell(
|
||||
colspan: 2,
|
||||
[$f(x) dot g(x) : f'(x)g(x) + f(x)g'(x) $]
|
||||
),
|
||||
grid.cell(
|
||||
colspan: 2,
|
||||
[#MathAlignLeft($ f(x)/g(x) : (f'(x)g(x) - f(x)g'(x))/(g(x)^2) $)]
|
||||
),
|
||||
[$f(x) = c : f'(x) = 0$],
|
||||
[$c dot f(x) : c dot f'(x)$],
|
||||
[$(x^(-n)) n in NN : n x^(n-1)$],
|
||||
[$e^(x) : e^(x)$],
|
||||
)
|
||||
- Kettenregel: $f(g(x)) : f'(g(x)) dot g'(x)$
|
||||
],
|
||||
|
||||
#block([
|
||||
#set text(size: 10pt)
|
||||
#table(
|
||||
align: horizon,
|
||||
columns: (1fr, 1fr, 1fr),
|
||||
table.header([*$F(x)$*], [*$f(x)$*], [*$f'(x)$*]),
|
||||
row-gutter: 1mm,
|
||||
fill: (x, y) => if x == 0 { color.hsl(180deg, 89.47%, 88.82%) }
|
||||
else if x == 1 { color.hsl(180deg, 100%, 93.14%) } else
|
||||
{ color.hsl(180deg, 81.82%, 95.69%) },
|
||||
[$1/(q + x) x^(q+1)$], [$x^q$], [$q x^(q-1)$],
|
||||
[$ln abs(x)$], [$1/x$], [$-1/x^2$],
|
||||
[$x ln(a x) - x$], [$ln(a x)$], [$1 / x$],
|
||||
[$2/3 sqrt(a x^3)$], [$sqrt(a x)$], [$a/(2 sqrt(a x))$],
|
||||
[$e^x$], [$e^x$], [$e^x$],
|
||||
[$a^x/ln(a)$], [$a^x$], [$a^x ln(a)$],
|
||||
|
||||
[$x arcsin(x) + sqrt(1 - x^2)$],
|
||||
[$arcsin(x)$], [$1/sqrt(1 - x^2)$],
|
||||
|
||||
[$x arccos(x) - sqrt(1 - x^2)$],
|
||||
[$arccos(x)$], [$-1/sqrt(1 - x^2)$],
|
||||
|
||||
[$x arctan(x) - 1/2 ln abs(1 + x^2)$],
|
||||
[$arctan(x)$], [$1/(1 + x^2)$],
|
||||
|
||||
[$x op("arccot")(x) + \ 1/2 ln abs(1 + x^2)$],
|
||||
[$op("arccot")(x)$], [$-1/(1 + x^2)$],
|
||||
|
||||
[$x op("arsinH")(x) + \ sqrt(1 + x^2)$],
|
||||
[$op("arsinH")(x)$], [$1/sqrt(1 + x^2)$],
|
||||
|
||||
[$x op("arcosH")(x) + \ sqrt(1 + x^2)$],
|
||||
[$op("arcosH")(x)$], [$1/sqrt(x^2-1)$],
|
||||
|
||||
[$x op("artanH")(x) + \ 1/2 ln(1 - x^2)$],
|
||||
[$op("artanH")(x)$], [$1/(1 - x^2)$],
|
||||
)
|
||||
])
|
||||
|
||||
|
||||
#bgBlock(fill: colorIntegral, [
|
||||
#subHeading(fill: colorIntegral, [Integral])
|
||||
|
||||
Summen: $integral f(x) + g(x) d x = integral f(x) d x + integral g(x)$
|
||||
|
||||
Vorfaktoren: $integral lambda f(x) d x = lambda f(x) d x$
|
||||
|
||||
*Partial Integration*
|
||||
|
||||
$integral u(x) dot v'(x) d x = u(x)v(x) - integral u'(x) dot v(x)$
|
||||
|
||||
*Subsitution*
|
||||
|
||||
$integral_(x_0)^(x_1) f\(underbrace(g(x), "t")\) dot g'(x) d x$
|
||||
|
||||
1. Ersetzung: $ d x := d t dot 1/(g'(x))$ und $t := g(x)$
|
||||
2. Grenzen: $t_0 = g(x_0)$, $t_1 = g(x_1)$
|
||||
3. $x$-kürzen sich weg
|
||||
])
|
||||
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorAllgemein, [
|
||||
#subHeading(fill: colorAllgemein, [Sin-Table])
|
||||
#sinTable
|
||||
])
|
||||
|
||||
#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) $
|
||||
#bgBlock(fill: colorIntegral, [
|
||||
#subHeading(fill: colorIntegral, [Integral])
|
||||
|
||||
Summen: $integral f(x) + g(x) d x = integral f(x) d x + integral g(x)$
|
||||
|
||||
Vorfaktoren: $integral lambda f(x) d x = lambda f(x) d x$
|
||||
|
||||
*Partial Integration*
|
||||
|
||||
$integral u(x) dot v'(x) d x = u(x)v(x) - integral u'(x) dot v(x)$
|
||||
|
||||
*Subsitution*
|
||||
|
||||
$integral_(x_0)^(x_1) f\(underbrace(g(x), "t")\) dot g'(x) d x$
|
||||
|
||||
1. Ersetzung: $ d x := d t dot 1/(g'(x))$ und $t := g(x)$
|
||||
2. Grenzen: $t_0 = g(x_0)$, $t_1 = g(x_1)$
|
||||
3. $x$-kürzen sich weg
|
||||
])
|
||||
|
||||
Reference in New Issue
Block a user