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Formelsammlung Mathe 1
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Der Inhalt beschränkt sich auf die wichtigsten Themen für Mathe 1 im ersten Semester im Studiengang Wirtschaftsingenieurwesen an der DHBW Stuttgart.
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1. Komplexe Zahlen: 1.5.2 Trigonometrisch/polar zurück in kartesisch𝑧 = |𝑧| ⋅ (cos 𝜑 + 𝑖 ⋅ sin 𝜑) → 𝑧 = 𝑎 + 𝑏𝑖
1.1 Komplexe Zahl / konjungiert komplexe Zahl mit 𝑎 = |𝑧| ∙ cos 𝜑 und 𝑏 = |𝑧| ∙ sin 𝜑
𝑧 = 𝑎 + 𝑏𝑖 𝑧̅ = 𝑎 − 𝑏𝑖 √−1 = ±𝑖 𝑖 2 = −1 1.6 Rechnen mit Komplexen Zahlen
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1.6.1 Addition: 𝑧1 + 𝑧2 = (𝑎 + 𝑏𝑖) + (𝑐 + ⅆ𝑖) = (𝑎 + 𝑐) + (𝑏 + ⅆ)𝑖
1.2 Gleichheit zweier Komplexer Zahlen
1.6.2 Subtraktion: 𝑧1 − 𝑧2 = (𝑎 + 𝑏𝑖) − (𝑐 + ⅆ𝑖) = (𝑎 − 𝑐) + (𝑏 − ⅆ)𝑖
𝑧1 = 𝑥1 + 𝑖𝑦1 und 𝑧𝑧 = 𝑥2 + 𝑖𝑦2 gilt: 𝑧1 = 𝑧2 , wenn 𝑥1 = 𝑥2 ; 𝑦1 = 𝑦2
1.6.3 Multiplikation:
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𝑧1 ∙ 𝑧2 = (𝑎 + 𝑏𝑖) ∙ (𝑐 + ⅆ𝑖) = 𝑎𝑐 + 𝑎ⅆ𝑖 + 𝑐𝑏𝑖 + 𝑏ⅆ𝑖 2 = (𝑎𝑐 − 𝑏ⅆ) + 𝑖 ∙ (𝑎ⅆ + 𝑏𝑐)
1.3 Betrag einer Komplexen Zahl
|𝑧| = |𝑎 + 𝑏𝑖| = √𝑎2 + 𝑏 2 = √𝑧 ⋅ 𝑧̅ 𝑧1 ∙ 𝑧2 = |𝑧1 | ∙ 𝑒 𝑖𝜑1 ∙ |𝑧2 | ∙ 𝑒 𝑖𝜑2 = |𝑧1 | ∙ |𝑧2 | ∙ 𝑒 𝑖(𝜑1 +𝜑2 ) (Drehstreckung)
________________________________________________________________ 1.6.4 Division:
1.4 Darstellungsformen 𝑧1 𝑎 + 𝑏𝑖 𝑎 + 𝑏𝑖 𝑐 − ⅆ𝑖 (𝑎𝑐 + 𝑏ⅆ) − 𝑖 ∙ (𝑎ⅆ + 𝑏ⅆ) 𝑎𝑐 + 𝑏ⅆ 𝑏𝑐 − 𝑎ⅆ
= = ∙ = 2 2
= 2 2
+𝑖∙ 2
1.4.1 Algebraische/kartesische Form (Normalform) 𝑧2 𝑐 + ⅆ𝑖 𝑐 + ⅆ𝑖 𝑐 − ⅆ𝑖 𝑐 +ⅆ 𝑐 +ⅆ 𝑐 + ⅆ²
𝑧 = 𝑎 + 𝑏𝑖 a: Realteil von z b: Imaginärteil von z 𝑧1 |𝑧 |∙𝑒 𝑖𝜑1 |𝑧 |
= |𝑧1 |∙𝑒 𝑖𝜑2 = |𝑧1 | ∙ 𝑒 𝑖(𝜑1 −𝜑2 )
𝑧2 2 2
1.4.2 Trigonometrische / Polarform _________________________________________________________________
𝑧 = |𝑧| ⋅ (cos 𝜑 + 𝑖 ⋅ sin 𝜑) konjungiert: 𝑧̅ = |𝑧| ⋅ (cos 𝜑 − 𝑖 ⋅ sin 𝜑) 1.7 Potenzen
1.4.3 Exponentialform 𝑧 𝑛 = (𝑎 + 𝑏𝑖)𝑛 = (|𝑧| ∙ 𝑒 𝑖𝜑 )𝑛 = |𝑧|𝑛 ∙ 𝑒 𝑖𝑛𝜑 = |𝑧|𝑛 ∙ (cos(𝑛𝜑) + 𝑖 ∙ sin(𝑛𝜑))
𝑧 = |𝑧| ⋅ 𝑒 𝑖𝜑 konjungiert: 𝑧̅ = |𝑧| ∙ 𝑒 −𝑖𝜑 De Moivresche Formel: (𝐜𝐨𝐬 𝝋 + 𝒊 ∙ 𝐬𝐢𝐧 𝝋)𝒏 = (𝐜𝐨𝐬(𝒏𝝋) + 𝒊 ∙ 𝐬𝐢𝐧(𝒏𝝋))
________________________________________________________________ _________________________________________________________________
1.5 Umrechnung zwischen den Darstellungsformen 1.8 Wurzel (Trigonometrische Tabelle)
𝜑+2𝜋𝑘
1.5.1 Kartesisch in Trigonometrisch 𝑛 𝑖( ) 𝑛 𝜑+2𝜋𝑘 𝜑+2𝜋𝑘
𝑧𝑘 = √|𝑧| ∙ 𝑒 𝑛 = √|𝑧| ∙ cos ( 𝑛
)+ 𝑖 ∙ sin ( 𝑛
) mit k = 0,1,2,…
𝑧 = 𝑎 + 𝑏𝑖 → 𝑧 = |𝑧| ∙ (cos 𝜑 + 𝑖 ∙ sin 𝜑) Bsp.: Alle 5ten Wurzeln von z²
mit |𝑧| = √𝑎2 + 𝑏 2 = √𝑧 ∙ 𝑧̅ und 𝜑=
𝑏
arctan ( ) 5 𝑖(
𝜑+2𝜋0
) 𝜑 + 2𝜋 ∗ 0 𝜑 + 2𝜋 ∗ 0
𝑎 𝑧0 = √|𝑧| ∙ 𝑒 5 = 5 ∙ cos ( ) + 𝑖 ∙ sin ( )
5 5
Korrekturwerte: 5 𝑖(
𝜑+2𝜋1
) 𝜑 + 2𝜋 ∗ 1 𝜑 + 2𝜋 ∗ 1
𝑧1 = √|𝑧| ∙ 𝑒 5 = 5 ∙ cos ( ) + 𝑖 ∙ sin ( )
1.-Quadrant: - 2/3 Quadrant: +𝜋 4. Quadrant: +2𝜋 5 5
a > 0, b > 0 a<0 a > 0, b < 0 _________________________________________________________________
𝜋 3𝜋
𝜑 = 2 , a = 0, b > 0 𝜑= , a = 0, b < 0 1.9 Verschiebesätze
2
𝜑 = 𝜋 , a <0 , b = 0 𝜑 = 0, a = 0, b = 0; a > 0, b = 0 𝜋 𝜋
cos 𝑥 = sin(𝑥 + 2 ) sin 𝑥 = cos(𝑥 − 2 )
, 1.10 Überlagerung gleichfrequenter Schwingungen 𝑌1 + 𝑌2 = 𝑌 2. Matrizen:
1. 𝑌1 𝑢. 𝑌2 in eine Form bringen (sin/cos)
2. 𝑌1 𝑢. 𝑌2in Exponentialform umschreiben und 𝑒 𝜔𝑡 ausklammern 𝑎 𝑏 𝑐 𝑎 ⅆ 𝑔
3. Exponentialformen in Polarformen umschreiben 2.1 Transponieren: (ⅆ 𝑒 𝑓) → (𝑏 𝑒 ℎ)
(negativer Winkel in sin & cos beachten und ggf. raus ziehen siehe 𝑔 ℎ 𝑖 𝑐 𝑓 𝑖
Additionstheorem) 2.2 Determinante:
4. Jeweils katesische Form berechnen 𝑎12 𝑎11
(2x2) – Matrix: det (𝑎
) = 𝑎11 ∙ 𝑎22 − 𝑎12 ∙ 𝑎21
5. Beide katesischen Formen addieren(=Y) 21 𝑎22
(3x3) – Matrix: Regel nach Sarrus
6. |𝑌| = A berechnen 𝑎 𝑏 𝑐 𝑎 𝑏
𝑏 180°∗ 𝜑
7. 𝜑 = arctan (𝑎) + Korrektur → 𝜋
= 𝜑° ⅆ 𝑒 𝑓 ⅆ 𝑒 𝑎 ∗ 𝑏 ∗ 𝑐 + ⋯− 𝑐 ∗ 𝑒 ∗ 𝑔 −⋯
8. Y = 𝐴 ∗ 𝑒 𝑖(𝜔𝑡+𝜑 )
= 𝐴 ∗ sin(𝜔𝑡 + 𝜑) 𝑔 ℎ 𝑖 𝑔 ℎ
> - Matrix: Mit Entwicklungssatz; Spalte od. Zeile mit meisten 0 suchen
1.11 Additionstheoreme beweisen Von übrigen Wert x Zeile und Spalte streichen und vor übrige Matrix
multiplizieren; beachte + - + - + Schema für Wert x!
𝑒 𝑖𝑥 = cos 𝑥 + 𝑖 ∙ sin 𝑥 cos(−𝑥) = cos 𝑥
sin 𝑥
𝑒 −𝑖𝑥 = cos 𝑥 − 𝑖 ∙ sin 𝑥 sin(−𝑥) = −sin 𝑥 tan 𝑥 =
cos 𝑥 det(𝐴𝜇 ) = 0: • Nullzeile / -spalte
↓ 𝐿 ∈ 𝑅\{𝜇1 } • Zeilen/Spalten linear abhängig / vielfaches v. einander
𝑒 𝑖𝑥 +𝑒 −𝑖𝑥 𝑒 𝑖𝑥 −𝑒 −𝑖𝑥 −𝑒 𝑖𝑥 +𝑒 −𝑖𝑥 • Summe zweier Zeile/Spalten = Summe anderer Zeile/Spalte
cos 𝑥 = sin 𝑥 = tan 𝑥 = 𝑒 𝑖𝑥 +𝑒 −𝑖𝑥
∙𝑖
2 2𝑖 • Identische Zeile/Spalten
Additionstheorem 1: 2 ∙ sin 𝑥 ∙ cos 𝑥 = 2 sin(2𝑥) 2.3 Inverse det(𝐴) ≠ 0 , so ist A invertierbar
Additionstheorem 2: cos² 𝑥 − sin2 𝑥 = cos(2𝑥)
1 ⅆ −𝑏 𝑎 𝑏
_________________________________________________________________ 𝐴−1 = 𝑎𝑑−𝑏𝑐 ∙ ( ) → 𝐴=( ) mit det(𝐴) ≠ (𝐴|𝐼) (𝐼|𝐴−1 )
−𝑐 𝑎 𝑐 ⅆ
1.12 Periodizität _________________________________________________________________
sin 𝑥 = sin(𝑥 + 2𝜋𝑘) cos 𝑥 = cos(𝑥 + 2𝜋𝑘) 2.4 Rechenregeln:
𝐴 ∙ 𝐴−1 = 𝐼
𝑒 𝑖(2𝜋𝑘) = 1 → cos(2𝜋𝑘) + 𝑖 ∙ sin(2𝜋𝑘) = 1 𝐴 ∙ 𝐵 = 𝐵 ∙ 𝐴 → 𝑘𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣
−1 −1 −1
_________________________________________________________________ (𝐴 ∙ 𝐵) = 𝐵 ∙𝐴
𝐴 ∙ 𝐼 = 𝐴 (Einheitsmatrix)
−1 −1 −1 −1
1.13 Gebietseinteilung (𝐴 ∙ 𝐵 ∙ 𝐶) = 𝐶 ∙𝐵 ∙𝐴
Umformen Bsp.:
𝑇 𝑇 𝑇
Kreisgleichung: )2
(𝑥 − 𝑚1 + (𝑥 − 𝑚2 = 𝑟² )2
M(x0/y0); r (𝐴 ∙ 𝐵) = 𝐵 ∙ 𝐴
2 2
𝐴−1 ∙ 𝑋 ∙ 𝐶 −1 + 𝐴−1 ∙ 𝑋 = 𝐷 |𝐴 ∙
Bsp.: 𝑥 + 𝑦 − 𝑥 + 2𝑦 = 4 −1 −1 𝑇 𝑇
(𝐴 ) = 𝐴 = (𝐴 )
Variablen auf eine Seite & Ganze Zahlen auf andere; dann Quadratisch ergänzen: 𝑋 ∙ 𝐶 −1 + 𝑋 = 𝐴 ∙ 𝐷 |( )
𝑇)
1 2 2 2 1 2 2 2 det(𝐴) = det(𝐴
= 𝑥2 − 𝑥 + (2) + 𝑦 2 + 2𝑦 + (2) =4+ (2) + (2) „ergänzen“ 𝑋 ∙ (𝐶 −1 + 𝐼) = 𝐴 ∙ 𝐷 | ∙ (𝐶 −1 + 𝐼)−1
−𝑛 (𝐴−1 )𝑛
2 2 𝐴 =
1 2 21 1 21
=(𝑥 − (2)) + (𝑦 + (2)) = → M( 2 / -1 ); r =√ 4 𝑋 = 𝐴 ∙ 𝐷 ∙ (𝐶 −1 + 𝐼)−1
4 (𝐴𝑛 )−1 = (𝐴−1 )𝑛
!=0 !=0
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