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Samenvatting Complexe getallen
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Samenvatting complexe getallen
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MAT1512 - Exam Questions PACK (2010-2022)
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Mathematics 144 Summaries
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1. Conventies2. Definities
3. Machten i
4. Bewijs voor Eulervorm
5. ei x = cos(x) + i sin(x) uit Taylor-reeksen
6. Geconjugeerd complexe
7. Verschil
8. Product
9. Machtsverheffen
10. Voor differentiëren en integreren van goniometrische machten
11. Lineaire functie
11.1. Inverse
12. Polynoom
12.1. Oplossingsmethoden
12.1.1. Trial and error
12.1.2. Kwadraat afsplitsen
12.1.3. Omzetten naar Eulervorm
12.2. Inverse goniometrische hoeken
1/6 © Peter Zomerdijk
, 1. Conventies
• voorbeelden zijn omkaderd
2. Definities
• complexe eenheid : i2 = ‒1 (NIET i = √‒ 1)
• verzameling complexe getallen : ℂ = { a + bi | a, b Є ℝ }
• normaalvorm : z = a + bi
• reële deel (x-as of Re-as) : a = Re(z)
• imaginaire deel (y-as of Im-as) : b = Im(z)
• modulus is de lengte : |z|= √a2 + b 2
b
• argument is de hoek met de Re-as : (z) = ϕ = arctan (a)
• modulo van de hoek
• als a > 0 : 2kπ
• als a < 0 : (2k+1)π
• uit de grafische weergave : a = |z| cos ϕ ꓥ b = |z| sin ϕ
• poolvorm : z = |z| (cos ϕ + i sin ϕ)
• uit Taylor-reeksen : ei ϕ = cos ϕ + i sin ϕ ⇒ ei π + 1 = 0
• Eulervorm : z = |z| ei ϕ
• complexe getallen zijn niet te ordenen, dus geen > of <
3. Machten i
n∈ℕ⇒
• i 4n = i 0 =1 = i 4n+4 = ‒ i · i = 1
• i 4n+1 = 1 · i =i etc.
• i 4n+2 =i·i =‒1
• i 4n+3 =‒1·i =‒i
4. Eulervorm
• bewijs: eia · eib = (cos(a) + i sin(a)) · (cos(b) + i sin(b))
= cos(a) cos(b) – sin(a)sin(b) + i (sin(a) cos(b) + cos(a) sin(b))
= cos(a + b) + i sin(a + b) =
e i(a+b)
b
i arctan( )
• van Eulervorm naar normaalvorm : z = √a2 + b 2 e a
2/6 © Peter Zomerdijk
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