Complex numbers Flashcards
what’s the modulus-argument form of a complex number z?
z = r(cos θ + i sin θ)
where r is the modulus of z and θ is the principal argument
how else can you express a complex number?
in the form z = r e^iθ
r is the modulus of z and a is the argument
cos (-θ) =
cos θ
sin (-θ) =
-sin(θ)
sin (θ±ρ) =
sin θ cos ρ ± sin ρ cos θ
cos (θ±ρ) =
cos θ cos ρ - /+ sin θ sin ρ
sin^2 θ + cos^2 θ =
1
multiplying z1= r1(cosθ +isinθ) and z2 = r2(cosρ+isinρ), z1 z2 =
z1z2= r1r2(cos(θ+ρ) + isin(θ+ρ))
modulus and argument of z1z2?
modz1z1 = r1r2
argz1z2 = θ + ρ
multiplying z1 = r1 e ^iθ and z2 = r2 e^iρ,
z1z2 = r1r2 e^i(θ+ρ)
how can you express a comolex number?
exponential form
modulus argument form
standard form
dividing complex numbers in modulus argument form
z1/z2 =
r1/r2 (cos(θ-ρ) +i sin (θ-ρ))
z1/z2 has modulus and argument…
r1/r2
argz1 - argz2
how do you prove de moivres’ theorem?
by induction
z^n =
[r(cos θ +i sin θ)] ^n = r^n(cos nθ + i sin nθ)
de moivres’ theorem in exponential form?
r^n e^inθ
(x + y)^n =
x^n + nC1 x^n-1 y + nC2 x^n-2 y^2 + … + y^n
nCk =
n!/k!(n-k)!
