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)!
z + 1/z =
2 cos θ
z - 1/z =
2i sin θ
z^n + 1/z^n =
2 cos nθ
z^n - 1/z^n =
2i sin nθ
cartesian equation of a circle center (a ,b) radius r?
(x-a)^2 + (y-b)^2 = r^2
Example of a locus
A circle
circle center (x1, y1) radius r?
where z1 = x1 + iy1
z - z1 | = r
what does | z - z1 | = | z - z2 | represent?
a perpendicular bisector of the line segment joining points z1 to z2.
angle in a semicircle =
90
angles subtended at an arc in the same segment are
equal
the angle subtended at the center of the circle is
twice the angle at the circumference
what does arg (z -z1) = a represent?
a half-line form the fixed point z1 making an angle a with a line from the point z1 parallel to the real axis
e^iθ =
cosθ + isinθ
cos 2θ =
= cos^2θ - sin^2θ
= 2cos^2θ -1
= 1 - 2sin^θ
sin2θ =
= 2sinθcosθ
what does w = z + a + ib represent?
a translation with vector [ a, b] where a and b are real
what does w = kz represent?
an enlargement scale factor k centre (0,0) where k > 0
what does w = kz + a +ib represent?
enlargement scale factor k centre (0,0) followed by a translation with vector [ a, b ] where k > 0 and a, b are real
three types of loci?
circle: | z - z1 | = r
perpendicular bisector of line segment joining z1 to z2:
| z - z1 | = | z - z2 |
half line: arg (z - z1) = θ
arg (z1 x z2) =
arg(z1) + arg(z2)
arg (z1 ÷ z2) =
arg(z1) - arg (z2)
|z1z2| =
|z1| |z2|
|z1 ÷ z2| =
|z1| ÷ |z2|
what does arg( z1/z1 + 4) = π/4 look like
arg(z1) - arg(z1 + 4) = π/4 looks like an arc of a circle
area of a sector A =
1/2 r^2 θ
θ in radians
area of a segment A =
1/2 r^2 (θ - sinθ)
θ in radians
area of a circle =
π r^2
