Complex Numbers Flashcards
De moivres theorum
(cosθ+isin θ)^n = cos nθ + isin nθ
(r(cosθ+isin θ))^n =
r^n(cos nθ + isin nθ)
Nth root of unity
where a^n = 1
finding the Nth root of unity
- apply de moivres to the complex number
- knowing 1 has modulus 1 and argument 0 equate 0 with nθ to get the n angles for θ
- put this angles back into the original complex number
what is always a root of unity?
1
Where do the complex roots of unity all lie?
equally spaced on the unit circle
finding the general Nth roots
- find 1 root (i.e. find the argument of one number and sub back in)
- split a circle into n+1 parts and work out the arguments of the other roots
- sub these arguments back into the complex number
OR
argument = argz +2πk/n sub these back into general de moivres form
uses of de moivre
- give multiple angle formula in terms of powers
- to express powers of sine and cosine in terms of multiple angles
sin nθ =
z^n - z^-n / 2i
cos nθ =
z^n + z^-n / 2
how do you come up with the sin nθ and cos nθ expressions
by finding a general z^n and z^-n and adding/ subracting the expression
e^iθ =
cosθ + isinθ
double angle formula cos
= cos^2θ - sin^2θ
= 2cos^2θ -1
= 1 - 2sin^θ
double angle formula sin
= 2cosθsinθ
using complex numbers to sum real series
- introduce a complex term to make it a known series
- manipulate to it becomes a de moivres expression
- equation real or imaginary parts to give the original value of the summation
express multiple angle formula in terms of powers
- use de moivres to show what it should be
- expand the expression using binomial expansion
- equate real and imaginary terms
useful method
equating the real and imaginary parts
dividing complex numbers
- set equal to x + yi then equate real/ imaginary parts
OR - multiply by the complex conjugate
y axis of argand diagram
imaginary
x axis of an argand diagram
real
|wz| =
|w||z|
|w/z|
|w|/|z|
arg(zw)
arg(z)+arg(w)
arg(z/w)
arg(z)-arg(w)
|z-z1|=r
|z-(a+bi)|=r circle centre (a,b) radius r
|z-z1| less than r
interior of the circle
|z-z1|>r
exterior of the circle
arg(z-z1) = θ
arg(z-(a+bi))= θ
consist of a half line from point a +bi in the direction θ
arg(z-z1) < θ
all the point below the half line and the horizontal line to the point z1
arg(z-z1) > θ
all the point above the half line and the horizontal line to the point z1
|z-z1|=|z-z2|
perpendicular bisector of the line joining z1 and z2
|z-z1| < |z-z2|
less than
represents the region closer to z1 from the bisector
|z-z1|>|z-z2|
represemts the region closer to z2 from the bisector
> =
line is included
>
line not included
how to show a line is not included in the region
dot it
expanding sinnθ or cosnθ
- use de moivres and take the imaginary part or real part depending on cos/sin i.e. im[]
- expand the power n from de moivres using binomial expansion
- after multiplying out the i’s keep only the real/imaginary expressions
- use identities to get in terms of just sin or cos
how to express powers of sin or cos in terms of multiple angles
- express sin/ cos in terms of z^n (the sin/ cos will have a 2 infront don’t forget the power this number)
- expand the z^n terms via bionomial expansion
- factorise out z^ns with their corresponding minus powers, replace these expressions with sin/cos
- divide by the 2^n
2cos nθ=
Z^n + Z^-n
2isin nθ =
Z^n - Z^-n
when equating real and imaginary parts and a real number on the bottom of the fraction
the bottom of the fraction appears for both the real and imaginary number
