Week 18 Flashcards
(18 cards)
Eq for r and arg?
r = ll x, y ll
x = rcostheta
y= rsintheta
Euelers formula?
z = x+iy -> r(costheta + isintheta) = re^i x theta
Polar exponential form?
re^ i x theta
Raising re^i theta to a power? e.g. z^8?
r^8 x e^i x theta x 8
Complex conjugate (x̄) ?
x - iy = re^-i theta
Fundamental theorem of algerba?
Non real roots of polynomials come in complex conjugate pairs
Root for complex?
complete the square
Difference eq notation?
P(E)yx = (Q)x
where P(E) is just the y terms in terms of E e.g. y x+3 = E^3
Difference eq homogenous and non homogenous?
Homogenous Q(x) = 0
Non Homogenous Q(X)≠0
Homogenous gen soln for real roots?
change P(E) -> P(M) and solve
then yx = A(root1)^x +B(root2)^x
Homogenous gen soln for repeated real roots?
yx = (A+Bx)(root)^x if A.M = 2 if more than just keep going to cx^2
Homogenous gen soln for conjugate imaginary roots Derivation?
do normally
make yx= make A (re^i theta)…. and make B = A bar
then yx= (r)^x(Ae^itheta + Abar…)
Make A = α+βi , and return to (r)^x( α+βi (cos + i sin x)) and solve
Homogenous gen soln for conjugate imaginary roots?
yx = (r)^x (C cos(θx)+D sin (θx))
Soln when difference eq with non homogenous?
Work out CS
Then PS mimic it so e.g. if 2x then (Ps)x= ax+b , (Ps)x+1 = a(x+1)+b
then sub into original eq
If (Q)x belongs to (CS)x?
mutliply by x
e.g. (Q)x =6(-3)^x
(PS)x has to be ax(-3)^x
To make alpha and beta real if don’t apply conjugate thing?
make alpha = a1+ia2
beta = b1+ib2
sub in and equal imaginary parts to 0 and we see they are complex conjugates
Only way to work out A and B of CS and a and b of PS?
using conditions given of yx = for CS and plug into gen soln
for a and b of PS just usual way
Shift operator (E)st?
E(st) = st+1
E(st+1)=st+2
where st+1 is basically yx+1
