Complex Numbers Flashcards
what is the exponential form for writing a complex number
z = re^iθ
what is eulers formula
e^iθ = cosθ + isinθ
what is the modulus of e^iθ
e^iθ = 1
what is the modulus of e^2piin
1
for 2 complex numbers z1 and z2, what does |z1*z2| equal
|z1*z2| = |z1| * |z2|
what does arg(z1*z2) equal
arg(z1*z2) = arg(z1) + arg(z2)
what is the effect of multiplying a complex number by i
rotation of pi/2 ACW
what is the complex form of cosθ
cosθ = (e^iθ + e^-iθ) / 2
what is the complex form of sinθ
sinθ = (e^iθ - e^-iθ) / 2i
why does (cosθ + isinθ)^n = cosnθ + isinnθ simply speaking
because e^inθ = (e^iθ)^n
what is the basis of demoivres theorem problems
using this idea of (cosθ + isinθ)^n = cosnθ + isinnθ to write one side in terms of the other
what does the equation 4 + 2e^iθ tell you about the circle drawn
- the centre is at (4,0)
- it has a radius of 2
how would you solve z^3 = -1
- write z^3 in its complex form
- add +2ipi*k to the power
- cube root both sides
- write all the solution for each k, starting from 0, until 2pi is reached
if a = a* (a* = complex conjugate of a) what does that mean about the imaginary part of a
- the imaginary part of a = 0
- so a is a real number
if p(z) = az^n + bz^n-1 + cz^n-2 +…+ qz + r, where all the coefficients are real, what does that mean for the roots of p(z)
the roots = 0
