Imaginary Numbers Flashcards
What is the modulus of z= x + iy?
The modulus is the length from the origin to z, defined as |z| = sqrt(x^2 + y^2).
What is the difference between the argument (arg(z)) and the principal value of the argument (Arg(z))?
arg(z) is the angle θ from the origin to z. Arg(z) is this value, θ, between -π<θ<π. Or, Arg(z) is the unique value of arg(z).
What is the polar form of a complex number?
A complex number in form z = rcis(θ) is described as polar.
What is an Argand diagram?
An Argand diagram is a visual representation of a complex number.
How to you divide two complex numbers?
Multiply the question by the complex conjugate of the denominator.
What is the complex conjugate of z = x + iy?
x - iy.
What do you do to the value of inverse tan(y/x) to find arg(z) if you know z lies within the second or third quadrants of an Argand diagram?
Add π.
What is the value of (cis θ1)(cis θ2)?
(cis θ1)(cis θ2) = cis(θ1 + θ2)
What is the value of 1/cis(θ)?
1/cis(θ) = cis(-θ)
|mod(z1) mod(z2)| = ?
arg(z1z2) = ?
|mod(z1)mod(z2)| = |mod(z1)||mod(z2)|.
arg(z1z2) = arg(z1) + arg(z2).
What is de Moivres theorom for an integral index?
(cis θ)^n = cis (nθ).
What is de Moivres theorom for a fractional index?
cis((p/q)θ) is one of the values of (cis θ)^(p/q).
With z = cis(θ), what is a trigonometric identity in regards to cos(θ)?
(optional help: cosθ has a plus)
cos(θ) = 1/2(z + 1/z).
With z = cis(θ), what is a trigonometric identity in regards to sin(θ)?
(optional help: sinθ has a minus)
sin(θ) = 1/2i(z - 1/z)
Consider the trigonometric identity of cos(θ) with z = cis(θ). What difference does having cos(mθ) make to the identity?
In the identity of cos(mθ), all references of z in the identity are to the power of m.
cos(θ) = 1/2(z + 1/z)
cos(mθ) = 1/2(z^m + 1/z^m)
How do you undergo synthetic division?
Divide the first value of the numerator by the first value of the denominator. This value is the first value of the result. Then, multiply this value by the denominator. Subtract this value from the numerator until you run out of values. the remaining values in the numerator are the remainder, and the values extracted is the completed division.
What is the term “the zeros of the polynomial” synonymous with?
The roots of the equation.
What term is given to the value n, where n is the highest power in a polynomial?
It is said that the polynomial is of degree n.
What does the Maclaurin series of some function f (i.e f(x) = sin(x)) describe?
The Maclaurin series of f describes an infinite expansion of values that some functions satisfy.
For f(x) = sin(x), the Maclaurin expansion is sin(x) = x - (x^3)/3! + (x^5)/5! + …
Define sinh(x) (pronounced shine x), and cosh(x).
Tip: involves e^x.
sinh(x) = 1/2(e^x - e^-x).
cosh(x) = 1/2(e^x + e^-x)
What is the value of log(z)?
log(z) = ln(|z|) + iarg(z)
How would you describe the principal value of the logarithm function?
Log(z) = ln(|z|) + iArg(z)
