Terminology & Notation
This symbol denotes the modulus of a complex number z = x+iy.
What is |z|?
The parallelogram law gives the geometric interpretation of this operation on two complex numbers.
What is addition?
The notation arg(z) refers to this mathematical object.
What is the set of all possible arguments of z?
This theorem provides a formula for raising complex numbers in polar form to integer powers.
What is De Moivre’s Theorem?
The open disk centered at z0 with radius ε is denoted by this symbol.
What is Bε(z0) = {z: |z−z0| < ε}?
This notation refers to the complex conjugate of z = x+iy.
What is z̄ = x - iy?
Multiplying two complex numbers multiplies their moduli and does this to their arguments.
What is add the arguments?
To define a single value for the argument, we restrict it to this interval.
What is (−π, π] (the principal argument, Arg(z))?
The n-th roots of a nonzero complex number always lie on this geometric figure.
What is a circle of radius r^(1/n) centered at the origin?
The set of all points in S that have a neighborhood entirely contained in S is called this.
What is the interior of S?
The generalized triangle inequality states that |z1 + z2| ≤ |z1| + |z2|. Equality holds under this condition.
What is when z1 and z2 point in the same direction (are positive multiples of each other)?
Geometrically, multiplying by a complex number can be described as a combination of these two transformations.
What is stretching (or squeezing) and rotation?
The ambiguity between defining principal argument in (−π, π] or [0, 2π) often leads to these.
What are misunderstandings and errors?
There are this many distinct n-th roots of a nonzero complex number.
What is exactly n?
An open, connected set in the complex plane is called this.
What is a domain?
This exponential identity connects trigonometric form with the complex exponential function.
What is Euler’s formula, e^{iθ} = cos θ + i sin
What is Euler’s formula, e^{iθ} = cos θ + i sin
The product z1z2 in polar form equals this expression.
What is r1 r2 (cos(φ1+φ2) + i sin(φ1+φ2))?
If z = −3i, its principal argument is this value.
What is −π/2?
The n-th roots of z = re^{iθ} are expressed with this formula.
What is z_k = r^(1/n) e^{i(θ + 2kπ)/n}, k = 0,1,…,n-1?
A set that contains all its boundary points is called this type of set.
What is a closed set?
In polar form, z = r(cosθ + i sinθ). This value corresponds to the distance from the origin.
What is r = |z|, the modulus?
When you multiply by the reciprocal 1/z, this happens to the modulus and argument.
What is modulus becomes reciprocal and argument changes sign?
The general relationship between argument and principal argument is given by this formula.
What is arg(z) = Arg(z) + 2nπ, n ∈ Z?
The n-th roots are separated by this angular measure around the circle.
What is 2π/n radians?
The fundamental theorem of topology states that the pair (C, τ) is this kind of mathematical structure.
What is a topological space?