Polar Form and the Geometry of C

Recall that a complex number z ∈ C with real part z1 and imaginary part z2 is dened to be the ordered pair (z1, z₂). This is clearly a point, the geometric image of z, in the plane R², as shown schematically in gure A.1 below. Upon inspection of Figure A.1, a geometric interpretation of modulus is immediate, it is simply the magnitude of the corresponding vector, the corresponding geometric image. We dene the angle θ, measured counter-clockwise from the positive real axis to the ray which extends from the origin to the geometric image of the complex number, and call it the argument, denoted arg z. With modulus and argument in hand, we may immediately dene a polar representation of the complex number z; the quantities r and θ, with r > 0, completely and uniquely characterize the polar form of the complex number. In particular, for the complex number z = z1+ iz₂, the relations between rectangular and polar co-ordinates are, of course,

r = |z| , tan θ = z₂

z₁, z1 = r cos θ, and

z₂ = r sin θ

z = z+iz Im

Re

1 2

z₁ z₂

θ r

z = z₁−iz₂

−θ

−z₂ ‒

Figure A.1: This is a schematic of the geometric image (z1, z₂) of the complex number z = z1+ iz2 in the complex plane. The horizontal axis is the real axis and is typically labelled by Re. Likewise, the vertical axis is known as the imaginary axis and is labelled Im. The modulus, |z| is labelled as r, and the argument of z is the angle θ taken positive counter-clockwise from the real axis to the ray which extends from the origin to the geometric image. The conjugate, z, is the reection of z in the real axis.

The relation tan θ = z2/z₁ denes θ implicitly. There is a subtlety which arises because tan is 2π-periodic, i.e., tan θ = tan (θ + 2π). Consequently, arg z is a multi-function, a function which takes a complex number z in its domain but returns a set of possible outcomes. That is to say that, for a given complex number z, arg z = {θ + 2πk | k ∈ Z} where θ is simply an angle measured from the positive real axis to a ray which corresponds to the ray which extends from the origin to the geometric image of z. The angle θ depicted in Figure A.1 is one amongst many possible such angles. Sometimes, to avoid this diculty, a principal argument is dened. We have, for complex number z with argument θ where θ is any element of the set arg z, i.e., any angle which satises tan θ = z2/z₁, the following. An angle α measured from the positive real axis is selected and xed. For this angle, dene Arg_αz to be the element of arg z which falls into the interval [α, α + 2π). Arg z is known as the principal argument of complex number z corresponding to the value of arg z which satises α ≤ arg z < α + 2π. We will drop the subscript α but understand implicitly that each time the principal argument is invoked, there is a particular range of values, known as the branch of arg, to which it corresponds, and that, because Argα 6= Arg_β when α 6= β, the principal arguments with respect to dierent branches may not be manipulated carelessly.

Before proceeding with geometric interpretations of the algebraic operations on C, we introduce the complex exponential, which allows the most expeditious expression of the polar form. Euler's formula, motivated by Maclaurin series for sin and cos but ultimately

taken as a denition, is

e^iθ = cos θ + i sin θ. (A.1)

Again, it should be apparent due to the 2π periodicity of sin and cos that e^iθ = e^i(θ+2πk)

for any θ ∈ R and any k ∈ Z. So, with the relations between polar and rectangular components and with Euler's formula, we must have

z = z1+ iz2 = r cos θ + ir sin θ = r (cos θ + i sin θ) = re^iθ (A.2) where θ = arg z. Equation (A.2) will be taken as the polar form of the complex number z.

The complex exponential, a function exp : C → C, is dened

exp z = e^z = eRe(z)+i Im(z) = e^Re(z)e^{i Im(z)}= e^Re(z)(cos (Im (z)) + i sin (Im (z))) . (A.3) Further discussion of complex functions of a complex variable and their geometry, including the geometry of the exponential function, follows in section A.3 subsequently. We are now prepared to discuss the geometry of the algebraic operations in C.

We have seen that the set of complex numbers C coincides with R² where the real part of z is the x coordinate, the imaginary part of z is the y coordinate in the plane. It should be clear that the addition of complex numbers, by the very denition, corresponds to vectorial addition in the plane, multiplication of complex numbers by a real number corresponds to a pure scaling as in scalar multiplication of vectors. We have seen that the modulus of a complex number is simply the distance from the origin of the complex plane to the geometric image of the complex number, and that the argument is simply one of the many angles winding around the origin k ∈ C times, and measured from the real axis to a ray which coincides with the ray generated by the geometric image. Again, θ is taken positive counter-clockwise. The conjugate of a complex number is the geometric image of a reection through the real axis of the number conjugated. What of multiplication and division? Consider the following for z = z1+ iz₂ = r₁e^iθ1 and w = w1+ iw₂ = r₂e^iθ2 in C,

zw = r₁e^iθ1r₂e^iθ2 = r₁r₂e^iθ1e^iθ2

= r₁r₂(cos θ₁+ i sin θ₁) (cos θ₂+ i sin θ₂)

= r₁r₂ cos θ₁cos θ₂− sin θ₁sin θ₂ + i (cos θ₁sin θ₂+ sin θ₁cos θ₂)

= r1r2 cos (θ1+ θ2) + i sin (θ1+ θ2)

= r₁r₂e^i(θ1+θ2)

= |z| |w| ei(arg z+arg w)

So obtaining the relations

|zw| = |z| |w| and arg (zw) = arg z + arg w.

A similar argument entails for division that

It is worthwhile to note that these relations do not hold for principal arguments because the sum of two principal arguments relative to the same branch of arg might actually fall outside of the branch and similarly for dierences of principal arguments. With these observations, it is abundantly clear that multiplication of w by a complex number z corresponds to rotation of the geometric image of w about an angle arg z and a scaling of w by an amount

|z|, whereas division of w by z corresponds to rotation of the geometric image of w about an angle |arg z| but oriented oppositely to arg z and a scaling of w by an amount 1/|z|. In particular, z⁻¹ is an inversion in the unit circle, z/|z| together with a reection through the real axis.

Now, prior to moving onwards to the discussion of complex functions of a complex variable, their geometry and analysis, in subsequent sections, a number of results are gathered and proven here. An important exercise is to keep the geometric interpretation in mind at all times. The proofs are algebraic, but the results themselves have signicant geometric content, i.e., that the relation is true should be apparent from the geometry alone.

Several of these have been proven, nevertheless, all of these results are proven again.

Proof. In the following, we take z = z1+ iz₂ = r₁e^iθ1 and w = w1+ iw₂ = r₂e^iθ2 1. z = z1+ iz₂ = z₁− iz₂ = z₁+ iz₂ = z

2. Re (z) = Re z1+ iz₂
= Re (z₁− iz₂) = z₁ = Re (z)

3. Im (z) = Im z1+ iz₂
= Im (z₁− iz₂) = −z₂ = − Im (z)

7. This result follows immediately from the previous by setting z to 1 and w to z.

8. zz = (z1+ iz₂) (z₁+ iz₂) = (z₁+ iz₂) (z₁− iz₂) = z₁²+ z₂² = |z|²

9. |z|² = zz = zz = zz = |z|² and, because the expression to be squared is positive, simply take square roots of both sides to obtain |z| = |z|, as required.

10. |zw|² = (zw) (zw) = zwzw = zzww = |z|²|w|² and, because the expression to be squared is positive, simply take square roots of both sides to obtain |zw| = |z| |w|, as required.

and, because the expression to be squared is positive, simply take square roots of both sides to obtain ^z

w

 = _|w|^|z|, as required.

12. This result follows immediately from the previous by setting z to 1 and w to z.

13. Re z ≤ |Re z| =q

(Re z)² ≤ q

(Re z)²+ (Im z)² = |Re z + i Im z| = |z|

14. Im z ≤ |Im z| =q

(Im z)² ≤ q

(Re z)²+ (Im z)² = |Re z + i Im z| = |z|

15. This is the triangle inequality. Consider |z + w|²

|z + w|² = (z + w) (z + w) = (z + w) (z + w) = zz + zw + wz + ww

= |z|²+ zw + zw
+ |w|² = |z|²+ 2 Re (zw) + |w|²

≤ |z|²+ 2 |Re (zw)| + |w|²

≤ |z|²+ 2 |zw| + |w|² = |z|²+ 2 |z| |w| + |w|² = |z| + |w|
2

so obtain |z + w|² ≤ (|z| + |w|)² and, because the expressions to be squared are positive, obtain |z + w| ≤ |z| + |w|. Replace w with its negative to obtain |z − w| ≤

|z| + |w|.

16. This is a variation on the triangle inequality. Begin by considering |z| then simul-taneously add and subtract w inside of the absolute values. |z| = |z − w + w| ≤

|z − w| + |w|, by triangle inequality, so |z| − |w| ≤ |z − w|. Now swap z and w to obtain |w| − |z| ≤ |w − z| = |z − w| so − |z − w| ≤ |z| − |w|. Combining, obtain

− |z − w| ≤ |z| − |w| ≤ |z − w| so ||z| − |w|| ≤ |z − w| as required. Replace w with its negative to obtain ||z| − |w|| ≤ |z + w|.