Complex Operations

Sid Su |

Introduction

Notes from chapter 1.1, 1.2 of Ahlfors. Better structured notes can be found in my handwritten notes. This is more meant as a quick reference for important theorems and equations, as well as a place for practice problems.

Quick Reference

There are times where it makes sense to have solutions to the equation 2⁢𝑥=3, so we invented the fractions—there are also times where it makes sense to have solutions to 𝑥2=−1, so we must invent the imaginary numbers1 (Gross).

1This moral argument is adapted from Herb Gross' video series. As he puts it, the foreman called down and asked ‘how many of you are down there?’ the workers responded ‘3!’. ‘OK, half of you come up!’ Notably it doesn’t make sense to have a solution to the foreman example, but there are obviously other problems that do benefit from having a solution. The same logic applies to the imaginary numbers.

Definition of 𝑖

Any solution to the equation

i2= −1 $$ i^2 = -1 $$ ⟹i=+ −1∨i= −−1 $$ \implies i = +\sqrt{-1} \lor i = -\sqrt{-1} $$

As a result, exponentiating 𝑖 creates the cycle

i1=i ,i2=−1 ,i3=−i ,i4=1 $$ i ^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 $$ ⟹in =inmod4 $$ \implies i^n = i^{n \bmod{4}} $$

Complex Addition

(α+i ⁢β)+( γ+i⁢δ )=(α+γ )+i(β+ δ) $$ \left( \alpha + i \beta \right) + \left( \gamma + i \delta \right) = \left( \alpha + \gamma \right) + i \left( \beta + \delta \right) $$

Complex Multiplication

(α+i ⁢β)×( γ+i⁢δ )=(α⁢ γ−β⁢ δ)+i(β ⁢γ+α ⁢δ) $$ \left( \alpha + i \beta \right) \times \left( \gamma + i \delta \right) = \left( \alpha \gamma - \beta \delta \right) + i \left( \alpha \delta + \beta \gamma \right) $$

Complex Division

α+i ⁢βγ+ i⁢δ= α⁢γ +β⁢δ γ2+δ 2+i⁢ β⁢γ −α⁢δ γ2+ δ2 $$ \frac{\alpha + i \beta}{\gamma + i \delta} = \frac{\alpha \gamma + \beta \delta}{\gamma^2 + \delta^2} + i \frac{\beta \gamma - \alpha \delta}{\gamma^2 + \delta^2} $$

Complex Square Root

α+i ⁢β=± (α+ α2+β2 2+i ⁢β|β |⁢ −α+α 2+β2 2) $$ \sqrt{\alpha + i \beta} = \pm \left(\sqrt{\frac{\alpha + \sqrt{\alpha^2 +\beta^2}}{2}} + i \frac{\beta}{|\beta|} \sqrt{\frac{-\alpha + \sqrt{\alpha^2 + \beta^2}}{2}} \right) $$

Complex Division Proof

Suppose α+i⁢β γ+i⁢ δ=x+i ⁢y( γ+i⁢δ ≠0) $$ \mathrm{Suppose } \frac{\alpha + i \beta}{\gamma + i \delta} = x + i y~~~\left(\gamma + i \delta \ne 0\right) $$ ⟹α+i ⁢β=(x +i⁢y)( γ+i⁢δ ) $$ \implies \alpha + i \beta = (x + iy)(\gamma + i\delta) $$ ⟹α+i ⁢β=(γ ⁢x-δy )+i⁢( δ⁢x+γ ⁢y) $$ \alpha + i \beta = (\gamma x - \delta y) + i(\delta x + \gamma y) $$ ⟹{ α=γ⁢x −δ⁢y β=δ⁢ x+γ⁢y $$ \implies \begin{cases} \alpha = \gamma x - \delta y \\ \beta = \delta x + \gamma y \end{cases} $$
⟹δ ⁢y=γ ⁢x−α $$ \implies \delta y = \gamma x - \alpha $$ ⟹y= γ⁢x− αδ $$ \implies y = \frac{\gamma x - \alpha}{\delta} $$ ⟹β= δ⁢x+γ (γ⁢x −αδ) $$ \implies \beta = \delta x + \gamma \left( \frac{\gamma x - a}{\delta} \right) $$ ⟹β⁢ δ=δ2 ⁢x+γ ⁢(γ⁢ x−α) $$ \implies \beta \delta = \delta^2 x + \gamma (\gamma x - \alpha) $$ ⟹β⁢ δ=δ2 ⁢x+γ2 ⁢x−γ ⁢α $$ \implies \beta \delta = \delta^2 x + \gamma^2 x - \gamma \alpha $$ ⟹β⁢ δ=x⁢( δ2+γ2 )−γ⁢ α $$ \implies \beta \delta = x \left(\delta^2 + \gamma^2 \right) - \gamma \alpha $$ ⟹x⁢ (δ2+ γ2)−γ ⁢α=β ⁢δ $$ \implies x \left(\delta^2 + \gamma^2 \right) - \gamma \alpha = \beta \delta $$ ⟹x= β⁢δ+ γ⁢α δ2+γ2 $$ \implies x = \frac{\beta \delta + \gamma \alpha}{\delta^2 + \gamma^2} $$
⟹γ ⁢x=α+ δ⁢y $$ \implies \gamma x = \alpha + \delta y $$ ⟹x= α+δ⁢y γ $$ \implies x = \frac{\alpha + \delta y}{\gamma} $$ ⟹β= δ⁢( α+δ⁢y γ)+γ ⁢y $$ \implies \beta = \delta \left(\frac{\alpha + \delta y}{\gamma}\right) + \gamma y $$ ⟹β⁢ γ=δ⁢( α+δ⁢y )+γ2 ⁢y $$ \implies \beta \gamma = \delta (\alpha + \delta y) + \gamma^2 y $$ ⟹β⁢ γ=δ⁢ α+δ2 ⁢y+γ2 ⁢y $$ \implies \beta \gamma = \delta \alpha + \delta^2 y + \gamma^2 y $$ ⟹β⁢ γ=δ⁢ α+y⁢( δ2+γ2 ) $$ \implies \beta \gamma = \delta \alpha + y \left( \delta^2 + \gamma^2 \right) $$ ⟹y⁢ (δ2+ γ2)+δ ⁢α=β ⁢γ $$ \implies y \left( \delta^2 + \gamma^2 \right) + \delta \alpha = \beta \gamma $$ ⟹y= β⁢γ− δ⁢α δ2+γ 2 $$ \implies y = \frac{\beta \gamma - \delta \alpha}{\delta^2 + \gamma^2} $$
⟹α+ i⁢βγ +i⁢δ =β⁢δ +γ⁢α δ2+γ 2+i⁢ (β⁢γ −δ⁢α δ2+ γ2) γ+ i⁢δ≠0 ,δ2 +γ2≠0 ∎ $$ \implies \frac{\alpha + i \beta}{\gamma + i \delta} = \frac{\beta \delta + \gamma \alpha}{\delta^2 + \gamma^2} + i \left( \frac{\beta \gamma - \delta \alpha}{\delta^2 + \gamma^2} \right) ~~~\gamma + i \delta \ne 0,\ \delta^2 + \gamma^2 \ne 0 $$

Complex Square Root Proof

Suppose (x+i⁢y )2=α+ i⁢β $$\mathrm{Suppose\ } \left(x + i y\right)^2 = \alpha + i \beta$$ Findx, yin termsof α,β $$\mathrm{Find\ } x, y \mathrm{\ in\ terms\ of\ } \alpha, \beta$$ x2− y2+2⁢ i⁢x⁢ y=α+i ⁢β $$x^2 - y^2 + 2 i x y = \alpha + i \beta$$ ⟹{ x2−y2 =α2 ⁢x⁢y =β $$ \implies \begin{cases} x^2 - y^2 = \alpha \\ 2xy = \beta \end{cases} $$ α2+ β2=( x2−y2 )2+(2 ⁢x⁢y )2 $$\alpha^2 + \beta^2 = \left(x^2 - y^2\right) + \left(2xy\right)^2$$ =x4 −2⁢x2 ⁢y2+ y4+4⁢ x2⁢y 2 $$= x^4 - 2x^{2}y^{2} + y^4 + 4x^{2}y^{2}$$ =x4 +2⁢x2 ⁢y2+ y4 $$= x^4 + 2x^{2}y^{2} + y^4$$ α2+ β2=( x2+y2 )2 $$\alpha^{2} + \beta^{2} = \left(x^2 + y^2\right)^2$$ ⟹x2 +y2= α2+β2 $$\implies x^2 + y^2 = \sqrt{\alpha^2 + \beta^2}$$
⟹x2 +(x2− α)=α2 +β2 $$\implies x^2 + \left(x^2 - \alpha\right) = \sqrt{\alpha^2 + \beta^2}$$ ⟹2⁢ x2−α= α2+β 2 $$\implies 2x^2 - \alpha = \sqrt{\alpha^2 + \beta^2}$$ ⟹x2 =12(α +α2+ β2) $$\implies x^2 = \frac{1}{2}(\alpha + \sqrt{\alpha^2 + \beta^2})$$
⟹(y2 +α)+y2 =α2+ β2 $$\implies \left(y^2 + \alpha\right) + y^2 = \sqrt{\alpha^2 + \beta^2}$$ ⟹2⁢ y2+α= α2+β 2 $$\implies 2y^2 + \alpha = \sqrt{\alpha^2 + \beta^2}$$ ⟹y2 =12(− α+α2 +β2) $$\implies y^2 = \frac{1}{2}\left(-\alpha + \sqrt{\alpha^2 + \beta^2}\right)$$
⟹α+ i⁢β= ±12( α+α2 +β2) ±i1 2(−α+ α2+β 2) $$\implies \sqrt{\alpha + i \beta} = \pm \sqrt{\frac{1}{2}\left(\alpha + \sqrt{\alpha^2 + \beta^2}\right)} \pm i \sqrt{\frac{1}{2} \left(-\alpha + \sqrt{\alpha^2 + \beta^2}\right)}$$ ⟹α+ i⁢β= ±(α+ α2+β 22+i ⁢β|β |⁢ −α+α 2+β2 2)∎ $$ \implies \sqrt{\alpha + i \beta} = \pm \sqrt{\frac{\alpha + \sqrt{\alpha^2 + \beta^2}}{2}} \pm i \frac{\beta}{|\beta|} \sqrt{\frac{-\alpha + \sqrt{\alpha^2 + \beta^2})}{2} $$ Note that β|β| is a result of satisfying β=2⁢x ⁢yfrom above. $$ \mathrm{Note\ that\ } \frac{\beta}{|\beta|} \mathrm{\ is\ a\ result\ of\ satisfying\ } \beta = 2xy \mathrm{\ from\ above} $$

Practice Problems

I. Find the values; identify α and β $$ \mathrm{I.\ Find\ the\ values;\ identify\ \alpha\ and\ \beta} $$
a)( 1+2⁢i) 3 $$ \mathrm{a)\ } (1+2i)^3 $$ This is a solution

Works Cited

Ahlfors, Lars V. Complex Analysis: An Introduction To The Theory Of Analytic Functions Of One Complex Variable. 1953. AMS Chelsea Publishing, 2021, pp 1-4.

Gross, Herbert. Part I: Complex Variables. Calculus Revisited: Complex Variables, Differential Equations, An Linear Algebra. 1972. MIT OpenCourseWare, 2011.