University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

A Visual Approach to Complex Analysis

Yuxuan Bao Yucheng Shi Justin Vorhees

University of Michigan

October 23, 2018

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Table of Contents

1 IntroductionVisualization ToolsResearch Motivation

2 Marden’s Theorem

3 Current Progress

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Graphing Functions

Real functions f : R→ R

Complex functions g : C→ C

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Graphing Functions

Real functions f : R→ R

Complex functions g : C→ C

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Graphing Functions

Real functions f : R→ R

Complex functions g : C→ C

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Graphing Functions

Real functions f : R→ R

Complex functions g : C→ C

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Phase Plots

z = re iθ ∈ C

Modulus = r , Phase = e iθ

Figure: Phase Color Wheel Figure: f (z) = z

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Phase Plots

z = re iθ ∈ CModulus = r , Phase = e iθ

Figure: Phase Color Wheel Figure: f (z) = z

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Phase Plots

z = re iθ ∈ CModulus = r , Phase = e iθ

Figure: Phase Color Wheel

Figure: f (z) = z

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Phase Plots

z = re iθ ∈ CModulus = r , Phase = e iθ

Figure: Phase Color Wheel Figure: f (z) = z

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Phase Plot Examples

Figure: f (z) = zFigure: f (z) = (z − 2− i)(z −2 + i)(z + 2− i)(z + 2 + i)

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Phase Plot Examples

Figure: f (z) = z Figure: f (z) = 1/z

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Roots of f , f ′

Research focus: relationship between roots of polynomialf and roots of f ′

Calculus I: Rolle’s Theorem

Complex Analysis: Gauss-Lucas Theorem

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Table of Contents

1 IntroductionVisualization ToolsResearch Motivation

2 Marden’s Theorem

3 Current Progress

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Gauss-Lucas Theorem

Definition (Convex Hull)

Let X be a bounded subset of the plane, the convex hull canbe visualized as the shape enclosed by a rubber band stretchedaround X .

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Gauss-Lucas Theorem

Theorem (Gauss-Lucas Theorem)

Let P be a polynomial, the roots of P ′ all lie within the convexhull of the roots of P.

Figure: f (z) = (z − 2− i)(z − 2 + i)(z + 2− i)(z + 2 + i)

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Marden’s Theorem

Theorem (Marden’s Theorem)

Suppose the roots of a third-degree polynomial f are z1, z2 andz3 and they form a triangle. There is a unique ellipse inscribedin the triangle and tangent to the sides at their midpoints. Thefoci of that ellipse are the zeroes of the derivative f ′.

Figure: f (z) = (z − 2− i)(z − 2 + i)(z + 2− i)(z + 2 + i)

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Table of Contents

1 IntroductionVisualization ToolsResearch Motivation

2 Marden’s Theorem

3 Current Progress

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Extension to Higher Degrees

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Extension to Higher Degrees

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Extension to Higher Degrees

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Extension to Higher Degrees

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

Extension to Higher Degrees

University ofMichiganLoG(M)

Yuxuan Bao,Yucheng Shi,

JustinVorhees

Introduction

VisualizationTools

ResearchMotivation

Marden’sTheorem

CurrentProgress

References

[1] Elias Wegert. Visual Complex Functions. 2012.

[2] Lab of Geometry at Michigan. LoG(M) Beamer Template.University of Michigan Department of Mathematics. 2018.

Special thanks to our mentors, Professor Luke Edholm andRachel Webb, and to the LoG(M) Leadership.