some complex analysis projects - westmont collegehowell/complex-analysis/... · 2014. 7. 17. ·...

SOME COMPLEX ANALYSIS PROJECTS

Robert Sachs

Department of Mathematical SciencesGeorge Mason University

Fairfax, Virginia 22030

rsachs@gmu.edu

June 26, 2014

R. Sachs (GMU) Complex Projects June 2014 1 / 1

Introduction

Student short projects during the course (time permitting) have workedwell for me

They allow students to explore questions that interest them andhopefully me

I’ve tended to do them near end of the course, but sooner might push aneed-to-know better

Complex analysis courses often end just when things can get reallyinteresting

R. Sachs (GMU) Complex Projects June 2014 2 / 1

Introduction

Student short projects during the course (time permitting) have workedwell for me

They allow students to explore questions that interest them andhopefully me

I’ve tended to do them near end of the course, but sooner might push aneed-to-know better

Complex analysis courses often end just when things can get reallyinteresting

R. Sachs (GMU) Complex Projects June 2014 2 / 1

Introduction

Student short projects during the course (time permitting) have workedwell for me

They allow students to explore questions that interest them andhopefully me

I’ve tended to do them near end of the course, but sooner might push aneed-to-know better

Complex analysis courses often end just when things can get reallyinteresting

R. Sachs (GMU) Complex Projects June 2014 2 / 1

Introduction

Student short projects during the course (time permitting) have workedwell for me

They allow students to explore questions that interest them andhopefully me

I’ve tended to do them near end of the course, but sooner might push aneed-to-know better

Complex analysis courses often end just when things can get reallyinteresting

R. Sachs (GMU) Complex Projects June 2014 2 / 1

Who are the high school kids?

They have had multivariable calculus and linear algebra

Neither course is taught at super-advanced level, nor the prior APcalculus course

Often but not always very computer savvy

Not much if any experience in proof

About 1/4 or less of the 20 to 36 will major in math at college

R. Sachs (GMU) Complex Projects June 2014 3 / 1

Who are the high school kids?

They have had multivariable calculus and linear algebra

Neither course is taught at super-advanced level, nor the prior APcalculus course

Often but not always very computer savvy

Not much if any experience in proof

About 1/4 or less of the 20 to 36 will major in math at college

R. Sachs (GMU) Complex Projects June 2014 3 / 1

Who are the high school kids?

They have had multivariable calculus and linear algebra

Neither course is taught at super-advanced level, nor the prior APcalculus course

Often but not always very computer savvy

Not much if any experience in proof

About 1/4 or less of the 20 to 36 will major in math at college

R. Sachs (GMU) Complex Projects June 2014 3 / 1

Who are the high school kids?

They have had multivariable calculus and linear algebra

Neither course is taught at super-advanced level, nor the prior APcalculus course

Often but not always very computer savvy

Not much if any experience in proof

About 1/4 or less of the 20 to 36 will major in math at college

R. Sachs (GMU) Complex Projects June 2014 3 / 1

Who are the high school kids?

They have had multivariable calculus and linear algebra

Neither course is taught at super-advanced level, nor the prior APcalculus course

Often but not always very computer savvy

Not much if any experience in proof

About 1/4 or less of the 20 to 36 will major in math at college

R. Sachs (GMU) Complex Projects June 2014 3 / 1

Some topics that arise organically but aren’t reallycomplex analysis

Some nice topics that are not complex analysis are topological

Contractible and non-contractible loops – fundamental group;homology version of Cauchy theorem

Topological degree of a mapping – important aspect of local degree incomplex analysis, ties back to fundamental theorem of algebra.

General idea of integer-valued continuous maps must be locallyconstant away from singularities

Vector fields and the Poincare index in the planar case

R. Sachs (GMU) Complex Projects June 2014 4 / 1

Some topics that arise organically but aren’t reallycomplex analysis

Some nice topics that are not complex analysis are topological

Contractible and non-contractible loops – fundamental group;homology version of Cauchy theorem

Topological degree of a mapping – important aspect of local degree incomplex analysis, ties back to fundamental theorem of algebra.

General idea of integer-valued continuous maps must be locallyconstant away from singularities

Vector fields and the Poincare index in the planar case

R. Sachs (GMU) Complex Projects June 2014 4 / 1

Some topics that arise organically but aren’t reallycomplex analysis

Some nice topics that are not complex analysis are topological

Contractible and non-contractible loops – fundamental group;homology version of Cauchy theorem

Topological degree of a mapping – important aspect of local degree incomplex analysis, ties back to fundamental theorem of algebra.

General idea of integer-valued continuous maps must be locallyconstant away from singularities

Vector fields and the Poincare index in the planar case

R. Sachs (GMU) Complex Projects June 2014 4 / 1

Some topics that arise organically but aren’t reallycomplex analysis

Some nice topics that are not complex analysis are topological

Contractible and non-contractible loops – fundamental group;homology version of Cauchy theorem

Topological degree of a mapping – important aspect of local degree incomplex analysis, ties back to fundamental theorem of algebra.

General idea of integer-valued continuous maps must be locallyconstant away from singularities

Vector fields and the Poincare index in the planar case

R. Sachs (GMU) Complex Projects June 2014 4 / 1

Some topics that arise organically but aren’t reallycomplex analysis

Some nice topics that are not complex analysis are topological

Contractible and non-contractible loops – fundamental group;homology version of Cauchy theorem

Topological degree of a mapping – important aspect of local degree incomplex analysis, ties back to fundamental theorem of algebra.

General idea of integer-valued continuous maps must be locallyconstant away from singularities

Vector fields and the Poincare index in the planar case

R. Sachs (GMU) Complex Projects June 2014 4 / 1

Other topics are about manifolds

Projective coordinates, the Riemann Sphere, and Mobiustransformations

Matrix groups

Non-Euclidean geometry

R. Sachs (GMU) Complex Projects June 2014 5 / 1

Other topics are about manifolds

Projective coordinates, the Riemann Sphere, and Mobiustransformations

Matrix groups

Non-Euclidean geometry

R. Sachs (GMU) Complex Projects June 2014 5 / 1

Other topics are about manifolds

Projective coordinates, the Riemann Sphere, and Mobiustransformations

Matrix groups

Non-Euclidean geometry

R. Sachs (GMU) Complex Projects June 2014 5 / 1

Algebra and Number Theory

Gaussian integers illuminates sums of squares – pretty basic, cando early

Lagrange and Jacobi on sums of four squares (interesting inquiryto find the Jacobi formula – not proof)

Prime number theorem (Newman version of proof) – maybecontrast with “elementary proofs”

Other roots of negative numbers adjoined (-17 for example) –non-unique factorization and Fermat’s last theorem

R. Sachs (GMU) Complex Projects June 2014 6 / 1

Algebra and Number Theory

Gaussian integers illuminates sums of squares – pretty basic, cando early

Lagrange and Jacobi on sums of four squares (interesting inquiryto find the Jacobi formula – not proof)

Prime number theorem (Newman version of proof) – maybecontrast with “elementary proofs”

Other roots of negative numbers adjoined (-17 for example) –non-unique factorization and Fermat’s last theorem

R. Sachs (GMU) Complex Projects June 2014 6 / 1

Algebra and Number Theory

Gaussian integers illuminates sums of squares – pretty basic, cando early

Lagrange and Jacobi on sums of four squares (interesting inquiryto find the Jacobi formula – not proof)

Prime number theorem (Newman version of proof) – maybecontrast with “elementary proofs”

Other roots of negative numbers adjoined (-17 for example) –non-unique factorization and Fermat’s last theorem

R. Sachs (GMU) Complex Projects June 2014 6 / 1

Algebra and Number Theory

Gaussian integers illuminates sums of squares – pretty basic, cando early

Lagrange and Jacobi on sums of four squares (interesting inquiryto find the Jacobi formula – not proof)

Prime number theorem (Newman version of proof) – maybecontrast with “elementary proofs”

Other roots of negative numbers adjoined (-17 for example) –non-unique factorization and Fermat’s last theorem

R. Sachs (GMU) Complex Projects June 2014 6 / 1

Complex Linear Algebra

Linear Algebra: Did inner product most likely already

Dimension counting real vs. complex

Cauchy integral theorem for square matrices A connects back tolots of topics, very powerful

Defining exp(At) and inverse Laplace transform.

Can prove Cayley Hamilton theorem in one line or so

R. Sachs (GMU) Complex Projects June 2014 7 / 1

Complex Linear Algebra

Linear Algebra: Did inner product most likely already

Dimension counting real vs. complex

Cauchy integral theorem for square matrices A connects back tolots of topics, very powerful

Defining exp(At) and inverse Laplace transform.

Can prove Cayley Hamilton theorem in one line or so

R. Sachs (GMU) Complex Projects June 2014 7 / 1

Complex Linear Algebra

Linear Algebra: Did inner product most likely already

Dimension counting real vs. complex

Cauchy integral theorem for square matrices A connects back tolots of topics, very powerful

Defining exp(At) and inverse Laplace transform.

Can prove Cayley Hamilton theorem in one line or so

R. Sachs (GMU) Complex Projects June 2014 7 / 1

Complex Linear Algebra

Linear Algebra: Did inner product most likely already

Dimension counting real vs. complex

Cauchy integral theorem for square matrices A connects back tolots of topics, very powerful

Defining exp(At) and inverse Laplace transform.

Can prove Cayley Hamilton theorem in one line or so

R. Sachs (GMU) Complex Projects June 2014 7 / 1

Complex Linear Algebra

Linear Algebra: Did inner product most likely already

Dimension counting real vs. complex

Cauchy integral theorem for square matrices A connects back tolots of topics, very powerful

Defining exp(At) and inverse Laplace transform.

Can prove Cayley Hamilton theorem in one line or so

R. Sachs (GMU) Complex Projects June 2014 7 / 1

Cauchy formula for square matrices

f (A) =1

2πi

∮f (λ)(λI − A)−1 dλ

where f is analytic on a set containing all eigenvalues of A and thecontour encloses all the eigenvalues as a simple curve (all windingnumbers are 1).

Analyst thinking: if A is diagonal or even diagonalizable, then true as1-D cases glued together. General case: Jordan canonical form with ε– i.e. diagonalizable case is dense.

R. Sachs (GMU) Complex Projects June 2014 8 / 1

Cauchy formula for square matrices

f (A) =1

2πi

∮f (λ)(λI − A)−1 dλ

where f is analytic on a set containing all eigenvalues of A and thecontour encloses all the eigenvalues as a simple curve (all windingnumbers are 1).

Analyst thinking: if A is diagonal or even diagonalizable, then true as1-D cases glued together. General case: Jordan canonical form with ε– i.e. diagonalizable case is dense.

R. Sachs (GMU) Complex Projects June 2014 8 / 1

Example: eAt uses f (λ) = eλt

Example: Cayley-Hamilton: Let χA(λ) = det(λI − A), thenχA(A) vanishing has a one line proof.

This introduces a big concept: resolvent of an operator in its mostbasic case as generalizing the Cauchy kernel.

R. Sachs (GMU) Complex Projects June 2014 9 / 1

Example: eAt uses f (λ) = eλt

Example: Cayley-Hamilton: Let χA(λ) = det(λI − A), thenχA(A) vanishing has a one line proof.

This introduces a big concept: resolvent of an operator in its mostbasic case as generalizing the Cauchy kernel.

R. Sachs (GMU) Complex Projects June 2014 9 / 1

Example: eAt uses f (λ) = eλt

Example: Cayley-Hamilton: Let χA(λ) = det(λI − A), thenχA(A) vanishing has a one line proof.

This introduces a big concept: resolvent of an operator in its mostbasic case as generalizing the Cauchy kernel.

R. Sachs (GMU) Complex Projects June 2014 9 / 1

Discrete Fourier

Fourier Series – infinite sum:∞∑

n=−∞cn eint

has unique Fourier recipe for ck as integral.

Discrete Fourier uses finite sum instead, so not “exact”.

Can do finite problem in several ways but a useful one is to use equallyspace points (usually roots of unity) and fit those values exactly(interpolation)

The discrete terms are unique but the mapping is not one-to-one(aliasing – wagon wheels in westerns can go backwards).

Basis for lots of modern things like digital music formats, some imageprocessing, etc.

R. Sachs (GMU) Complex Projects June 2014 10 / 1

Discrete Fourier

Fourier Series – infinite sum:∞∑

n=−∞cn eint

has unique Fourier recipe for ck as integral.

Discrete Fourier uses finite sum instead, so not “exact”.

Can do finite problem in several ways but a useful one is to use equallyspace points (usually roots of unity) and fit those values exactly(interpolation)

The discrete terms are unique but the mapping is not one-to-one(aliasing – wagon wheels in westerns can go backwards).

Basis for lots of modern things like digital music formats, some imageprocessing, etc.

R. Sachs (GMU) Complex Projects June 2014 10 / 1

Discrete Fourier

Fourier Series – infinite sum:∞∑

n=−∞cn eint

has unique Fourier recipe for ck as integral.

Discrete Fourier uses finite sum instead, so not “exact”.

Can do finite problem in several ways but a useful one is to use equallyspace points (usually roots of unity) and fit those values exactly(interpolation)

The discrete terms are unique but the mapping is not one-to-one(aliasing – wagon wheels in westerns can go backwards).

Basis for lots of modern things like digital music formats, some imageprocessing, etc.

R. Sachs (GMU) Complex Projects June 2014 10 / 1

Discrete Fourier

Fourier Series – infinite sum:∞∑

n=−∞cn eint

has unique Fourier recipe for ck as integral.

Discrete Fourier uses finite sum instead, so not “exact”.

Can do finite problem in several ways but a useful one is to use equallyspace points (usually roots of unity) and fit those values exactly(interpolation)

The discrete terms are unique but the mapping is not one-to-one(aliasing – wagon wheels in westerns can go backwards).

Basis for lots of modern things like digital music formats, some imageprocessing, etc.

R. Sachs (GMU) Complex Projects June 2014 10 / 1

Discrete Fourier

Fourier Series – infinite sum:∞∑

n=−∞cn eint

has unique Fourier recipe for ck as integral.

Discrete Fourier uses finite sum instead, so not “exact”.

Can do finite problem in several ways but a useful one is to use equallyspace points (usually roots of unity) and fit those values exactly(interpolation)

The discrete terms are unique but the mapping is not one-to-one(aliasing – wagon wheels in westerns can go backwards).

Basis for lots of modern things like digital music formats, some imageprocessing, etc.

R. Sachs (GMU) Complex Projects June 2014 10 / 1

More discrete Fourier

Fast finite Fourier makes this even faster – done by Gauss,rediscovered by Cooley and Tukey in 1960s.

Split into pieces (often two) then iterate – faster calculations.

Lots more interesting and useful things here – Signals and transferfunctions, Wiener filter, other bases, wavelets, uncertainty principle,Chebyshev interpolation and Trefethen et al package ChebFun.

R. Sachs (GMU) Complex Projects June 2014 11 / 1

More discrete Fourier

Fast finite Fourier makes this even faster – done by Gauss,rediscovered by Cooley and Tukey in 1960s.

Split into pieces (often two) then iterate – faster calculations.

Lots more interesting and useful things here – Signals and transferfunctions, Wiener filter, other bases, wavelets, uncertainty principle,Chebyshev interpolation and Trefethen et al package ChebFun.

R. Sachs (GMU) Complex Projects June 2014 11 / 1

More discrete Fourier

Fast finite Fourier makes this even faster – done by Gauss,rediscovered by Cooley and Tukey in 1960s.

Split into pieces (often two) then iterate – faster calculations.

Lots more interesting and useful things here – Signals and transferfunctions, Wiener filter, other bases, wavelets, uncertainty principle,Chebyshev interpolation and Trefethen et al package ChebFun.

R. Sachs (GMU) Complex Projects June 2014 11 / 1

Complex version of classical Hamiltonian systems

A very useful topic for physics: a classical Hamiltonian system with ndegrees of freedom has phase space of size R2n

As in John’s talk, a matrix J comes into play to write the system in“canonical form” – will try to go to a document online from our localUCSB physics – in any case this is big when it comes to going fromclassical to quantum system.

R. Sachs (GMU) Complex Projects June 2014 12 / 1

Complex version of classical Hamiltonian systems

A very useful topic for physics: a classical Hamiltonian system with ndegrees of freedom has phase space of size R2n

As in John’s talk, a matrix J comes into play to write the system in“canonical form” – will try to go to a document online from our localUCSB physics – in any case this is big when it comes to going fromclassical to quantum system.

R. Sachs (GMU) Complex Projects June 2014 12 / 1

Complex Analysis topics

We can discuss more topics, but at the end of every book is a varietyof topics

Usually not varieties, but that could work too!

R. Sachs (GMU) Complex Projects June 2014 13 / 1

Complex Analysis topics

We can discuss more topics, but at the end of every book is a varietyof topics

Usually not varieties, but that could work too!

R. Sachs (GMU) Complex Projects June 2014 13 / 1

Concluding remarks

I didn’t mention lots of other topics (but maybe we hit them duringdiscussions): algebraic curves and homogeneous coordinates; ellipticcurves; uniformization; growth rates of entire functions; Picardtheorems; Weierstrass product; Gamma function; Zeta function;complex ODEs; modular forms; etc.

Also historical topics!

R. Sachs (GMU) Complex Projects June 2014 14 / 1

Concluding remarks

I didn’t mention lots of other topics (but maybe we hit them duringdiscussions): algebraic curves and homogeneous coordinates; ellipticcurves; uniformization; growth rates of entire functions; Picardtheorems; Weierstrass product; Gamma function; Zeta function;complex ODEs; modular forms; etc.

Also historical topics!

R. Sachs (GMU) Complex Projects June 2014 14 / 1