Teaching Complex Analysis to...

Teaching Complex Analysis to Engineers

John P. D’AngeloUniversity of Illinois

Advanced Calculus for Engineers as a PostDoc at MIT (Lots ofresidues)

Courses at UIUC: 446, 448, 540

Honors Course at UIUC: 198 (I wrote my own book for this course:AMS Sally series)

ECE 493 (Spring 2014) (I will focus on this course)

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IMSE initative: support for creating interactions betweenmathematicians and engineers.

I got support for two TAs and for a math grad student to putprinted notes in latex.

We had a seminar with two ECE professors and grad students.

I attended two curriculum committee meetings in ECE department.

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Course content

I Review of linear algebra

I Review of complex variables

I Review of vector analysis

I Complex Analysis (Cauchy theory and residues)

I Transform methods (Laplace, Fourier, generating functions)

I Hilbert spaces including Riesz lemma, least squares regression,Sturm-Liouville theory.

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Some highlights

I Kakuro, Dec. 31 game, and linear algebra.

I transmission lines and linear fractional transformations.(ABCD matrices)

I summation by parts, conditionally convergent series

I Green’s function for Sturm-Liouville (py ′)′ + qy + λwy = 0.

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Course structure

I 2 diagnostic exams

I 2 midterms

I seven Homework assignments

I final exam

Diagnostic exams were widely praised by students.

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y ′ − 2y = 1 for x > 0y ′ − 2y = 0 for x < 0y continuous at 0.This problem caused considerable difficulty!Wrong answer if Laplace transform is used incorrectly.

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I prefer generating functions to the Z -transform.Given an, put f (z) =

∑anz

n. f is the Generating function.Put F (z) = f (1z ). F is the Z -transform.

Inversion: an is the Taylor coefficient.Easy inversion formula:

an =1

2πi

∫|z|=ε

f (z)

zn+1dz .

Formula using F is a bit different. Work on Riemann sphere,replace z by 1

z .

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I prefer generating functions to the Z -transform.Given an, put f (z) =

∑anz

n. f is the Generating function.Put F (z) = f (1z ). F is the Z -transform.Inversion: an is the Taylor coefficient.Easy inversion formula:

an =1

2πi

∫|z|=ε

f (z)

zn+1dz .

Formula using F is a bit different. Work on Riemann sphere,replace z by 1

z .

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I prefer generating functions to the Z -transform.Given an, put f (z) =

∑anz

n. f is the Generating function.Put F (z) = f (1z ). F is the Z -transform.Inversion: an is the Taylor coefficient.Easy inversion formula:

an =1

2πi

∫|z|=ε

f (z)

zn+1dz .

Formula using F is a bit different. Work on Riemann sphere,replace z by 1

z .

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Does anybody really know what day it is? Does anybodyreally care?

To me: The winding number n(γ, p) of a closed curve about apoint in C is an integer.n(γ, p) = 1

2πi

∫γ

dzz−p .

To people in ECE (I found this stuff on line)Phase UnwrappingIn working with phase delay, it is often necessary to “unwrap” thephase response Θ(ω). Phase unwrapping ensures that allappropriate multiples of 2π have been included in Θ(ω). Wedefined Θ(ω) simply as the complex angle of the frequencyresponse H(e jωT ), and this is not sufficient for obtaining a phaseresponse which can be converted to true time delay. If multiples of2π are discarded, as is done in the definition of complex angle, thephase delay is modified by multiples of the sinusoidal period.ETC.

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Does anybody really know what day it is? Does anybodyreally care?

To me: The winding number n(γ, p) of a closed curve about apoint in C is an integer.n(γ, p) = 1

2πi

∫γ

dzz−p .

To people in ECE (I found this stuff on line)Phase UnwrappingIn working with phase delay, it is often necessary to “unwrap” thephase response Θ(ω). Phase unwrapping ensures that allappropriate multiples of 2π have been included in Θ(ω). Wedefined Θ(ω) simply as the complex angle of the frequencyresponse H(e jωT ), and this is not sufficient for obtaining a phaseresponse which can be converted to true time delay. If multiples of2π are discarded, as is done in the definition of complex angle, thephase delay is modified by multiples of the sinusoidal period.ETC.

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Final Exam

There are 240 total points. There are 12 short problems, worth 15each. Then there is a choice of harder problems. Choose two ofthem; each is worth 30 points. Make it clear which of these twoyou want graded.

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1. By any valid method, solve the differential equationy ′ − 2y = H. Assume y is continuous at 0. Here H(t) = 1 fort ≥ 0 and H(t) = 0 for t < 0.

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2. Suppose that an is a sequence of real numbers such that

an = c1λn1 + c2λ

n2.

Here c1 and c2 are unknown constants, and 0 < λ1 < λ2. Find(with justification)

limn→∞

an+1

an.

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3. Consider the vector v = (4, 9, 3) ∈ R3. Find its orthogonalprojection onto the subspace generated by (2, 1, 0) and (0, 0, 1).

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4. Suppose H is a Hilbert space, and L : H → H is linear. Let Rdenote the range of L. Assume that f is not in R(L). Let g = Lube the unique element of R(L) such that ||f − g || is smallest.Show that L∗f = L∗Lu.

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5. Suppose V is a real or complex vector space, and L : V → V islinear. Give the precise definition of eigenvalue and eigenvector.Find the eigenvalues and corresponding eigenvectors of the linearmap L : R2 → R2 with matrix(

26 −1836 −25

).

The numbers work out easily. Note that 25× 26 = 650 and18× 36 = 648.

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6. By any valid method (but indicate what it is), find the inverseLaplace transform of s

s2+4. Express your answer in real variables.

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7. Suppose H is a Hilbert space and φn (for n ≥ 0) is acomplete orthonormal system. Find a simple formula for

||∞∑n=0

φntn||2.

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8. Let Ω be an open subset of C, let f : Ω→ C, and let p ∈ Ω.

I Give the definition that f is complex analytic at p.

I Assume that f is complex analytic on Ω, and f = u + iv forreal functions u, v as usual. State the Cauchy-Riemannequations for u, v , and use them to show that u and v areharmonic. (Recall that w is harmonic if wxx + wyy = 0.)

I If γ is a closed curve and f is analytic on and inside γ, showthat ∫

γf (z) dz =

∫γ

(u + iv)(dx + i dy) = 0.

Use Green’s theorem.

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9. Put the differential equation xy ′′ + 5y ′ + λxy = 0 inSturm-Liouville form

(Py ′)′ + Qy + λwy = 0.

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10. Suppose f is complex analytic near p and f (p) = 0. If f is notidentically zero, then 1

f has a pole at p. What is the residue of 1f

at p? Check your answer by finding the residue of 1e3z−1 at 0.

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11. Let H be a Hilbert space, and let L : H → H be linear.Assume that f is an eigenvector for L with correspondingeigenvalue λ.

I If L = L∗, show that λ is real.

I If L∗ = L−1, what can you say about λ? (Justify)

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12. Suppose f is a differentiable function on Rn and γ is a curve.What is the value of ∫

γ∇f · ds? (∗)

Either directly or by using (*), find∫γ e

izdz if γ is the line segmentfrom 0 to i in C.

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II. Longer Problems.1. Assume that f is complex analytic on Ω.

I Use Cauchy’s theorem (from problem 8) to verify the Cauchyintegral formula.

I Use the Cauchy integral formula to verify that, for eachp ∈ Ω, we can write

f (z) =∞∑n=0

an(z − p)n,

where the series converges in a disk about p.

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2. Using the method described, compute the integral

J =

∫ ∞0

dx

1 + xm.

Assume m > 1. Let γR,ε denote the piecewise smooth simpleclosed curve: from ε to R on the positive real axis, followed by a

circular arc to the point Re2πim , followed by a line segment toward

the origin to the point εe2πim , followed by a circular arc back to the

point ε.

I Find∫γR ,ε

. (Use problem 10 to compute the residue.)

I Determine what happens as R →∞ and ε→ 0.

I Simplify your answer to express J in terms of sin and or sinc .

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3. Consider the differential equation

x2y ′′ + xy ′ + λy = 0

on the interval [1, e] with boundary conditions y(1) = y(e) = 0.First put it in Sturm-Liouville form. Then verify that sin(nπlogx)are eigenfunctions. Find the corresponding eigenvalues.

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4. Prove that∑∞

n=1sin(nx)

n converges for all real x .

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5. Consider the differential equation

I ′′ +R

LI ′ +

1

LCI = 0

governing an L,R,C circuit. Here I (t) is the current at time t.

The expression ζ =√

CLR2 is called the damping factor. Describe

the solutions in the three cases ζ < 1, ζ > 1, and ζ = 1.

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The language barrier is large but can be overcome. Workingtogether is possible, albeit on occasion painful for both sides. Inthe end one cannot feel anything but deep appreciation of theunderlying mathematics.QUESTIONS?

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