Qualifying Exams Study Sheet

Jesse Adams

August 10, 2014

Contents

1) Previous Material 31.a) Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.a.i) Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.a.ii) Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.a.iii) Trig Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.a.iv) Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.a.v) Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.a.vi) Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.a.vii) Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.a.viii)Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.a.ix) Integration Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.a.x) Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.a.xi) Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.a.xii) Trig Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.b) Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2) Numerical Analysis 62.a) Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.a.i) Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.a.ii) Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.a.iii) Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.a.iv) Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.a.v) QR Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.a.vi) Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.a.vii) Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.a.viii)Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.a.ix) Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.a.x) Cholesky Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.a.xi) Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.a.xii) Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.b) Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.b.i) Functional Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.b.ii) Polynomial Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.b.iii) ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.b.iv) Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.b.v) Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.b.vi) BVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3) Analysis 143.a) Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.a.i) `p Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.a.ii) Lebesgue (Lp) Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.a.iii) Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.b) Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1

3.b.i) Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.b.ii) Defining Topologies and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.c) Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.c.i) Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.c.ii) Specific Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.c.iii) Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.d) Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.d.i) Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.d.ii) Measurable Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.d.iii) Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.e) Convergence and Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.e.i) Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.f) Uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.f.i) Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.f.ii) Interchanging Limits and Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.f.iii) More on Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.g) Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.g.i) Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.g.ii) Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4) Principals and Methods 214.a) Dynamics of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.a.i) Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.a.ii) Phase Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.a.iii) Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.b) Contour Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.b.i) Complex Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.b.ii) Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.c) Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.c.i) Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.c.ii) Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.d) Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.d.i) Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.e) Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.e.i) Common Functions/Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.e.ii) Other stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.e.iii) Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.e.iv) Other Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.f) Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.f.i) Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.f.ii) Sturm-Liouville . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.g) Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.g.i) Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.g.ii) Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2

1) Previous Material

1.a) Calculus

1.a.i) Fundamental Theorem of Calculus

Given f(x) continuous and Riemann integrable on [a, b],

1. F (x) =∫ xaf(t)dt is continuously differentiable on (a, b), with F ′(x) = f(x).

2.∫ baf(x)dx = [F (x)]

ba = F (b)− F (a).

1.a.ii) Sums

1.

n∑k=1

k =n(n+ 1)

2;

n∑k=1

k2 =n(n+ 1)(2n+ 1)

6;

n∑k=1

k3 =

(n(n+ 1)

2

)2

2.

∞∑n=1

1

n2=π2

6

1.a.iii) Trig Substitutions

1.√b2x2 − a2 ⇒ x =

a

bsec (θ)

2.√a2 − b2x2 ⇒ x =

a

bsin (θ)

3.√a2 + b2x2 ⇒ x =

a

btan (θ)

4. x = tan (θ/2)⇒ sin(θ) =2x

1 + x2, cos(θ) =

1− x2

1 + x2, dθ =

2 dx

1 + x2

1.a.iv) Integrals

1. Arc Length: L =∫ds =

∫ x=b

x=a

√1 +

(dy

dx

)2

dx =

∫ y=d

y=c

√1 +

(dx

dy

)2

dy =

∫ t=f

t=e

√(dx

dt

)2

+

(dy

dt

)2

dt =∫ θ=h

θ=g

√r2 +

(dr

dθ

)2

dθ

2. Surface Area: A =∫

2πy ds about x-axis, A =∫

2πx ds about y-axis, with ds as defined in arc length.

1.a.v) Sequences

1. Integral test: Given continuous, positive, and decreasing f(x) on [k,∞) with f(n) = an, then if∫∞kf(x) dx

is convergent/divergent, then so is∑∞n=k an.

2. Comparison test: Given two series with 0 ≤ an ≤ bn ∀ n, then∑bn < ∞ ⇒

∑an < ∞;

∑an = ∞ ⇒∑

bn =∞.

3. Limit comparison test: c = limn→∞

anbn

. If 0 < c <∞, then either both converge or both diverge.

4. Alternating series test: Given an = (−1)nbn or an = (−1)n+1bn with bn ≥ 0, if limn→∞

bn = 0 and {bn}decreasing, then

∑an is convergent.

5. Ratio test: L = limn→∞

∣∣∣∣an+1

an

∣∣∣∣. L < 1⇒ convergence, L > 1⇒ divergence.

6. Root test: L = limn→∞

n√|an|. L < 1⇒ absolute convergence, L > 1⇒ divergence.

7. Absolute convergence:∑|an| <∞. Implies convergence, otherwise the series is conditionally convergent.

3

1.a.vi) Series

1. Power series:1

a− f(x)=

1

a

∞∑n=0

(f(x)

a

)nprovided |f(x)| < |a|, and a 6= 0.

2. Taylor series: f(x) =

∞∑n=0

f (n)(x0)

n!(x− x0)n

3. Binomial series: (a+ b)n =

n∑k=0

(n

k

)an−kbk

4. (1 + z)k =

∞∑n=0

(k

n

)zn for |z| < 1, and

(k

n

)=k(k − 1) · · · (k − n+ 1)

n!.

5. ez =

∞∑n=0

zn

n!; sin(z) =

∞∑n=0

z2n+1

(2n+ 1)!(−1)n; cos(z) =

∞∑n=0

z2n

(2n)!(−1)n; ln(z) =

∞∑n=1

(z − 1)n

n(−1)n+1

1.a.vii) Vectors

Given a vector function r(t) with r′(t) 6= 0,

1. Unit tangent vector: T(t) =r′(t)

||r′(t)||.

2. Unit normal vector: N(t) =T′(t)

||T′(t)||.

3. Binormal vector: B(t) = T(t)×N(t).

4. Arc length: L =∫ ba||r′(t)|| dt =

∫ t0||r′(u)|| du.

5. Curvature: κ =||T′(t)||||r′(t)||

=||r′(t)× r′′(t)||||r′(t)||3

.

1.a.viii) Coordinate Systems

1. Cylindrical: r2 = x2 + y2, θ = tan−1(y/x), z = z; x = r cos(θ), y = r sin(θ), z = z.

2. Spherical: r = ρ sin(φ), θ = θ, z = ρ cos(φ); ρ2 = r2 + z2; x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), z =ρ cos(φ); ρ2 = x2 + y2 + z2

1.a.ix) Integration Elements

1. In general, take the determinant: dx =

∣∣∣∣∂(x)

∂u

∣∣∣∣ du =

∂x1

∂u1

∂x1

∂u2

∂x1

∂u3∂x2

∂u1

∂x2

∂u2

∂x2

∂u3∂x3

∂u1

∂x3

∂u2

∂x3

∂u3

du

2. Cylindrical:

(a) Constant radius: dA = r dθ dz

(b) Constant angle: dA = dr dz

(c) Constant height: dA = r dr dθ.

(d) Volume: dV = r dr dθ dz

3. Spherical:

(a) Constant radius: dA = ρ2 sin(φ) dφ dθ

(b) Constant φ: dA = ρ sin(φ) dθ dr

(c) Constant θ: dA = ρ dρ dφ

(d) Volume: dV = ρ2 sin(φ) dρ dφ dθ

4

1.a.x) Multiple Integrals

1. Change of variables:

∫∫D

f(x, y) dA =

∫∫S

f(g(u, v), h(u, v))

∣∣∣∣∂(x, y)

∂(u, v)

∣∣∣∣ dudv (similarly for triple integrals).

2. Cylindrical: dA = r drdθ, dV = r dzdrdθ, Spherical: dV = ρ2 sin(φ) dρdθdφ.

1.a.xi) Vector Fields

1. Gradient: ∇f(x, y, z) = 〈fx, fy, fz〉.

2. Conservative vector field: F such that F = ∇f , where f is called the potential function.

3. Given vector field F = P i+Qj on open, simply connected D. If P,Q have continuous 1st order derivatives in

D and∂P

∂y=∂Q

∂xthen F is conservative.

4. Green’s Theorem: C positively oriented, piecewise smooth closed curve enclosing D. P,Q have continuous

1st order partials, then

∫C

P dx+Q dy =

∫∫D

(∂Q

∂x− ∂P

∂y

)dA.

5. Curl: Let F = P i+Qj +Rk. Then

curl(F) = ∇× F =

∣∣∣∣∣∣i j k∂∂x

∂∂y

∂∂z

P Q R

∣∣∣∣∣∣(a) If f(x, y, z) has continuous 2nd order partials, then curl(∇f) = 0.

(b) If F is a conservative vector field, then curl(F) = 0.

6. Divergence:

div(F) = ∇ · F =∂P

∂x+∂Q

∂y+∂R

∂z

7. Stoke’s Theorem: Given smooth surface S bounded by simple, closed, smooth curve C and vector field F,∫C

F · dr =

∫∫S

curl(F) · dS

8. Divergence Theorem: Given simple solid region E, boundary surface S, and vector field F with continuous1st order partials,∫∫

S

F · dS =

∫∫∫E

div(F)dV

1.a.xii) Trig Identities

1. cos(a± b) = cos(a) cos(b)∓ sin(a) sin(b)

2. sin(a± b) = sin(a) cos(b)± sin(b) cos(a)

3. sin(2θ) = 2 sin(θ) cos(θ)

4. cos(2θ) = 1− 2 sin2(θ) = 2 cos2(θ)− 1

5

1.b) Differential Equations

2) Numerical Analysis

2.a) Linear Algebra

2.a.i) Basics

For a matrix A ∈ Cm×n,

1. Range: Space spanned by the columns of A, i.e. im(A) = range(A) = {y : Ax = y}.

2. Nullspace: ker(A) = null(A) = {x : Ax = 0}.

3. In Rn, null(A) = (range(A>))⊥, and null(A>) = (range(A))⊥.

4. Unitary: Q∗ = Q−1.

5. Symmetric: A> = A.

(a) A real ⇒ real eigenvalues, and is diagonalizable by real, orthogonal Q (i.e. D = Q>AQ).

(b) A−1 symmetric iff A symmetric.

(c) If A, B both symmetric, then AB symmetric iff AB = BA (i.e. they commute).

6. Hermitian: A∗ = A.

(a) Main diagonal is real.

(b) Has real eigenvalues.

(c) Normal, i.e. A∗A = AA∗.

7. Skew Hermitian: A∗ = −A. If A ∈ Rm×n, then it’s skew symmetric.

8. Determinant: det(A) =∏j λj

9. Trace: trace(A) =∑j ajj =

∑j λj

10. Positive definite: x∗Ax > 0 ∀ x 6= 0,x ∈ Cm.

(a) Hermitian pos. def.: Positive definite with A∗ = A.

(b) All eigenvalues real and positive

(c) For λi 6= λj , vi ⊥ vj (eigenvectors are orthogonal).

(d) For HPD A, full rank X ∈ Cm×n, then X∗AX is HPD

11. Spectral Radius: ρ(A) = maxi(|λi|), and ρ(A) ≤ ||A||.

2.a.ii) Norms

1. Must satisfy

(a) ||x|| ≥ 0, with ||x|| = 0⇔ x = 0

(b) ||αx|| = |α| ||x||(c) ||x + y|| ≤ ||x||+ ||y||

2. All vector norms on Cn are equivalent, i.e. ∃ 0 < c1 < c2 <∞ : c1 ||·||∗ ≤ ||·||∗∗ ≤ c2 ||·||∗

3. Induced Matrix Norms: ||A||(m,n) = supx∈Cn

x 6=0

||Ax||(m)

||x||(n)

= sup||x||(n)=1

||Ax||(m)

4. ||ABx|| ≤ ||A|| ||Bx|| ≤ ||A|| ||B|| ||x|| ⇒ ||AB|| ≤ ||A|| ||B||

6

5. Frobenius Norm: ||A||F =

∑i,j

|aij |21/2

=

(∑i

||ai||22

)1/2

=√trace(A∗A) =

√∑i

σ2i

6. Nuclear Norm: ||A||∗ = trace(√A∗A) =

∑i σi

7. Unitary matrices Q satisfy ||QA||2 = ||A||2, ||QA||F = ||A||F

2.a.iii) Singular Value Decomposition

Given A ∈ Cm×n, with m ≥ n

1. Reduced SVD: A = U ΣV ∗, with U ∈ Cm×n, Σ ∈ Rn×n≥0 , and V ∈ Cn×n; U , V unitary, and Σ diagonal withdecreasing elements.

2. Full SVD: A = UΣV with

U =[U U⊥

], V ∗ =

[V ∗

(V ∗)⊥

], Σ =

[Σ

0(m−n)×n

]3. Finding the SVD:

(a) σj ’s are the square roots of the eigenvalues of A∗A or AA∗.

(b) Find eigenvectors: (λjI −A∗A)vj = 0, or (λjI −AA∗)uj = 0.

(c) Find the other matrix: U = Σ−1AV , or V = A∗U Σ−1.

4. Properties:

(a) r = rank(A) = #{σj > 0}.(b) range(A) = span〈u1, . . . , ur〉, null(A∗) = span〈ur+1, . . . , um〉.(c) null(A) = span〈vr+1, . . . , vn〉, range(A∗) = span〈v1, . . . , vr〉.(d) A =

∑rj=1 σjujv

∗j (for rank k < r approx, use the first k; gives min error in 2 and Frobenius norms).

2.a.iv) Projectors

1. Projector: P 2 = P

2. Complimentary Projector: I − P ; range(I − P ) = null(P ) and null(I − P ) = range(P )

3. Orthogonal Projector: P 2 = P and P ∗ = P .

(a) Rank 1 orthogonal projectors (a 6= 0): Pa =aa∗

a∗a; P⊥a = I − Pa.

(b) Onto range of A (arbitrary basis): P = A(A∗A)−1A∗ = AA+

2.a.v) QR Factorization

For a matrix A ∈ Cm×n, and m ≥ n

1. Reduced QR: A = QR, with Q ∈ Cm×n unitary, R ∈ Cn×n upper triangular.

2. Full QR: A = QR with

Q =[Q Q⊥

], R =

[R

0(m−n)×n

]3. Finding Q and R: Gram-Schmidt. By hand, use classical:

qj =aj −

∑j−1i=1 rijqirjj

rij = q∗i aj rjj =

∣∣∣∣∣∣∣∣∣∣aj −

j−1∑i=1

rijqi

∣∣∣∣∣∣∣∣∣∣2

7

4. Modified Gram-Schmidt: AR1R2 · · ·Rn︸ ︷︷ ︸R−1

= Q

5. Householder Triangularization: QnQn−1 · · ·Q1︸ ︷︷ ︸Q∗

A = R, with Qk =

[I 00 F

]where F = I − 2

vv∗

v∗v.

6. Algorithms: Pseudocode.

Classical Gram-Schmidt:

f o r j = 1 : nQ[ : , j ] = A[ : , j ]f o r i = 1 : j−1

R[ i , j ] = Q[ : , i ] ’ ∗ A[ : , j ]Q[ : , j ] = Q[ : , j ] − R[ i , j ] ∗ Q[ : , i ]

R[ j , j ] = norm(Q[ : , j ] , 2)Q[ : , j ] = Q[ : , j ] / R[ j , j ]

Modified Gram-Schmidt:

f o r i = 1 : nR[ i , i ] = norm(A[ : , i ] , 2)Q[ : , i ] = A[ : , i ] / R[ i , i ]f o r j = i +1:n

R[ i , j ] = Q[ : , i ] ’ ∗ A[ : , j ]A[ : , j ] = A[ : , j ] − R[ i , j ] ∗ Q[ : , i ]

Householder QR:

% Note that the output i s R = A, and W matr i ce sf o r i = 1 : n

x = A[ i : , i ]x [ 1 ] = s i gn (x [ 1 ] ) ∗ norm(x ) + x [ 1 ]W[ i : , i ] = x / norm(x , 2)A[ i : , i : ] = A[ i : , i : ] − 2 ∗ W[ i : , i ] ∗ (W[ i : , i ] ’ ∗ A[ i : , i : ] )

% To so lve , note that% Rx = Q∗b ,f o r i = 1 : n

b [ i : ] = b [ i : ] − 2 ∗ W[ i : , i ] ∗ (W[ i : , i ] ’ ∗ b [ i : ] )

% Use back s ub s t i t u t i o n to s o l v e f o r xx = R\b

2.a.vi) Least Squares

1. Normal Equations: A∗Ax = A∗b

2. Pseudoinverse: A+ = (A∗A)−1A∗ = R−1Q∗ = V Σ−1U

3. Solution methods:

(a) Cholesky: A∗A = R∗R, where R is upper triangular (requires full rank A). Best for speed, bad for error.

(b) QR: Reduces to Rx = Q∗b, use back substitution. Good method unless A is close to rank deficient.

(c) SVD: ΣV ∗x = U∗b, then solve. Stable even if A close to rank deficient, but more time/memory consuming.

2.a.vii) Conditioning

1. Absolute condition number: κ = supδx

||δf ||||δx||

= ||J(x)||

2. Relative condition number: κ = supδx

(||δf ||||f(x)||

/||δx||||x||

)=

||J(x)||||f(x)|| / ||x||

3. Matrix-vector condition: κ = ||A||||x||||Ax||

4. Matrix condition number: κ(A) = ||A||∣∣∣∣A−1

∣∣∣∣ or ||A|| ||A+|| in rank deficient case.

8

2.a.viii) Stability

Given problem f : X → Y , and algorithm f : X → Y , for all x ∈ X,

1. Accuracy:

∣∣∣∣∣∣f(x)− f(x)∣∣∣∣∣∣

||f(x)||= O(εm)

2. Stability: ∃x :||x− x||||x||

= O(εm) and

∣∣∣∣∣∣f(x)− f(x)∣∣∣∣∣∣

||f(x)||= O(εm)

3. Backward stability: f(x) = f(x) for some x as above.

4. Both Householder triangularization is backward stable, and so is MGS provided Q∗b is formed implicitly.

2.a.ix) Gaussian Elimination

1. No pivoting: Use for hand calcs, not stable. A = LU where L = L−11 L−1

2 · · ·L−1m−1 is lower triangular, U is

upper triangular. We have `jk =xjk

xkkfor k < j ≤ m, and

Lk =

1. . .

1−`k+1,k 1

.... . .

−`mk 1

⇒ L =

1 0 · · · 0

`21 1. . .

......

. . .. . . 0

`m1 · · · `m,m−1 1

2. Partial pivoting: Do row interchanges to maximize |xkk|.

U = Lm−1Pm−1 · · ·L2P2L1P1A = (L′m−1 · · ·L′2L′1)(Pm−1 · · ·P2P1)A = L−1PA

where L′k = Pm−1 · · ·Pk+1LkP−1k+1 · · ·P

−1m−1. Solve to get PA = LU .

3. Full pivoting: (L′m−1 · · ·L′2L′1)(Pm−1 · · ·P2P1)A(Q1Q2 · · ·Qm−1) = L−1PAQ = U . Here, L and P are asbefore, and Q = Q1Q2 · · ·Qm is another permutation matrix. Solve for PAQ = LU .

4. Algorithms:

GE No Pivot:

U = A; L = If o r k = 1 :m−1

f o r j = k+1:mL [ j , k ] = U[ j , k ] / U[ k , k ]U[ j , k : ] = U[ j , k : ] − L [ j , k ] ∗ U[ k , k : ]

GE Partial Pivot:

U = A; L = I ; P = If o r k = 1 :m−1

f o r j = k+1:mSe l e c t i >= k : abs (U[ i , k ] ) i s maximizedU[ k , k : ] , U[ i , k : ] = U[ i , k : ] , U[ k , k : ]P [ k , : ] , P [ i , : ] = P[ i , : ] , P [ k , : ]f o r j = k+1:m

L [ j , k ] = U[ j , k ] / U[ k , k ]U[ j , k : ] = U[ j , k : ] − L [ j , k ] ∗ U[ k , k : ]

2.a.x) Cholesky Factorization

1. Given Hermitian positive definite A, solve for A = R∗R, where R is upper triangular.

A =

[a11 w∗

w K

]=

[α 0∗

w/α I

]︸ ︷︷ ︸

R∗1

[1 0∗

0 K − ww∗

a11

]︸ ︷︷ ︸

A1

[α w∗/α0 I

]︸ ︷︷ ︸

R1

= R∗1 · · ·R∗n︸ ︷︷ ︸R∗

I Rn · · ·R1︸ ︷︷ ︸R

where α =√a11.

9

2. Algorithm:

Cholesky:

R = Afo r k = 1 : n

f o r j = k+1:nR[ j , j : ] = R[ j , j : ] − R[ k , j : ] ∗ R[ k , j : ] ’ / R[ k , k ]

R[ k , k : ] = R[ k , k : ] / s q r t (R[ k , k ] )

2.a.xi) Eigenvalues

1. Eigenvalue Decomposition: A = XΛX−1, with X nonsingular, Λ diagonal.

2. Characteristic Polynomial: PA(z) = det(zI −A).

3. Similarity Transform: X ∈ Cm×m nonsingular. A, B similar if A = X−1BX.

4. Eigenvalue Multiplicity:

(a) Geometric: # of lin. indep. eigenvectors for λ.

(b) Algebraic: multiplicity of root of char. poly. for λ

(c) algebraic ≥ geometric

5. Defective/Degenerate: λ if algebraic mult. > geometric mult.

6. Nondefective: iff it has an eigenvalue decomposition.

7. Unitary Diagonalization: A = QΛQ∗, with Q unitary. ⇔ A normal.

8. Schur Factorization: A = QTQ∗, with T upper triangular. Every square matrix has one.

2.a.xii) Iterative Methods

1. Want to solve Ax = b starting with initial guess.

2. Jacobi: (D − L− U)x = b⇒ Dx = (L+ U)x + b⇒ x = D−1(L+ U)︸ ︷︷ ︸TJ

x +D−1b︸ ︷︷ ︸cJ

3. Gauss-Seidel: (D − L− U)x = b⇒ (D − L)x = Ux + b⇒ x = (D − L)−1U︸ ︷︷ ︸TGS

x + (D − L)−1b︸ ︷︷ ︸cGS

4. Algorithms:

Jacobi:

D = diag ( diag (A) )f o r i = 1 : c I t e r

x = D \ ( (D − A)x + b)

Gauss-Seidel:

D = diag ( diag (A) ) ; L = − t r i l (A) ; U = −t r i u (A)f o r i = 1 : c I t e r

x = (D − L) \ (b + Ux)

5. Converges if T converges, i.e. ρ(T ) < 1.

10

2.b) Numerical Methods

2.b.i) Functional Iteration

Casting nonlinear system as a fixed point problem: x = g(x) for x ∈ Rn, and g(x) = (g1(x), g2(x), . . . , gn(x))>.Iterating x(k+1) = g(x(k)).

1. Contraction Mapping: If ||g(x)− g(y)|| ≤ λ ||x− y|| ∀x, y :∣∣∣∣x− x(0)

∣∣∣∣ ≤ ρ, ∣∣∣∣y − x(0)∣∣∣∣ ≤ ρ with 0 ≤ λ <

1, and∣∣∣∣g(x(0) − x(0)

∣∣∣∣ ≤ (1− λ)ρ, then limk→∞ x(k) = a where g(a) = a, and a unique (in this region).

2. If gi(x) has continuous 1st order ∂s:

∣∣∣∣∂gi(x)

∂xj

∣∣∣∣ ≤ λ

n∀ i, j = 1 : n and x ∈ Bρ(a) = {x ∈ Rn : ||x− a||∞ ≤ ρ},

then x(0) ∈ Bρ(a)⇒ x(k) → a (unique).

3. Nonlinear Systems: g(x) = x−A(x)f(x).

(a) Easy: A(x) = A.

(b) Newton: A(x) = J−1(x) (inverse Jacobian of f).

2.b.ii) Polynomial Interpolation

1. Lagrange: Pn(x) =

n∑j=0

f(xj)φnj(x), where φnj(x) =∏i 6=j

(x− xi)/∏i 6=j

(xj − xi), where xj , j = 0 : n are known

points.

2. Pointwise Error: Rn(x)def= f(x)− Pn(x) =

∏nj=0

(x−xj)(n+1)! f

(n+1)(ξ) for some x0 < ξ < xn.

3. Rolle’s Theorem: ∃ a point between 2 zeros where f ′(z) = 0 for continuous f .

4. Numerical Differentiation/Integration: For f ∈ C([a, b]), xi ∈ [a, b], i = 0 : n then

f(x) = Pn(x) +ωn(x)

(n+ 1)!f (n+1)(ξ) ωn(x) =

n∏j=0

(x− xj)

f ′(xi) =

n∑j=0

f(xj)φ′nj(xi)︸ ︷︷ ︸

approx

+ω′n(xi)

(n+ 1)!f (n+1)(ξ(xi))︸ ︷︷ ︸

err

f (k)(xi) ≈n∑j=0

f(xj)φ(k)nj (xi)

∫ b

a

f(x) dx ≈n∑j=0

(∫ b

a

φnj(x) dx

)f(xj)

5. Weighted Least Squares: f ∈ L2([a, b]). Then Qn(x) =∑nj=0 cjPj(x) with cj =

∫ baf(x)Pj(x)w(x) dx mini-

mizes ||f(x)−Qn(x)||22 =∫ ba|f(x)−Qn(x)|2 w(x) dx, with inner product 〈f(x), g(x)〉w =

∫ baf(x)g(x)w(x) dx.

w(x) ≥ 0 on [a, b], and∫ baw(x) dx > 0.

6. G-S Orthonormalization: Given {gi(x)}ni=0

f0(x) = d0g0(x) d0 = 1/ ||g0||L2

f1(x) = d1 [g1(x)− c01f0(x)] d1 = 1/ ||g1 − c01f0||L2

fn(x) = dn

gn(x)−n−1∑j=0

cjnfj(x)

cij = 〈fi, gj〉

11

7. Trig. Interpolation: On [−π, π] xk = kh, k = −n : n, h = π/n.

Un(x) = −1

2

(cne

inx + c−ne−inx)+

n∑j=−n

cjeijx

cj =1

2n

(−1

2

(f(xn)e−ijxn + f(x−ne

−ijx−n)

+

n∑k=−n

f(xk)e−ijxk

)j = −n : n

2.b.iii) ODEs

1. Initial Value Problems:{y′ = f(t,y) y = [y1, y2, . . . , ym]>, f = [f1, f2, . . . , fm]>, c = [c1, c2, . . . , cm]>

y(0) = c autonomous: f(t,y) = g(y)

2. Given a system u(m) = g(t, u, u′, . . . , u(m−1)), let y = (u, u′, . . . , u(m−1)), and{y′n = yn+1 ∀ n = 1 : m− 1

y′m = g(t, y1, y2, . . . , ym)

3. Stability: Test equation y′ = λy, λ ∈ C. Solution: eλty(0), t ≥ 0. Then |y(t)− y(t)| = |y(0)− y(0)| e<(λ)t.

(a) Stable: <(λ) ≤ 0.

(b) Assympytotically stable: <(λ) < 0.

(c) Unstable: <(λ) > 0.

2.b.iv) Basic Concepts

1. Local Truncation Error: dn = Nhy(tn)

(a) Consistent (accurate) if dn → 0 as hn → 0 for all n

(b) dn = O(hpn), p ∈ Z+ ⇒ Nh is accurate order p

2. 0-Stability: If ∃h0, k > 0 : for all mesh fns xn, zn with h ≤ h0

|xn − zn| ≤ k[|x0 − z0|+ max

1≤j≤N|Nhxn(tj)−Nhzn(tj)|

]3. Absolute Stability: Test fn y′ = λy,y(0) = c. Region in C satisfying |yn| ≤ |yn−1|

4. Stiffness: Require extremely small step size for explicit methods. Specifically, if b<λ << −1, where b is theinterval length.

5. A-Stable: Region of absolute stability includes {<(z) ≤ 0}.

6. Rough Problems: Can’t bound derivatives by const. of moderate size. Need to break up solution atdiscontinuities.

2.b.v) Methods

1. Forward Euler: yn = yn−1 + hnf(tn−1,yn−1)

2. Backward Euler: yn = yn−1 + hnf(tn,yn)

3. Trapezoidal Method: yn = yn−1 +hn2

(f(tn−1,yn−1) + f(tn,yn))

4. Taylor Series Method: y′ = f(t,y), yn = yn−1 + hy′n−1 + h2

2! y′′n−1 + · · ·

12

5. Explicit Midpoint:{yn−1/2 = yn−1 + h

2 f(tn−1,yn−1)

yn = yn−1 + hf(tn−1/2, yn−1/2)

6. RK Methods: For 1 ≤ i ≤ s

{Yi = yn−1 + h

∑sj=1 aijf(tn−1 + cj , h,Yj)

yn = yn−1 + h∑si=1 bif(tn−1 + cih,Yi)

{Ki = f

(tn−1 + cih,yn−1 + h

∑sj=1 aijKj

)yn = yn−1 + h

∑si=1 biKi

(a) Tableau:

c1 a11 a12 · · · a1s

c2 a21 a22 · · · a2s

......

. . . · · ·...

cs as1 as2 · · · assb1 b2 · · · bs

(b) Require: ci =∑sj=1 aij for i = 1 : s, and

∑sj=1 bj = 1

(c) Explicit if aij = 0 for j ≥ i.

(d) Order p if b>AkC`−11 =(`− 1)!

(`+ k)!, 1 ≤ `+ k ≤ p, where C = diag(c).

(e) L-Stable: A-stable and |R(z)| → 0 as |z| → ∞.

7. Linear Multistep Methods:

k∑j=0

αjyn−j = h

k∑j=0

βjfn−j

(a) Adams Family: α0 = 1, α1 = −1, αj = 0 for j > 1.

(b) BDF: Derived from interpolating polynomial. With α0 = 1, this is

k∑i=0

αiyn−i = hβ0f(tn, yn)

(c) Order: Order p if 0 = C0 = · · · = Cp 6= Cp+1, where

C0 =

k∑j=0

αj , Ci = (−1)i

1

i!

k∑j=0

jiαj +1

(i− 1)!

k∑j=0

ji−1βj

(d) Char. Poly.s: ρ(ξ) =

∑kj=0 αjξ

k−j , σ(ξ) =∑kj=0 βjξ

k−j .

(e) 0-Stable: All roots of ρ(ξ) satisfy |ξi| ≤ 1, and if |ξi| = 1 it is simple. Strongly stable if all |ξi| < 1(except ξ = 1), weakly stable if 0-stable but not strongly stable.

(f) Stability Region: z =ρ(eiθ)

σ(eiθ)

8. Predictor Corrector Methods: Use explicit multistep method to predict, then implicit method to correct.

2.b.vi) BVPs

{y′ = A(t)y + q(t) 0 < t < b

B0y(0) +Bby(b) = b

13

1. Fundamental solution: Y ′ = A(t)Y , Y (0) = I. Gives general solution:

y(t) = Y (t)

[c +

∫ t

0

Y −1(s)q(s) ds

], Qc = b−BbY (b)

∫ b

0

Y −1(s)q(s) ds, Q = B0 +BbY (b)

unique solution iff Q nonsingular.

2. Green’s Functions:

3. Shooting:

4. Finite Difference:

3) Analysis

1. Pointwise Convergence: ∀ε > 0, x ∈ I, ∃N(ε, x) : |fn(x)− f(x)| < ε, ∀n > N(x, ε).

2. Uniform Convergence: ∀ε > 0, ∃N(ε) : |fn(x)− f(x)| < ε ∀x and n > N(ε). ALT: limn→∞ ||fn − f ||∞ = 0.

3.a) Metric Spaces

1. A pair (M,d), M a set, d : M ×M → [0,∞) a function, satisfying

(i) d(x, y) = 0⇔ x = y

(ii) d(x, y) = d(y, x)

(iii) d(x, z) ≤ d(x, y) + d(y, z)

2. Convergence: Given (M,d), {xn}, x. xn → x if ∀ε > 0 ∃N(ε) : n > N(ε)⇒ d(xn, x) < ε.

3. Equivalence of Metrics: Given (M,d), (M,d′), equivalent if ∀ε > 0, ∃δ : B′(x, δ) ⊂ B(x, ε), and ∀ε′ > 0,∃δ′ : B(x, δ′) ⊂ B′(x, ε′).

3.a.i) `p Spaces

For 1 ≤ p ≤ ∞ the norm is given by

||x||p =

∞∑j=0

|xj |p1/p

, ||x||∞ = sup0≤j<∞

|xj |

`p = {x : ||x||p <∞}. If 1 ≤ p < q ≤ ∞, then `p ⊂ `q (strict).

1. Holder’s Inequality: 1 ≤ p, q ≤ ∞ with 1/p+ 1/q = 1. Then for x ∈ `p, y ∈ `q,

∞∑j=0

|xjyj | ≤

∞∑j=0

|xj |p1/p ∞∑

j=0

|yj |q1/q

2. Jensen’s Inequality: f : [a, b] → R convex (i.e. f(px1 + (1 − p)x2) ≤ pf(x1) + (1 − p)f(x2) for x1 < x2,0 < p < 1), a ≤ x1 ≤ · · · ≤ xn ≤ b, and pi ∈ (0, 1) with

∑i pi = 1 then

f

n∑j=1

pjxj

≤ n∑j=1

pjf(xj)

3. Minkowski Inequality: For x,y ∈ `p, ||x + y||p ≤ ||x||p + ||y||p.

4. “Converse” of Holder: If ∃c > 0 :∑akxk ≤ c ||x||p ∀x ∈ Rn, then ||a||q ≤ c.

5. Continuity: Metric spaces (M,d), (N, d′), function f : M → N . f is continuous at x0 ∈M if ∀ε > 0, ∃ δ > 0 :d(x, x0) < δ ⇒ d′(f(x), f(x0)) < ε.

14

3.a.ii) Lebesgue (Lp) Spaces

For 1 ≤ p ≤ ∞ and interval X, the norm is given by

||f ||p =

(∫X

|f(x)|p dx)1/p

, ||f ||∞ = supx∈X|f(x)|

Lp(X) = {f : ||f ||p <∞}. If 1 ≤ p < q ≤ ∞ and X finite, then Lq ⊂ Lp (strict).

1. If 1 ≤ r < s ≤ ∞ and X finite, {fn} ∈ Ls(X), then ||fn − f ||s → 0 ⇒ ||fn − f ||r → 0.

2. If fn → f pointwise and ||fn||p → ||f ||p, then fn → f in Lp.

3.a.iii) Normed Linear Spaces

1. Norms: See norms. A seminorm doesn’t require x = 0 for ρ(x) = 0.

2. Inner Product Space: A function F (x, y) : X ×X → R is an inner product if

(a) F is linear in each argument

(b) F (x, x) ≥ 0 and F (x, x) = 0 iff x = 0

(c) F (x, y) = F (y, x)

A norm is derived from an inner product iff the parallelogram law holds, i.e. ||x+ y||2 + ||x− y||2 =

2 ||x||2 + 2 ||y||2.

3.b) Topology

Topological Space: Set X, collection of open subsets T that satisfy

(i) ∅, X ∈ T

(ii) U, V ∈ T ⇒ U ∩ V ∈ T

(iii) {Uα}α∈I ⊂ T ⇒⋃α

Uα ∈ T

3.b.i) Basic Definitions

1. Open Ball: B(x, ε) = {y : d(x, y) < ε}.

2. Open: If ∀x ∈ U, ∃ε : B(x, ε) ⊂ U , then U open.

(a) Finite intersections of open sets are open

(b) All unions of open sets are open

3. Closed: If complement is open.

(a) Finite unions of closed sets are closed

(b) All intersections of closed sets are closed

4. Interior Point: x ∈ A ⊆ X. x is an interior point if ∃ open Ux ⊆ A.

5. Interior: A◦ = { all interior points of A }.

6. Point of Closure: A ∈ X, x ∈ X is a point of closure of A if ∀ open Ux, Ux ∩A 6= ∅.

7. Accumulation Point: (or limit point) if Ux ∩ (A\{x}) 6= ∅.

8. Closure: A ∈ X, and F(A) = {F : F closed, A ⊆ F}. Then the closure is A = ∩F∈F(A)F .

ALT: A = { all points of closure }

(a) A ⊆ A

15

(b) A ⊆ B ⇒ A ⊆ B

(c) A = A

(d) A ∪B = A ∪B(e) ∅ = ∅

9. Gδ: Countable intersection of open sets.

10. Fσ: Countable union of closed sets.

3.b.ii) Defining Topologies and Continuity

1. Base: Collection B = {B(x, ε) : x ∈ X, ε > 0} with

(a) ∪B∈B = X

(b) x ∈ B1, B2 ⇒ ∃B3 ⊂ B1 ∩B2 : x ∈ B3

2. Sub-base: B0 ⊂ X with ∪B∈B0B = X. Then B =

{∩nk=1B

0k ∈ B0

}is a base for the topology.

3. Weak topology: X has two topologies T ,S. S weaker than T means S ⊂ T .

4. Local base:

5. Continuity: Two topological spaces (X, T ), (Y,S), function f : X → Y . f is continuous at x0 if ∀V ∈ Yopen, with f(x0) ∈ V , ∃U ⊂ X with x0 ∈ U , and x ∈ U ⇒ f(x) ∈ V .

(a) A function is continuous iff xn → x⇒ f(xn)→ f(x).

(b) If f continuous, then f−1(open) = open.

6. Convergence: xn → x if ∀U 3 x, ∃N : n ≥ N ⇒ xn ∈ U .

7. Connected: (X, T ) connected if no nonempty sets A,B ⊂ X with A ∪B = X, and A ∩B = ∅.(X, T ) connected and f : X → Y continuous ⇒ (Y,S) connected.

8. Hausdorff Space: For every x, y ∈ X, x 6= y, ∃ open U, V , with x ∈ U , y ∈ V , U ∩ V = ∅.

(a) Convergent sequences have unique limits

(b) Complement of {x} is open.

3.c) Distributions

Topological linear space of functions D. Distributions are complex-valued continuous (with respect to D) linearfunctions D′. I.e. if ϕn → ϕ in D, then 〈T, ϕn〉 → 〈T, ϕ〉.

3.c.i) Basics

1. Taylor’s Formula: ϕ (N + 1)-times differentiable on [−M,M ], ϕN+1 continuous.

ϕ(x) =

N∑j=0

1

j!ϕ(j)(0)xj +

xN+1

N !

∫ 1

0

(1− u)Nϕ(N+1)(xu) du =

N∑j=0

1

j!ϕ(j)(0)xj + xN+1ψ(x)

where ψ(x) is continuous, and has as many derivatives as ϕ.

2. Improper Integrals: f ∈ C(R). Then∫R f(x) dx converges if

limA→−∞

limB→∞

∫ B

A

f(x) dx or limA→−∞

∫ C

A

f(x) dx+ limB→∞

∫ B

C

f(x) dx

exist and are finite, with C arbitrary.

16

3. Principal Value: Converges if

limR→∞

∫ R

−Rf(x) dx

exists and is finite.

4. Support: supp(ϕ) = {x : ϕ(x) 6= 0}. Support is compact if ∃M : supp(ϕ) ⊆ [−M,M ].

5. Bump Function: f(x) = exp

{−1

(x− a)(x− b)

}on (a, b) and 0 else.

6. Distribution: T : D → C satisfies

(a) T is complex linear: T (αϕ1 + βϕ2) = αT (ϕ1) + βT (ϕ2) ∀ α, β ∈ C and ϕ1, ϕ2 ∈ D.

(b) T is continuous WRT the topology on D.

3.c.ii) Specific Distributions

1. Principal Value: PV

(1

x

)(ϕ)

def= lim

ε↓0

∫|x|>ε

ϕ(x)

xdx = lim

ε↓0

[∫|x|>ε

ϕ(0)

xdx+

∫|x|>ε

ϕ(x)− ϕ(0)

xdx

]

2.1

x± i0= PV

(1

x

)∓ iπδ(x)

3.c.iii) Function Spaces

1. C∞: infinitely differentiable functions (smooth).

2. C∞0 : subset of C∞ with compact support.

3. CkN : k times continuously differentiable with support in [−N,N ].

4. Schwartz space S : subset of C∞ with limabsx→∞ |x|k |Dαϕ(x)| = 0 for all k, α.

5. C∞0 Convergence: ϕn → ϕ if:

(a) supp(ϕn) ⊆ [−N,N ] independent of n.

(b) Derivatives ϕ(m)n

n→∞−−−−→ ϕ(m) uniformly, i.e. limn→∞

supx∈[−N,N ]

∣∣∣ϕ(m)n (x)− ϕ(m)(x)

∣∣∣→ 0

ALT: T ∈ D′ iff ∀N , ∃B(N), k(N): |〈T, ϕ〉| ≤ B ||ϕ||N,k ∀ϕ ∈ CkN .

6. Order: Smallest k independent of N : above convergence definition holds.

7. Norm: ||ϕ||N,kdef=

k∑j=0

sup[−N,N ]

∣∣∣ϕ(j)(t)∣∣∣

8. Topology: d(ϕ1, ϕ2)def=

∞∑k=0

||ϕ1 − ϕ2||N,k1 + ||ϕ1 − ϕ2||N,k

· 1

2k

9. Convergence: {Tj} ∈ D′. If limj→∞

〈Tj , ϕ〉 = 〈T, ϕ〉 ∀ ϕ ∈ D, then Tj converges to T .

10. Derivatives:⟨T (n), ϕ

⟩= (−1)n

⟨T, ϕ(n)

⟩.

17

3.d) Measure Theory

3.d.i) Basic Definitions

1. Dense: A ⊂ B. A is dense in B if B ⊆ A. A set is dense iff (Ac)◦ = ∅.

2. Nowhere Dense: Interior of closure is empty: (A)◦ = ∅. A nowhere dense iff (Ac)◦ = X.

3. (A◦)c = Ac and (Ac)◦ = (A)c.

4. First Category: Countable union of nowhere dense sets (also meager).

5. Residual: Complement of first category (also comeager). Residual sets are dense.

6. Second Category: Not first category.

7. Measure Zero: ∀ ε > 0, ∃ {Bn} countable collection of open balls Bn = B(xn, δn) such that

A ⊂∞⋃n=1

Bn and

∞∑n=1

volume of Bn ≤ ε

8. Baire Category Theorem in R:

(a) The complement of a 1st category set is dense.

(b) The intersection of countably many open dense sets is dense.

9. Cantor Intersection Theorem: Given nonempty closed and bounded {Cn} with C0 ⊇ C1 ⊇ · · · ⊇ Cn ⊇ · · · ,then ∩Ck 6= ∅.

3.d.ii) Measurable Spaces

1. (X,B), set X, collection of subsets B satisfying

(i) X ∈ B, ∅ ∈ B(ii) A ∈ B ⇒ Ac ∈ B(iii) {Aj} ∈ B ⇒

⋃nj=1 ∈ B

This defines an algebra of sets, and if n =∞ a σ−algebra.

2. Measurable: (X,B), (Y, C) measurable spaces. Then f : X → Y is measurable if A ∈ C ⇒ f−1(A) ∈ B.

3. Borel σ−algebra: Given (X, T ), the σ−algebra generated by T .

4. Additive Measure: on (X,B). µ : B → [0,∞]

(a) µ(∅) = 0

(b) {Ak} ∈ B mutually disjoint ⇒ µ (⋃nk=1Ak) =

∑nk=1 µ(Ak) where n can by ∞.

3.d.iii) Probability Spaces

1. A measurable space + a measure, (X,B, P ), with P : B → [0, 1].

2. Borel-Cantelli:

(a) Given (X,B, P ) (P (X) = 1). Let E1, E2, . . . be events (∈ B). Then if

∞∑m=1

P (Em) <∞, P (Emi.o.) = P (lim supEm) = 0

(b) If E1, E2, . . . independent and∑∞n=1 P (En) =∞, then P (Eni.o.) = 1.

3. Independent: E1, . . . , En are independent if P (Ei1 ∩ . . . Eik) =∏kj=1 P (Eij ).

4. Borel Zero-One Law: En independently often (i.o.) can only have probability 0 or 1.

18

3.e) Convergence and Compactness

3.e.i) Basic Definitions

In a metric space (M,d).

1. Cauchy Sequence: If ∀ ε > 0, ∃N(ε) : m,n > N ⇒ d(xm, xn) < ε, then {xn} is Cauchy. If xn → x, then{xn} is Cauchy.

2. Complete: Every Cauchy sequence converges to a limit x ∈M .

(a) Rn is complete.

(b) `p is complete.

(c) Lp is complete.

(d) Closed subspace of complete space is complete.

3. Contraction Mapping Theorem: (M,d) complete metric space. f : M → M continuous, and ∃ 0 < k < 1such that d(f(x), f(y)) ≤ kd(x, y) ∀ x, y ∈M . Then there exists a unique z ∈M for which f(z) = z.

4. Sequential Compactness: In (X, T ). A ∈ X is compact if every sequence {xn} ∈ A has a convergentsubsequence with limit in A.

5. Compactness:

(a) In Rn, compact = closed and bounded.

(b) In (M,d), compact = complete and totally bounded or = sequentially compact.

(c) In (X, T ) compact if every open cover has a finite subcover.

(d) Compact sets are closed and bounded.

(e) Closed subsets of compact sets are compact.

(f) Compact sets are complete.

(g) f : M → R continuous and M compact, then f assumes its max and min values.

6. ε−Net: ε > 0, finite collection x1, . . . , xn such that ∪nj=1B(xj , ε) ⊇M .

7. Totally Bounded: M is totally bounded if there is an ε−net for every ε > 0.

8. Open Cover: (X, T ), A ⊆ X, O = {Uα}α∈I (open sets), and A ⊆ ∪α∈IUα.

9. Finite Subcover: Finite subset of O that covers A.

3.f) Uniformity

3.f.i) Basic Definitions

1. Uniformly Continuous: f : (M,d)→ (M ′, d′). If ∀ ε > 0, ∃ δ > 0 : d(x, y) < δ ⇒ d′(f(x), f(y)) < ε.If A ⊂M is compact and f : M →M ′ is continuous, then f is uniformly continuous on A.

2. Uniform Convergence: fn : (M,d)→ (M ′, d′).

(a) fn converges to f if ∀ ε, ∀ x, ∃N(x, ε) : n > N(x, ε)⇒ d′(f(x), fn(x)) < ε.

(b) fn converges uniformly to f if ∀ ε, ∃ N(ε) : n > N(ε)⇒ d′(f(x), fn(x)) < ε ∀x.

(c) {fn} is Cauchy if ∀ ε, ∀ x, ∃N(ε) : n,m > N(x, ε)⇒ d′(fn(x), fm(x)) < ε.

(d) {fn} is uniformly Cauchy if ∀ ε, ∃ N(ε) : n,m > N(ε)⇒ d′(fn(x), fm(x)) < ε ∀x.

3. fn : M →M ′, M ′ complete. fn converges Uniformly iff it is uniformly Cauchy.

4. A ⊂M , fn : A→ R.∑∞n=0 fn(x) converges uniformly on A if the sequence of partial sums converges uniformly.

5. Weirstrass M-test: (for uniform convergence). If ∃Mn ≥ 0 : |fn(x)| ≤ Mn ∀x ∈ A, and∑∞n=0Mn < ∞,

then∑∞n=0 fn converges uniformly.

19

3.f.ii) Interchanging Limits and Integrals

Given fn → f uniformly, fn continuous.

1. limh→0 limn→∞ fn(x0 + h) = limn→∞ limh→0 fn(x0 + h), i.e. f is continuous.

2.∫ ∑

fn =∑∫

fn.

3. Given fn → f (uniform not required) on [a, b], f ′n continuous, and f ′n → g uniformly. Then f ′ = g.

3.f.iii) More on Integrals

1. Riemann integrable: Define Mi = max{f(t) : ti ≤ t ≤ ti+1}, mi = min{f(t) : ti ≤ t ≤ ti+1}, and U(f ; ∆) =n∑i=0

Mi(ti+1 − ti), L(f ; ∆) =

n∑i=0

mi(ti+1 − ti) for some partition ∆. If infall ∆ U(f ; ∆) = supall ∆ L(f ; ∆) then

f is Riemann integrable.

(a) f on [a, b] is R-integrable if it is continuous.

(b) f on [a, b] is R-integrable iff the set of points of discontinuity of f has Lebesgue measure zero.

(c) Riemann integral exists ⇒ Lebesgue integral exists, and they are equal (not converse).

2. Lebesgue integrable: Suppose f bounded, −M,≤ f(t) ≤M and partition range: −M −1 = y0 < y1 < · · · <

yn < yn+1 = M + 1. Let µi be the length of {t : yi ≤ f(t) < yi+1}. Then the Lebesgue sum is:

n∑i=0

yiµi.

Alternatively, define

f+(x) =

{f(x) if f(x) ≥ 0

0 elsef−(x) =

{−f(x) textiff(x) ≤ 0

0 else

Then f(x) = f+(x)− f−(x), and

∫X

f dµ =

∫X

f+ dµ−∫X

f− dµ (provided at least 1 of the two integrals on

the right is finite).

3. Simple Function: A1, . . . , An ∈ X pairwise disjoint measurable sets. ϕ(x) =

n∑j=1

αjcAj(x) with αj ≥ 0 is a

non-negative simple function, and has integral

∫X

ϕ dµ =

n∑j=1

αjµ(Aj).

4. Measurable Function: (X,B) a measurable space. f : X → [−∞,∞] is measurable if {t : f(t) < α} ∈ B foreach α ∈ R. Limit of measurable functions is measurable.

3.g) Convergence Theorems

3.g.i) Basic Definitions

1. Lim inf: lim infn→∞

xndef= limn→∞

(infm≥n

xm

), alternatively the leftmost limit point (or ±∞).

2. Lim sup: lim supn→∞

xndef= limn→∞

(supm≥n

xm

), alternatively the rightmost limit point (or ±∞).

3. Markov’s Inequality: (X,B, µ), f : X → [−∞,∞] measurable. ∀ ε > 0, µ ({x : |f(x)| > ε) ≤ 1

ε

∫X

|f | dµ.

4. Chebyshev’s Inequality: (X,B, µ), f : X → [−∞,∞] measurable. ∀ ε > 0, µ ({x : |f(x)| > ε) ≤ 1

ε2

∫X

|f |2 dµ.

20

3.g.ii) Theorems

1. Fatou’s Lemma: {fn} sequence of non-negative measurable functions. Then

∫X

lim inf fn dµ = lim inf

∫X

fn dµ.

2. Monotone Convergence: {fn} sequence of non-negative measurable functions, and for almost every x,

{fn(x)} is nondecreasing with limit f(x). Then limn→∞

∫X

fn dµ =

∫X

limn→∞

fn dµ =

∫X

f dµ.

3. Corollary: gn ≥ 0 measurable. Then

∫X

∞∑n=1

gn dµ =

∞∑n=1

∫X

gn dµ.

4. Lebesgue Dominated Convergence: (X,B, µ), {fn} measurable, and fn → f a.e.. Suppose ∃g ∈ L1:

|fn(x)| ≤ |g(x)| a.e., ∀ n. Then limn→∞

∫X

fn dµ =

∫X

limn→∞

fn dµ =

∫X

f dµ.

4) Principals and Methods

4.a) Dynamics of Nonlinear Systems

4.a.i) Dimensional Analysis

1. Set x = Lx, t = T t, u = Cu, where · is dimensionless. Then ∂x = ∂xdxdx = 1

L∂x, ∂t = ∂tdtdt = 1

T ∂t.

2. Discrete Symmetries: Example: sign invariance, i.e. given solution u to differential equation, −u is also asolution.

3. Continuous Symmetries: Translation invariance, e.g. t→ t+ τ , x→ x+ λ.

4. Traveling Wave: Both time and spatially invariant. u(x, t) = u(x− ct) = u(z).

5. Scaling Symmetry: uλ(x, t) = λcu(λax, λbt), with z(x, t) = z(λax, λbt).

6. Buckingham Pi: Given f(x1, . . . , xn) = 0 of n physical variables in k physical units, can restate as F (Π1, . . . ,Πp) =

0, where Πi =

n∏j=0

xajj , and p = n− k.

7. Symmetries → Reductions:

(a) Time translation (t→ t+ τ) → steady solutions (u(x, t) = u(x))

(b) Space translation (x→ x+ λ) → homogeneous solutions (u(x, t) = u(t))

(c) Rescaling symmetries (u→ uλ) → self similar solutions (u(x, t) = tc/au(x/ta/b))

4.a.ii) Phase Plane

1. Potential Systems: uz = v, vz = f(u).

(a) Potential V (u) = −∫f(u) du = −

∫uzz du. Get phase plane from graph of potential.

(b) Energy E(z) = V (u) + 12u

2z

2. Linearization of FPs: ddz

[uv

]= A

[uv

]where A is the linearized Jacobian evaluated at the fixed point.

General solution for x = Ax is x(t) =∑ni=1 cie

λitvi where Avi = λivi.

21

(a) det(A) = λ1λ2, tr(A) = λ1 + λ2 for 2× 2 case.

(b) 0 < λ1 < λ2 < 0⇒ unstable/stable node

(c) λ1 < 0 < λ2 ⇒ saddle

(d) 0 < λ1 = λ2 < 0⇒ unstable/stable improper node

(e) λ1 = λ2, and 0 < <(λ) < 0⇒ unstable/stable spiral

(f) λ1 = λ2, and <(λ) = 0⇒ elliptic FP/center

3. Heteroclinic Orbit: connects two fixed points

4. Homoclinic Orbit: connects a fixed point to itself

4.a.iii) Dispersion Relations

4.b) Contour Integration

4.b.i) Complex Basics

1. Cauchy-Riemann: The function f(z) = u(x, y) + iv(x, y) is differentiable at z = x + iy iff ux = vy andvx = −uy, and all partials are continuous in a neighborhood of z.

2. Analytic: At a point if differentiable at that point; in a region if analytic at every point in the region.

3. Entire: Analytic at every point in C except ∞.

4. Singular Point: Where f is not analytic.

4.b.ii) Integration

1. Cauchy’s Integral Formula: f (k)(z0) =k!

2πi

∮C

f(z)

(z − z0)k+1dz.

2. Laurent Series: f(z) =

∞∑n=−∞

cn(z − z0)n where cn =1

2πi

∮C

f(z)

(z − z0)n+1dz.

(a) Strength: c−n for pole of order n.

(b) Residue: c−1 = 12π

∮Cf(z) dz

(c) Essential singularity: ∞ number of c−n terms.

3. Residue: Res (f(z), z0) = limz→z0

1

(k − 1)!

dk−1

dzk−1

[f(z)(z − z0)k

]for a pole of order k.

4. Residue at ∞: Res (f(z),∞) = Res

(− 1

z2f

(1

z

), 0

)

5. Cauchy’s Residue Theorem: For a simple closed contour C,

∮C

f(z) dz = 2πi

N∑j=1

rj , where rj are residues

of poles in the interior of C.

6. Jordan’s Lemma: If f(z)→ 0 uniformly on Reiθ, 0 ≤ θ ≤ π, and∣∣f(Reiθ

∣∣ ≤ G(R), and limR→∞G(R) = 0,

then limR→∞

∫CR

eikzf(z) dz = 0, for k > 0. Take lower arc for k < 0.

7. Branch cuts: Let x = ze2πi.

22

4.c) Fourier Series

Given a 2L periodic function f

f(x) =a0

2

∞∑n=1

an cos(πnLx)

+ bn sin(πnLx)

or =

∞∑n=−∞

cn exp(πnLx)

an =1

L

∫ L

−Lf(x) cos(nx) dx bn =

1

L

∫ L

−Lf(x) sin(nx) dx cn =

1

2L

∫ L

−Lf(x)einx dx

a0 = 2c0; an = (cn + c−n); bn = i(cn − c−n); cn = (an − ibn)/2

4.c.i) Hilbert Space

A complete, normed linear space whose norm comes from an inner product.

1. Inner Product: Satisfies 〈f, f〉 ≥ 0 with = iff f = 0, 〈f, αg〉 = α 〈f, g〉, 〈αf + βg, h〉 = α 〈f, h〉 + β 〈g, h〉,〈f, g〉 = 〈g, f〉.

2. For L2[a, b], 〈f, g〉 =∫ baf(x)g(x) dx.

3. Norm: ||f ||2 = 〈f, f〉.

4. Has a dense orthonormal basis, for L2[a, b] : ϕn = e2πinx/(b−a)/√b− a

4.c.ii) Theorems

1. Riemann-Lebesgue: Given f ∈ L1[a, b], limn→∞ |cn| = 0. More specifically, for f ∈ Cn[a, b] (actually only

need f (n) ∈ L1[a, b]), |ck| ≤c

|k|nfor |k| ≥ 1, and constant c.

2. Parseval’s Identity: For f ∈ L2[a, b], ||f ||2L2 = 2π

∞∑k=−∞

|ck|2.

3. Carlson’s Theorem: f ∈ L2 ⇒ Sn(f)→ f pointwise almost everywhere, where Sn is the partial sum of theFourier series.

4. For f(x) periodic, piecewise smooth on [a, b], and integrable, then in (a, b), limn→∞ Sn(f) = f(x) where f iscontinuous, and limn→∞ Sn(f) = 1

2 (f(x+) + f(x−)) where f is discontinuous (convergence may break down atendpoints).

5. For f(x) periodic, continuous, and piecwise smooth on [a, b], then Sn(f)→ f uniformly. The convergence is atthe rate ||Sn(f)− f ||∞ = O(n1−p), where f (p) ∈ L1[a, b] (or O(n−p) for f ∈ Cp[a, b]).

6. Gibb’s Phenomenon: Apporximately 10% error near discontinuities in function.

7. Can integrate term by term to get Fourier series of F (x) (but need to make it periodic), and differentiate termby term to get Fourier series of f ′(x) on the condition that f ∈ C, f ′ ∈ L1.

4.d) Distributions

Also see distributions.

4.d.i) Basics

1. Null sequence: In C∞0 (R), {ϕm(x)} such that limm→∞

supx∈[−K,K]

∣∣∣∣ dndxnϕm(x)

∣∣∣∣ = 0 for all n ∈ Z+. In S(R),

{ϕm(x)} such that limm→∞

supx∈R

∣∣∣∣xk dndxnϕm(x)

∣∣∣∣ = 0 for all k, n ∈ Z+.

2. Seminorms: Examples are pm,k(ϕ) = supx∈R

∣∣∣∣xk dmϕdxm

∣∣∣∣, and qm,k(ϕ) = supx∈R

∣∣∣∣(1 + x2)kdmϕ

dxm

∣∣∣∣.23

3. Continuity: Given {ϕm} → ϕ, then T continuous if T [ϕm]→ T [ϕ].

4. To prove T is a distribution, show that T [αϕ + βψ] = αT [ϕ] + βT [ψ] (linearity) and T [ϕm] → 0 for all nullsequences (continuity). Break up integral and use seminorms to prove.

5. If f ∈ Lloc1 , then Tf is a distribution in C∞0 and S.

6. For ψ smooth, then ψT [ϕ] = T [ψϕ].

7. Change of Variables: T [ϕ] ≡∫T (y(x))ϕ(x) dx ≡

∫T (z)ϕ(y−1(z))

|y′(y−1(z))|dy, where y(x) is the change of

variables.

8. Convolution: Tf ? Tg = Tf?g. If Tfn → Tf , Tgn → Tg, then Tfn ? Tgn → S where S depends only on Tf , Tg(not n).

4.e) Fourier Transforms

F [f(x)] = f(k) =

∫ ∞−∞

f(x)e−ikx dx F−1[f(k)

]= f(x) =

1

2π

∫ ∞−∞

f(k)eikx dk

4.e.i) Common Functions/Distributions

1. Sinc and box: F [sinc(x)] = πχ[−1,1], and F[χ[−1,1]

]= 2 sinc(k)

2. Gaussian: F[e−x

2/a]

=√aπ exp

(−ak2

4

)3. Sin and cos: F [sin(ax)] = iπ [δ(k + a)− δ(k − a)] and F [cos(ax)] = π [δ(k + a) + δ(k − a)]

4. Delta: F [δ(x)] = 1, and F [1] = 2πδ(k)

5. Heaviside: F [H(x)] = πδ(k)− iPV(

1k

)6. Principal Value and sign: F

[PV

(1x

)]= −iπsign(k), and F [sign(x)] = −2iPV

(1k

)4.e.ii) Other stuff

1. Poisson Sum:

∞∑n=−∞

δ(x− 2nπ) =1

2π

∞∑k=−∞

e−ikx ⇔∞∑

n=−∞ϕ(nT ) =

1

T

∞∑k=−∞

ϕ

(2πk

T

)

2. f(k) = f(−k), f ∈ R ⇒ f Hermitian, i.e. f(k) = f(−k);f(x) = 2πf(−x)

3. F [f(x− b)] = e−ikbf(k); F [f(ax)] = 1|a| f

(ka

)4. F

[f (n)

]= (ik)nf(k); F

[∫ x−∞ f(t) dt

]= 1

ik f(k) + πf(0)δ(k)

5. F [xnf(x)] = inf (n)(k)

6. f ∈ L1 ⇒ f is bounded.

7. F : L1(R)→ C(R), F : L2(R)→ L2(R)

8. Parseval’s/Plancherel’s Theorem: If f, g ∈ L1(R), then⟨f , g⟩

= 2π 〈f, g〉, so ||f ||22 = 12π

∣∣∣∣∣∣f ∣∣∣∣∣∣22

(for

f ∈ L2(R)).

9. Convolution: F [f ? g] = f · g, and F−1[f ? g

]= 2πf · g.

10. Multi dimensions: F [f(x)] = f(k) =

∫Rn

f(x)e−ik·x dx; F−1[f(k)

]= f(x) =

1

(2π)n

∫Rn

f(k)eikx dk

11. FT of Distributions: Tf [ϕ] = Tf [ϕ] = Tf [ϕ]

24

4.e.iii) Sampling

1. Nyquist Rate: ω0/π, where f(ω) = 0 for |ω| > ω0

2. Sampling frequency: 1/τ > Nyquist rate, i.e. τ < π/ω0.

3. Shannon sampling:

4.e.iv) Other Transforms

1. Hilbert Transform: H[f ] = 1πf ? PV (1/x), so F [H[f ]] = 1

π f(k) · F [PV (1/x)] = −isign(k)f(k).

2. Laplace Transform: L [f ] = F (s) =

∫ ∞0

f(t)e−st dt; f(t) =1

2πi

∫ c+i∞

c−i∞F (s)est ds

4.f) Spectral Theory

4.f.i) Basics

1. Operator: L : X → X. Defined on the domain D(L).

2. Bounded: If ||L|| ≤ ∞, where ||L|| = supu∈X

||Lu||||u||

. This implies ||Lu|| ≤ ||L|| ||u||. Bounded ⇔ continuous.

Find bound by bounding ||Lu||2 = 〈Lu,Lu〉 ≤ c ||u||2, or show unbounded via sequence of uns.

3. Adjoint: An operator L∗ such that 〈Lu, v〉 = 〈u, L∗v〉. If L is bounded and linear, then L∗ exists and isbounded and linear. Also, L∗∗ = L, and ||L|| = ||L∗||.

4. Normal: L∗L = LL∗.

5. Self Adjoint: L∗ = L. formally if they have different domains. All eigenvalues are real. No residualspectrum.

6. Resolvent: Complete normed space X 6= {0} and linear operator L : D → X, D ⊂ X. The resolvent isRλ(L) = (L− λI)−1.

7. Regular value: λ such that Rλ(L) exists, is bounded, and is defined on a dense set in X.

8. Point Spectrum: σp(L) = {λ : (L− λI)−1 d.n.e.}. Eigenvalues of L, Lu = λu. λ ∈ σp(L)⇒ λ ∈ σp(L∗).

9. Continuous Spectrum: σc(L) = {λ : (L− λI)−1 exists and is unbounded}.

10. Residual Spectrum: σc(L) = {λ : (L−λI)−1 exists, possibly bounded, but not defined on a dense set of X}.

11. Rayleigh Quotient: λ =〈Lu, u〉〈u, u〉

.

4.f.ii) Sturm-Liouville

1. Lu =−1

σ(x)((p(x)ux)x + q(x)u(x)) on some domain [a, b]

2. Always self adjoint.

3. Eigenvalues λ1 < λ2 < · · · , and eigenfunctions are orthogonal, complete basis.

25

4.g) Green’s Functions

4.g.i) Variation of Parameters

1. Wronskian: W (x) = det

∣∣∣∣u1 u2

u′1 u′2

∣∣∣∣ = u1u′2 − u2u

′1, for solutions Lu1 = Lu2 = 0 of the S-L operator.

2. v′1(x) = −u2(x)f(x)

p(x)W (x); v′2(x) =

u1(x)f(x)

p(x)W (x).

3. up(x) = u1v1 + u2v2 = −u1(x)

∫ x

0

u2(t)f(t)

p(t)W (t)dt+ u2(x)

∫ x

0

u1(t)f(t)

p(t)W (t)dt

4. u(x) = uh(x) + up(x).

4.g.ii) Green’s Functions

Assuming S-L operator L = (pu′)′ + qu.

1. Satisfies LG(x, t) = δ(x− t) as well as boundary conditions of L, and G is continuous at x = t.

2. Jump condition: Gx(t+, t)−Gx(t1, t) = 1p(t)

3. General solution: find 2 linearly indep. homogenous solutions u1, u2. Then

G(x, t) =

{a1(t)u1(x) + a2(t)u2(x) x < t

b1(t)u2(x) + b2(t)u2(x) x > t

Use BCs, continuity, and jump to find a1, a2, b1, b2. Solution of Lu = f on [c, d] is

u(x) =

∫ d

c

G(x, t)f(t) dt =

∫ x

c

Gb(x, t)f(t) dt+

∫ d

x

Ga(x, t)f(t) dt

26