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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]
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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.
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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