Pierson Guthrey
pguthrey@iastate.edu
1
Important Fourier Transforms
1
fˆ(k) = √
2π
Z
∞
f (x)e−ikx dx
−∞
General f
f (x)
fˆ(k)
1
f (x − a)
e−ak fˆ(k)
δ(x)
e2πiax f (x)
eiax
f (ax)
fˆ(k − 2πa)
1 ˆ k
|a| f a
cos(ax)
fˆ(x)
f (−k)
sin(ax)
d f (x)
dxn
(ik)n fˆ(k)
xn
xn f (x)
in d dkf (k)
n
n
(f ∗ g)(x)
2
Distributions
√
1
x
2π fˆ(k)ĝ(k)
sign (x)
f (x)g(x)
(fˆ∗ĝ)(k)
√
2π
f (x) = f (x) (real)
fˆ(−k) = fˆ(k)
f (x) (real, even)
fˆ(k) (real, even)
f (x) (real, odd)
fˆ(k) (imaginary, odd)
f (x)
fˆ(−k)
√1
2π
√
2π
2
√
2π
2i
(δ(k − a) + δ(k + a))
√
in 2πδ (n) (k)
√
− 2i 2πsign (k)
√2 1
2π ik
√
H(x)
2π
2
∞
X
x3
x5
(−1)n 2n+1
x
=x−
+
− ...
sin(x) =
(2n + 1)!
3!
5!
n=0
∞
X
(−1)n 2n
x2
x4
cos(x) =
x =1−
+
− ...
(2n)!
2!
4!
n=0
Hyperbolic Functions
sinh(x) =
∞
X
x3
x5
x2n+1
=x+
+
− ...
(2n + 1)!
3!
5!
n=0
cosh(x) =
Exponential Function
ex =
2πδ(k − a)
(δ(k − a) + δ(k + a))
Important Maclaurin Series
Trigonometric Functions
2πδ(k)
√
ˆ
n
√
∞
X
x2n
x2
x4
=1+
+
− ...
(2n)!
2!
4!
n=0
∞
X
xn
x2
x3
=1+x+
+
+ ...
n!
2
6
n=0
1
1
iπk
+ δ(k)
Pierson Guthrey
pguthrey@iastate.edu
Natural Logarithm (for |x| < 1)
log(1 − x) = −
log(1 + x) =
∞
X
∞
X
x2
x3
xn
= −x −
−
− ...
n
2
3
n=1
(−1)n+1
n=1
xn
x2
x3
=x−
+
− ...
n
2
3
Geometric Series (for |x| < 1)
∞
X
1
=
xn = 1 + x + x2 + x3 + ...
1 − x n=0
Binomial Series (for |x| < 1, α ∈ C)
(1 + x)α =
∞ X
α
xn ,
n
n=0
This includes the square root series for α =
1
2
α
n
=
α(α − 1)...(α − n + 1)
n!
and the infinite geometric series for α = −1.
1
1
1
(1 + x) 2 = 1 + x − x2 + ...
2
8
3
3.1
Calculus Techniques
Coordinate Transformations
Making of linear transformation of coordinates
α
a
=
β
c
Involves the Jacobian
∂(α, β)
= det
∂(x, t)
so
3.2
a
dxdt = det
c
b
d
a
c
b
d
x
t
b
d
−1 dαdβ
Integration and Derivatives
Mean Value Theorem for Integrals
Z
b
f (x)dx = (b − a)f (θ),
θ ∈ (a, b)
a
Directional Derivative
∇~v f (~x) = ∇f (~x) · ~v,
kvk2 = 1
This operator is linear, obeys the product rule ∇~v (f g) = f ∇~v f + g∇~v g and obeys the chain rule ∇~v (h ◦
df
= ∇~n f
g)(~x) = h0 (g(~x))∇~v g(~x). Example: Unit normal vectors dn
2
3.3
3.3
Pierson Guthrey
pguthrey@iastate.edu
Divergence Theorem
Divergence Theorem
Z
Z
~ · ~gdx =
∇
~g · ~nds
Ω
∂Ω
The first integral is an integral over Ω, the second integral is a line integral around the boundary of Ω
3.4
Green’s Identity
Z Z
Z
(f ∆g − g∆f )dx =
f
ΩN
∂Ω
∂f
∂h
−h
∂n
∂n
ds
The first integral is an integral over Ω, the second integral is a line integral around the boundary of Ω
3.5
3.5.1
Laplace Operator
Polar Coordinates
1 ∂
∇ =∆=
r ∂r
2
1 ∂2
∂2
+
r2 ∂φ2
∂z 2
∂
r
∂r
+
If the system is radially symmetric, this becomes
1 ∂
∇ =∆=
r ∂r
2
3.5.2
∂
r
∂r
Spherical Coordinates
1 ∂
∇ =∆= 2
ρ ∂ρ
2
∂
ρ
∂ρ
2
1
∂2
+ 2
ρ sin(θ) ∂θ
∂
1
∂2
sin(θ)
+ 2 2
∂θ
ρ sin (θ) ∂φ2
If the system is radially symmetric, this becomes
1 ∂
∇ =∆= 2
ρ ∂ρ
2
3.6
Jacobian Factors
3.6.1
Polar Coordinates
Z
f (x, y, z)rdrdθ
Ω
Ω
Spherical Coordinates
Z
Z
f (x, y, z)dxdydz =
Ω
3.7
∂
ρ
∂ρ
2
Z
f (x, y)dxdy =
3.6.2
Ω
Divergence and Curl
....
3
f (x, y, z)ρ2 dρdθdφ
3.8
Pierson Guthrey
pguthrey@iastate.edu
Sequences and Series
3.8
Sequences and Series
3.8.1
Sequences
3.8.2
Series
The partial sum of a geometric series is given by (r 6= 1)
a + ar + ar2 + ar4 + ... + arn−1 =
n−1
X
ark = a
k=0
1 − rn
1−r
If and only if |r| < 1, then as n → ∞,
a + ar + ar2 + ar4 + ... =
∞
X
ark =
k=0
a
1−r
Convergence A series S converges to a limit L if and only if the sequence of partial sums SK converges
to L.
∞
P
1
• The p-series
nr converges for r > 1 and diverges for r ≤ 1.
n=1
∞
P
• The harmonic series
n=1
1
n
diverges.
• If the sequence {bn } converges to the limit L as n → ∞, then the telescoping series
∞
P
(bn − bn+1 )
n=1
converges to b1 − L
For function series,
• A function series converges pointwise on Ω if it converges for each x ∈ Ω. That is, pointwise convergence is defined as
N
∞
X
X
SN (x) =
fn (x) → S(x) =
fn (x) ∀ x ∈ Ω
n=1
n=1
• A function series converges uniformly on Ω if it converges pointwise and remainder from the partial
series sum converges to 0 as n → ∞ independent of x. That is, it converges if
∀ > 0, ∃ N s.t. n > N =⇒ |Sn (x) − f (x)| < 4
Trigonometric Functions
Z
cos(αx)eβx dx = β 2 + α2 (β cos(αx) + α sin(αx)) eβx
Ω
Z
sin(αx)eβx dx = β 2 + α2 (β sin(αx) − α cos(αx)) eβx
Ω
eix = cos(x) + i sin(x)
4.1
cos(x) =
1 −ix
e
+ eix
2
sin(x) =
i −ix
e
− eix
2
Pythagorean Identities
sin2 x + cos2 x = 1
1 + tan2 x = sec2 x
4
1 + cot2 x = csc2 x
4.2
4.2
Pierson Guthrey
pguthrey@iastate.edu
Sum- Difference Formulas
Sum- Difference Formulas
sin(u ± v) = sin u cos v ± cos u sin v
cos(u ± v) = cos u cos v ∓ sin u sin v
tan(u ± v) =
4.3
tan u ± tan v
1 ∓ tan u tan v
Double Angle Formula
sin(2u) = 2 sin u cos u
cos(2u) = cos2 u − sin2 u = 2 cos2 u − 1 = 1 − 2 sin2 u
2 tan u
tan(2u) =
1 − tan2 u
4.4
Sum to Product Formulas
u±v
u∓v
cos
2
2
u+v
u−v
cos u + cos v = 2 cos
cos
2
2
u+v
u−v
cos u − cos v = −2 sin
sin
2
2
sin u ± sin v = 2 sin
4.5
Differentiation
d
(tan x) = sec2 x
dx
d
dx
d
dx
4.6
sin−1 x =
csc−1 x =
du
√ 1
1−u2 dx
−1
du
√
|u| u2 −1 dx
d
−1 du
−1
x = √1−u
2 dx
dx cos
d
−1
du
−1
√
sec
x
=
dx
|u| u2 −1 dx
d
−1
dx tan
d
−1
dx cot
x=
x=
1
1+u2
−1
1+u2
Integration
Z
5
d
(cot x) = − csc2 x
dx
Z
sec2 x = tan x + C
csc2 x = − cot x + C
Hyperbolic Functions
sinh x =
ex − e−x
2
cosh x =
ex + e−x
2
5
tanh x =
sinh x
ex − e−x
= x
cosh x
e + e−x
du
dx
du
dx
Pierson Guthrey
pguthrey@iastate.edu
6
6.1
Areas and Volumes
Two Dimensions
Shape
Trapezoid
6.2
Area
b1 +b2
2 h
Perimeter
sum of sides
Three Dimensions
For shapes with height h, base b, radius r,
Shape
Cone
Pyramid
Sphere
6.3
Volume
1
2
3 πr
1
3 bh
4
3
3 πr
Surface Area
√
πr + πrs = πr2 πr r2 + h2
2
4πr2
N Dimensions
Shape
Sphere
Volume
π n\2
rN
Γ( N
2 +1)
6
Surface Area
N π n\2 N −1
r
Γ( N
2 +1)
Pierson Guthrey
pguthrey@iastate.edu
7
Named Functions
7.1
Gamma Function
For a positive integer n,
Γ(n) = (n − 1)!
This function is also defined for all complex numbers except negative integers and zero. For complex
numbers with a positive real part, the Gamma function is defined as the improper integral
Z ∞
Γ(t) =
xt−1 e−x dx
0
8
Fourier
8.1
Parseval’s Identity
Given the Fourier coefficients of f , cn ,
∞
X
1
2π
2
|cn | =
n=−∞
8.2
Z
π
2
|f (x)| dx
−π
Plancherel Theorem
If f ∈ L1 (R) ∩ L2 (R), then fˆ ∈ L2 (R), and the FT is an isometry wrt k·kL2 (R)
Z
Z ˆ 2
2
|f (x)| dx =
f (k) dk
RN
8.3
RN
Poisson Summation Formula
For φ ∈ S
√
n
X
2π
φ(2πn) =
n=−∞
9
n
X
φ̂(n)
n=−∞
Famous Inequalities
9.1
Jensen’s Inequality
If φ is convex on R then
φ
1
b−a
Z
1
b−a
Z
!
b
f (t)dt
a
1
≤
b−a
Z
1
b−a
Z
b
φ(f (t))dt
a
If φ is concave on R then
φ
9.2
!
b
f (t)dt
≥
a
b
φ(f (t))dt
a
Normed Linear Space Inequalities
X is a vector space with inner product h·, ·i and norm kxk =
7
p
hx, xi. Take any x, y ∈ X.
9.3
9.2.1
Pierson Guthrey
pguthrey@iastate.edu
Young’s Inequality
Cauchy Schwartz Inequality
2
|hx, yi| ≤ hx, xi hy, yi or equivalently, |hx, yi| ≤ kxk kyk
9.2.2
Parallelogram Law
X is an normed inner product space if and only if
2
2
2
kx + yk + kx − yk = 2kxk + 2kyk
9.2.3
2
Pythagorean Theorem
2
x + y 2 = x2 + y 2 9.2.4
Bessel’s Inequality
For an infinite dimensional basis, SN =
P
hx, en i en = PMN x where MN = L {e1 , e2 , ..., en }, which implies
∞
X
2
2
|hx, en i| ≤ kxk
n=1
9.3
Young’s Inequality
For , a, b > 0, 1 ≤ p, q < ∞ such that
1
p
+
1
q
= 1, then
ab ≤ q
−q b
ap
+ p
p
q
In particular, if p = q = 2,
ab ≤ a2
1 b2
+
2
2
ab ≤
a2
b2
+
2
2
Further, if = 1,
9.4
Young’s Convolution Inequality
If φ, ψ ∈ C0∞ , then for 1 +
1
r
=
1
p
+
1
q
and 1 ≤ p, q, r ≤ ∞
kφ ∗ ψkLr (RN ) ≤ kφkLp (RN ) kψkLq (RN )
Example: kφ ∗ ψkLp (RN ) ≤ kφkLp (RN ) kψkL1 (RN )
9.5
Holder Inequality
Integral version
For u, v measurable, and 1 ≤ p, q ≤ ∞ such that
1
p
1
q
Z
Z
|u(x)v(x)| dx ≤
Ω
+
=1
p
p1 Z
|u(x)| dx
Ω
q1
|v(x)| dx
q
Ω
ku(x)v(x)kL1 (Ω) = ku(x)kLp (Ω) kv(x)kLq (Ω)
8
9.6
Pierson Guthrey
pguthrey@iastate.edu
Minkawski Inequality
Sum Version
For 1 ≤ p, q ≤ ∞ such that
1
p
+
1
q
= 1,
X
9.6
|ak bK | ≤
X
p
|ak |
p1 X
|bk |
q
q1
Minkawski Inequality
For u, v measurable, and 1 ≤ p ≤ ∞
Z
p1 Z
p1
Z
p1
p
p
p
≤
|u(x)| dx
+
|v(x)| dx
|u(x) + v(x)| dx
Ω
Ω
Ω
ku + vkLp (Ω) ≤ kukLp (Ω) + kvkLp (Ω)
Sum Version
For 1 ≤ p ≤ ∞
X
10
10.1
p
|ak + bk |
p1
≤
X
p
|ak |
p1
+
X
p
|bk |
p1
Common Theorems
Stone-Weierstrass Theorem
(also known as Weierstrass Approximation Theorem)
Every continuous function on a closed interval can be uniformly approximated by a polynomial function.
10.2
Heine-Borel Theorem
Every closed and totally bounded subset of a complete matrix space is compact.
For a subset S ⊂ RN , the following are equivalent
• S is closed and bounded
• S is compact
10.3
Arzela Ascoli Theorem
Consider a sequence of real-valued continuous functions {fn } defined on a closed and bounded interval
[a, b] of the real line. There exists a subsequence {fnk } that converges uniformly if and only if this sequence
is uniformly bounded and equicontinuous.
10.4
Fubini’s Theorem
Given measureable spaces A, B, and if f is A × B measureable, and if the integral with respect to a product
measure satisfies
Z
|f (x, y)| d(x, y) < ∞
A×B
then the integral with respect to a product measure is equal to the iterated integrals
Z
Z Z
Z Z
f (x, y)d(x, y) =
f (x, y)dy dx =
f (x, y)dx dy
A×B
A
B
B
9
A
10.5
Pierson Guthrey
pguthrey@iastate.edu
Lax Milgram Theorem
Corollary
If f satisfies the above conditions and additionally f (x, y) = h(x)g(y), then
Z
Z
Z
f (x, y)d(x, y) =
h(x)dx
g(y)dy
A×B
10.5
A
B
Lax Milgram Theorem
If a(·, ·) be a bilinear form on H which is
• bounded: |a(u, v)| ≤ CkukH kvkH
2
• coercive: |a(u, u)| ≥ ckukH
2
then for any f ∈ H∗ there is a unique solution u ∈ H to the equation a(u, v) = hf, vi and also kuk ≤ 1c kuk .
10.6
Fredholm Alternative
10.6.1
Operator Version
Given a compact integral operator K, a nonzero λ is either an eigenvalue of K of lies in the domain of the
resolvent.
Rλ (K) = (K − λI)−1
10.6.2
Integral Equation Version
Let K(x, y) be an kernel of the integral operator T u = λu − hK, ui. If K(x, y) yields a compact integral
operator, then the following theorem holds: For any nonzero λ ∈ C, either the integral equation
b
Z
λφ(x) −
K(x, y)φ(y)dy = f (x)
a
has a solution for all f (x) OR the associated homogenous case f (x) = 0 has only trivial solutions.
K(x, y) being Hilbert Schmidt is a sufficient but not necessary condition.
10.6.3
Linear Algebra Version
For A ∈ Cn×m and b ∈ Cm×1 ,
• Either A~x = ~b has a solution ~x
• OR: AT ~y = 0 has a solution ~y with ~yT ~b 6= 0.
That is, A~x = ~b has a solution if and only if for any ~y s.t. AT ~y = 0, ~yT ~b = 0.
10.7
Riesz Representation Theorem
Given a Hilbert space H and its dual space H0 . For all y ∈ H0 , there exists a unique φy such that
φy (x) = hx, yi
10
10.8
10.8
Pierson Guthrey
pguthrey@iastate.edu
Riemmann Lebesgue Lemma
Riemmann Lebesgue Lemma
The Fourier Transform of any L1 function vanishes at infinity.
Let f ∈ L1 (R) and since f ∈ L1 there exists a smooth function (say g), compactly supported (say on [a, b])
that approximates f . Thus let kf − gkL1 < . Since g is smooth,
Z
ĝ(k) =
b
g(x)e−ixk dx =
a
g(a)e−iak
g(b)e−ibk
−
+
−ik
−ik
Z
b
g 0 (x)e−ixk −ikdx
a
So |ĝ(k)| → 0 at at k → ±∞. Then
Z
Z
Z
ˆ f (k) = f (x)e−ixk dx ≤ (f (x) − g(x))e−ixk dx + |ĝ(k)| ≤ |f (x) − g(x)| dx + |ĝ(k)| < + |ĝ(k)|
So as k → ±∞, lim supk→±∞ = 0
10.9
Eigenfunction Expansion Theorem
Let K be a self adjoint compact operator and let (λk , ek ) be the set of eigenpairs for K where λk 6= 0 and
ek are the eigenfunctions orthonormalized to kek k = 1.
Any function in the range of K can be expanded in a Fourier series in the eigenfunctions of K corresponding
to nonzero eigenvalues. There eigenfunctions form an orthonormal basis for R(K) (but necessarily for H).
Thus, for all f ∈ H,
X
X
X
Kf =
hKf, ek i ek =
hf, Kek i ek =
λk hf, ek i ek
where equality is in the L2 sense.
If we include the eigenfunctions for λ = 0, we have a basis for H. If h is the projection if f onto the nullspace
of K, then an arbitrary function can be decomposed uniquely as
X
f =h+
hf, ek i ek
11