Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 8 docx

I.5 Table of Fourier Cosine Transforms
f(x)
1
π

∞
0
f(x) cos (ωx) dx
2

∞
0
C(ω) cos (ωx) dω C(ω)
f

(x) ωS(ω) −
1
π
f(0)
f

(x) −ω
2
C(ω) −
1
π
f

(0)
xf(x)

∂
∂ω
F
s
[f(x)]
f(cx), c > 0
1
c
C

ω
c

2c
x
2
+ c
2
e
−cω
e
−cx
c/π
ω
2
+ c
2
e
−cx
2

1
√
4πc
e
−ω
2
/(4c)

π
c
e
−x
2
/(4c)
e
−cω
2
2254
I.6 Table of Fourier Sine Transforms
f(x)
1
π

∞
0
f(x) sin (ωx) dx
2

∞
0

S(ω) sin (ωx) dω S(ω)
f

(x) −ωC(ω)
f

(x) −ω
2
S(ω) +
1
π
ωf(0)
xf(x) −
∂
∂ω
F
c
[f(x)]
f(cx), c > 0
1
c
S

ω
c

2x
x
2
+ c

2
e
−cω
e
−cx
ω/π
ω
2
+ c
2
2 arctan

x
c

1
ω
e
−cω
1
x
e
−cx
1
π
arctan

ω
c


2255
1
1
πω
2
x
1
x
e
−cx
2
ω
4c
3/2
√
π
e
−ω
2
/(4c)
√
πx
2c
3/2
e
−x
2
/(4c)
ω
e

−cω
2
2256
Appendix J
Table of Wronskians
W [x − a, x − b] b − a
W

e
ax
,
e
bx

(b − a)
e
(a+b)x
W [cos(ax), sin(ax)] a
W [cosh(ax), sinh(ax)] a
W [
e
ax
cos(bx),
e
ax
sin(bx)] b
e
2ax
W [
e

ax
cosh(bx),
e
ax
sinh(bx)] b
e
2ax
W [sin(c(x − a)), sin(c(x − b))] c sin(c(b − a))
W [cos(c(x − a)), cos(c(x − b))] c sin(c(b − a))
W [sin(c(x − a)), cos(c(x − b))] −c cos(c(b − a))
2257
W [sinh(c(x − a)), sinh(c(x − b))] c sinh(c(b − a))
W [cosh(c(x − a)), cosh(c(x − b))] c cosh(c(b − a))
W [sinh(c(x − a)), cosh(c(x − b))] −c cosh(c(b − a))
W

e
dx
sin(c(x − a)),
e
dx
sin(c(x − b))

c
e
2dx
sin(c(b − a))
W

e

dx
cos(c(x − a)),
e
dx
cos(c(x − b))

c
e
2dx
sin(c(b − a))
W

e
dx
sin(c(x − a)),
e
dx
cos(c(x − b))

−c
e
2dx
cos(c(b − a))
W

e
dx
sinh(c(x − a)),
e
dx

sinh(c(x − b))

c
e
2dx
sinh(c(b − a))
W

e
dx
cosh(c(x − a)),
e
dx
cosh(c(x − b))

−c
e
2dx
sinh(c(b − a))
W

e
dx
sinh(c(x − a)),
e
dx
cosh(c(x − b))

−c
e

2dx
cosh(c(b − a))
W [(x − a)
e
cx
, (x − b)
e
cx
] (b − a)
e
2cx
2258
Appendix K
Sturm-Liouville Eigenvalue Problems
• y

+ λ
2
y = 0, y(a) = y(b) = 0
λ
n
=
nπ
b − a
, y
n
= sin

nπ(x − a)
b − a


, n ∈ N
y
n
, y
n
 =
b − a
2
• y

+ λ
2
y = 0, y(a) = y

(b) = 0
λ
n
=
(2n − 1)π
2(b − a)
, y
n
= sin

(2n − 1)π(x − a)
2(b − a)

, n ∈ N
y

n
, y
n
 =
b − a
2
• y

+ λ
2
y = 0, y

(a) = y(b) = 0
λ
n
=
(2n − 1)π
2(b − a)
, y
n
= cos

(2n − 1)π(x − a)
2(b − a)

, n ∈ N
2259
y
n
, y

n
 =
b − a
2
• y

+ λ
2
y = 0, y

(a) = y

(b) = 0
λ
n
=
nπ
b − a
, y
n
= cos

nπ(x − a)
b − a

, n = 0, 1, 2, . . .
y
0
, y
0

 = b − a, y
n
, y
n
 =
b − a
2
for n ∈ N
2260
Appendix L
Green Functions for Ordinary Differential
Equations
• G

+ p(x)G = δ(x − ξ), G(ξ
−
: ξ) = 0
G(x|ξ) = exp

−

x
ξ
p(t) dt

H(x − ξ)
• y

= 0, y(a) = y(b) = 0
G(x|ξ) =

(x
<
− a)(x
>
− b)
b − a
• y

= 0, y(a) = y

(b) = 0
G(x|ξ) = a − x
<
• y

= 0, y

(a) = y(b) = 0
G(x|ξ) = x
>
− b
2261
• y

− c
2
y = 0, y(a) = y(b) = 0
G(x|ξ) =
sinh(c(x
<

− a)) sinh(c(x
>
− b))
c sinh(c(b − a))
• y

− c
2
y = 0, y(a) = y

(b) = 0
G(x|ξ) = −
sinh(c(x
<
− a)) cosh(c(x
>
− b))
c cosh(c(b − a))
• y

− c
2
y = 0, y

(a) = y(b) = 0
G(x|ξ) =
cosh(c(x
<
− a)) sinh(c(x
>

− b))
c cosh(c(b − a))
• y

+ c
2
y = 0, y(a) = y(b) = 0, c =
npi
b−a
, n ∈ N
G(x|ξ) =
sin(c(x
<
− a)) sin(c(x
>
− b))
c sin(c(b − a))
• y

+ c
2
y = 0, y(a) = y

(b) = 0, c =
(2n−1)pi
2(b−a)
, n ∈ N
G(x|ξ) = −
sin(c(x
<

− a)) cos(c(x
>
− b))
c cos(c(b − a))
• y

+ c
2
y = 0, y

(a) = y(b) = 0, c =
(2n−1)pi
2(b−a)
, n ∈ N
G(x|ξ) =
cos(c(x
<
− a)) sin(c(x
>
− b))
c cos(c(b − a))
2262
• y

+ 2cy

+ dy = 0, y(a) = y(b) = 0, c
2
> d
G(x|ξ) =

e
−cx
<
sinh(
√
c
2
− d(x
<
− a))
e
−cx
<
sinh(
√
c
2
− d(x
>
− b))
√
c
2
− d
e
−2cξ
sinh(
√
c
2

− d(b − a))
• y

+ 2cy

+ dy = 0, y(a) = y(b) = 0, c
2
< d,
√
d − c
2
=
nπ
b−a
, n ∈ N
G(x|ξ) =
e
−cx
<
sin(
√
d − c
2
(x
<
− a))
e
−cx
<
sin(

√
d − c
2
(x
>
− b))
√
d − c
2
e
−2cξ
sin(
√
d − c
2
(b − a))
• y

+ 2cy

+ dy = 0, y(a) = y(b) = 0, c
2
= d
G(x|ξ) =
(x
<
− a)
e
−cx
<

(x
>
− b)
e
−cx
<
(b − a)
e
−2cξ
2263
Appendix M
Trigonometric Identities
M.1 Circular Functions
Pythagorean Identities
sin
2
x + cos
2
x = 1, 1 + tan
2
x = sec
2
x, 1 + cot
2
x = csc
2
x
Angle Sum and Difference Identities
sin(x + y) = sin x cos y + cos x sin y
sin(x − y) = sin x cos y −cos x sin y

cos(x + y) = cos x cos y −sin x sin y
cos(x − y) = cos x cos y + sin x sin y
2264
Function Sum and Difference Identities
sin x + sin y = 2 sin
1
2
(x + y) cos
1
2
(x − y)
sin x − sin y = 2 cos
1
2
(x + y) sin
1
2
(x − y)
cos x + cos y = 2 cos
1
2
(x + y) cos
1
2
(x − y)
cos x − cos y = −2 sin
1
2
(x + y) sin
1

2
(x − y)
Double Angle Identities
sin 2x = 2 sin x cos x, cos 2x = cos
2
x − sin
2
x
Half Angle Identities
sin
2
x
2
=
1 − cos x
2
, cos
2
x
2
=
1 + cos x
2
Function Product Identities
sin x sin y =
1
2
cos(x − y) −
1
2

cos(x + y)
cos x cos y =
1
2
cos(x − y) +
1
2
cos(x + y)
sin x cos y =
1
2
sin(x + y) +
1
2
sin(x − y)
cos x sin y =
1
2
sin(x + y) −
1
2
sin(x − y)
Exponential Identities
e
ıx
= cos x + ı sin x, sin x =
e
ıx
−
e

−ıx
ı2
, cos x =
e
ıx
+
e
−ıx
2
2265
M.2 Hyperbolic Functions
Exponential Identities
sinh x =
e
x
−
e
−x
2
, cosh x =
e
x
+
e
−x
2
tanh x =
sinh x
cosh x
=

e
x
−
e
−x
e
x
+
e
−x
Reciprocal Identities
csch x =
1
sinh x
, sech x =
1
cosh x
, coth x =
1
tanh x
Pythagorean Identities
cosh
2
x − sinh
2
x = 1, tanh
2
x + sech
2
x = 1

Relation to Circular Functions
sinh(ıx) = ı sin x sinh x = −ı sin(ıx)
cosh(ıx) = cos x cosh x = cos(ıx)
tanh(ıx) = ı tan x tanh x = −ı tan(ıx)
Angle Sum and Difference Identities
sinh(x ± y) = sinh x cosh y ±cosh x sinh y
cosh(x ± y) = cosh x cosh y ±sinh x sinh y
tanh(x ± y) =
tanh x ± tanh y
1 ± tanh x tanh y
=
sinh 2x ± sinh 2y
cosh 2x ± cosh 2y
coth(x ± y) =
1 ± coth x coth y
coth x ± coth y
=
sinh 2x ∓ sinh 2y
cosh 2x − cosh 2y
2266
Function Sum and Difference Identities
sinh x ± sinh y = 2 sinh
1
2
(x ± y) cosh
1
2
(x ∓ y)
cosh x + cosh y = 2 cosh
1

2
(x + y) cosh
1
2
(x − y)
cosh x − cosh y = 2 sinh
1
2
(x + y) sinh
1
2
(x − y)
tanh x ± tanh y =
sinh(x ± y)
cosh x cosh y
coth x ± coth y =
sinh(x ± y)
sinh x sinh y
Double Angle Identities
sinh 2x = 2 sinh x cosh x, cosh 2x = cosh
2
x + sinh
2
x
Half Angle Identities
sinh
2
x
2
=

cosh x − 1
2
, cosh
2
x
2
=
cosh x + 1
2
Function Product Identities
sinh x sinh y =
1
2
cosh(x + y) −
1
2
cosh(x − y)
cosh x cosh y =
1
2
cosh(x + y) +
1
2
cosh(x − y)
sinh x cosh y =
1
2
sinh(x + y) +
1
2

sinh(x − y)
See Figure M.1 for plots of the hyperbolic circular functions.
2267
-2
-1 1
2
-3
-2
-1
1
2
3
-2
-1 1
2
-1
-0.5
0.5
1
Figure M.1: cosh x, sinh x and then tanh x
2268
Appendix N
Bessel Functions
N.1 Definite Integrals
Let ν > −1.

1
0
rJ
ν

(j
ν,m
r)J
ν
(j
ν,n
r) dr =
1
2
(J

ν
(j
ν,n
))
2
δ
mn

1
0
rJ
ν
(j

ν,m
r)J
ν
(j


ν,n
r) dr =
j

2
ν,n
− ν
2
2j

2
ν,n

J
ν
(j

ν,n
)

2
δ
mn

1
0
rJ
ν
(α
m

r)J
ν
(α
n
r) dr =
1
2α
2
n

a
2
b
2
+ α
2
n
− ν
2

(J
ν
(α
n
))
2
δ
mn
Here α
n

is the n
th
positive root of aJ
ν
(r) + brJ

ν
(r), where a, b ∈ R.
2269
Appendix O
Formulas from Linear Algebra
Kramer’s Rule. Consider the matrix equation
Ax =

b.
This equation has a unique solution if and only if det(A) = 0. If the determinant vanishes then there are either no
solutions or an infinite number of solutions. If the determinant is nonzero, the solution for each x
j
can be written
x
j
=
det A
j
det A
where A
j
is the matrix formed by replacing the j
th
column of A with b.

Example O.0.1 The matrix equation

1 2
3 4

x
1
x
2

=

5
6

,
has the solution
x
1
=




5 2
6 4









1 2
3 4




=
8
−2
= −4, x
2
=




1 5
3 6









1 2
3 4




=
−9
−2
=
9
2
.
2270
Appendix P
Vector Analysis
Rectangular Coordinates
f = f(x, y, z), g = g
x
i + g
y
j + g
z
k
∇f =
∂f
∂x
i +
∂f
∂y

j +
∂f
∂z
k
∇ ·g =
∂g
x
∂x
+
∂g
y
∂y
+
∂g
z
∂z
∇ ×g =






i j k
∂
∂x
∂
∂y
∂
∂z

g
x
g
y
g
z






∆f = ∇
2
f =
∂
2
f
∂x
2
+
∂
2
f
∂y
2
+
∂
2
f

∂z
2
2271
Spherical Coordinates
x = r cos θ sin φ, y = r sin θ sin φ, z = r cos φ
f = f(r, θ, φ), g = g
r
r + g
θ
θ + g
φ
φ
Divergence Theorem.

∇ · u dx dy =

u · n ds
Stoke’s Theorem.

(∇ × u) · ds =

u · dr
2272
Appendix Q
Partial Fractions
A proper rational function
p(x)
q(x)
=
p(x)

(x − a)
n
r(x)
Can be written in the form
p(x)
(x − α)
n
r(x)
=

a
0
(x − α)
n
+
a
1
(x − α)
n−1
+ ··· +
a
n−1
x − α

+ (···)
where the a
k
’s are constants and the last ellipses represents the partial fractions expansion of the roots of r(x). The
coefficients are
a

k
=
1
k!
d
k
dx
k

p(x)
r(x)





x=α
.
Example Q.0.2 Consider the partial fraction expansion of
1 + x + x
2
(x − 1)
3
.
The expansion has the form
a
0
(x − 1)
3
+

a
1
(x − 1)
2
+
a
2
x − 1
.
2273
The coefficients are
a
0
=
1
0!
(1 + x + x
2
)|
x=1
= 3,
a
1
=
1
1!
d
dx
(1 + x + x
2

)|
x=1
= (1 + 2x)|
x=1
= 3,
a
2
=
1
2!
d
2
dx
2
(1 + x + x
2
)|
x=1
=
1
2
(2)|
x=1
= 1.
Thus we have
1 + x + x
2
(x − 1)
3
=

3
(x − 1)
3
+
3
(x − 1)
2
+
1
x − 1
.
Example Q.0.3 Consider the partial fraction expansion of
1 + x + x
2
x
2
(x − 1)
2
.
The expansion has the form
a
0
x
2
+
a
1
x
+
b

0
(x − 1)
2
+
b
1
x − 1
.
The coefficients are
a
0
=
1
0!

1 + x + x
2
(x − 1)
2





x=0
= 1,
a
1
=
1

1!
d
dx

1 + x + x
2
(x − 1)
2





x=0
=

1 + 2x
(x − 1)
2
−
2(1 + x + x
2
)
(x − 1)
3






x=0
= 3,
b
0
=
1
0!

1 + x + x
2
x
2





x=1
= 3,
b
1
=
1
1!
d
dx

1 + x + x
2
x

2





x=1
=

1 + 2x
x
2
−
2(1 + x + x
2
)
x
3





x=1
= −3,
2274
Thus we have
1 + x + x
2
x

2
(x − 1)
2
=
1
x
2
+
3
x
+
3
(x − 1)
2
−
3
x − 1
.
If the rational function has real coefficients and the denominator has complex roots, then you can reduce the work
in finding the partial fraction expansion with the following trick: Let α and α be compl ex conjugate pairs of roots of
the denominator.
p(x)
(x − α)
n
(x − α)
n
r(x)
=

a

0
(x − α)
n
+
a
1
(x − α)
n−1
+ ··· +
a
n−1
x − α

+

a
0
(x − α)
n
+
a
1
(x − α)
n−1
+ ··· +
a
n−1
x − α

+ (···)

Thus we don’t have to calculate the coefficients for the root at α. We just take the complex conjugate of the coefficients
for α.
Example Q.0.4 Consider the partial fraction expansion of
1 + x
x
2
+ 1
.
The expansion has the form
a
0
x − i
+
a
0
x + i
The coefficients are
a
0
=
1
0!

1 + x
x + i






x=i
=
1
2
(1 − i),
a
0
=
1
2
(1 − i) =
1
2
(1 + i)
Thus we have
1 + x
x
2
+ 1
=
1 − i
2(x − i)
+
1 + i
2(x + i)
.
2275
Appendix R
Finite Math
Newton’s Binomial Formula.

(a + b)
n
=
n

k=0

k
n

a
n−k
b
k
= a
n
+ na
n−1
b +
n(n − 1)
2
a
n−2
b
2
+ ··· + nab
n−1
+ b
n
,

The binomial coefficients are,

k
n

=
n!
k!(n − k)!
.
2276
Appendix S
Physics
In order to reduce processing costs, a chicken farmer wished to acquire a plucking machine. Since there was no such
machine on the market, he hired a mechanical engineer to design one. After extensive research and testing, the professor
concluded that it was impossible to build such a machine with current technology. The farmer was disappointed, but
not wanting to abandon his dream of an automatic plucker, he consulted a physicist. After a single afternoon of work,
the physicist reported that not only could a plucking machine be built, bu t that the design was simple. The elated
farmer asked him to describe his method. The physicist replied, “First, assume a spherical chicken . . . ”.
The problems in this text will implicitly make certain simplifying assumptions about chickens. For example, a
problem might assume a perfectly elastic, frictionless, spherical chicken. In two-dimensional problems, we will assume
that chickens are circular.
2277