and Erd´elyi, A., Tables of Integral Transforms.. Oberhettinger, F., Tables of Fourier Transforms and Fourier Transforms of Distributions, Springer-Verlag, Berlin, 1980.. Oberhettinger,
Trang 1T3.6 T ABLES OF I NVERSE M ELLIN T RANSFORMS 1193
No Direct transform, ˆf (s) Inverse transform, f (x) = 1
2πi
σ+i∞
σ–i∞
ˆ
f (s)x–s ds
11 cos s2/a
2
a
π cos 14a|ln x| 2 –14π
12 arctan
a
s + b
, Re s > –b
⎧
⎨
⎩
x b
|ln x|sin a|ln x|
if 0< x <1 ,
T3.6.4 Expressions with Special Functions
No Direct transform, ˆf (s) Inverse transform, f (x) = 1
2πi
σ+i∞
σ–i∞
ˆ
f (s)x–s ds
2 Γ(s), –1 < Re s <0 e–x– 1
3 sin 12πs
Γ(s), –1 < Re s <1 sin x
4 sin(as) Γ(s),
Re s > –1 , |a| < π
2 exp(–x cos a) sin(x sin a)
5 cos 12πs
Γ(s), 0< Re s <1 cos x
6 cos 12πs
Γ(s), –2 < Re s <0 – 2 sin2(x/2 )
7 cos(as) Γ(s), Re s >0 , |a| < π2 exp(–x cos a) cos(x sin a)
8 Γ(s)
cos(πs), 0< Re s < 1
9 Γ(a + s)Γ(b – s),
–a < Re s < b, a + b >0 Γ(a + b)x a (x +1)–a–b
10 Γ(a + s)Γ(b + s),
Re s > –a, –b 2x(a+b)/2K a–b 2√ x
11 Γ(s)
Γ(s + ν), Re s >0, ν >0
( 1– x) ν–1
Γ(ν) if 0< x <1,
0 if 1< x
12
Γ(1– ν – s)
Γ(1– s) ,
Re s <1– ν, ν >0
0 if 0< x <1,
(x –1 )ν–1
Γ(ν) if 1< x
13
Γ(s)
Γ(ν – s +1 ),
0< Re s < ν2 +
3 4
x–ν/2J ν 2√ x
14 Γ(s + ν)Γ(s – ν)
Γ(s +1/2 ) , Re s >|ν| π– 1/2
e–x/2K ν (x/2 ) 15
Γ(s + ν)Γ(1 /2– s)
Γ(1+ ν – s) ,
–ν < Re s < 12
π1/2e–x/2I (x/2 )
16 ψ (s + a) – ψ(s + b),
Re s > –a, –b
x b – x a
1– x if 0< x <1 ,
0 if 1< x
17 Γ(s)ψ(s), Re s >0 e–x ln x
18 Γ(s, a), a >0 0 if 0< x < a,
e–x if a < x
19 Γ(s)Γ(1 – s, a), Re s >0, a >0 (x +1 )–1e–a(x+1)
Trang 21194 INTEGRALTRANSFORMS
No Direct transform, ˆf (s) Inverse transform, f (x) = 1
2πi
σ+i∞
σ–i∞
ˆ
f (s)x–s ds
20 γ (s, a), Re s >0, a >0 e–x if 0< x < a,
0 if a < x
21 J0 a √
b2– s2
, a> 0
⎧
⎪
⎪
0 if 0< x < e–a, cos b √
a2– ln2x
π √
a2– ln2x if e–a < x < e a,
22 s–1I0 (s), Re s >0
1 if 0< x < e–1,
π–1arccos(ln x) if e– 1< x < e,
23 I (s), Re s >0
⎧
⎪
⎪
⎪
⎪
– 2ν sin(πν)
πF (x) √
ln2x– 1 if 0< x < e
– 1 , cos
ν arccos(ln x)
π √
1 – ln2x if e–1< x < e,
where F (x) = – 1– ln x + √
1– ln x 2ν
24 s–1I (s), Re s >0
⎧
⎪
⎨
⎪
⎩
2ν sin(πν)
πνF (x) if 0< x < e–1, sin
ν arccos(ln x)
πν if e– 1< x < e,
where F (x) = – 1– ln x + √
1– ln x 2ν
25 s–ν ν (s), Re s > –12
⎧
⎪
⎪
0 if 0< x < e– 1 , ( 1 – ln2x)ν–1/2
√
π2ν Γ(ν +1/2 ) if e
– 1< x < e,
26 s–1K0 (s), Re s >0
arccosh(– ln x) if 0< x < e– 1 ,
0 if e– 1< x
27 s–1K1(s), Re s >0 √
ln2x– 1 if 0< x < e–1,
0 if e– 1< x
28 K ν (s), Re s >0
⎧
⎨
⎩
cosh
ν arccosh(– ln x)
√
ln2x– 1 if 0< x < e
– 1 ,
29 s–1K ν (s), Re s >0
1
νsinh
ν arccosh(– ln x)
if 0< x < e– 1 ,
30 s–ν K ν (s), Re s >0, ν > –12
⎧
⎨
⎩
√
π(ln2x– 1 )ν–1/2
2ν Γ(ν +1/2 ) if 0< x < e– 1 ,
0 if e– 1< x
References for Chapter T3
Bateman, H and Erd´elyi, A., Tables of Integral Transforms Vol 1, McGraw-Hill, New York, 1954 Bateman, H and Erd´elyi, A., Tables of Integral Transforms Vol 2, McGraw-Hill, New York, 1954 Ditkin, V A and Prudnikov, A P., Integral Transforms and Operational Calculus, Pergamon Press, New
York, 1965.
Oberhettinger, F., Tables of Fourier Transforms and Fourier Transforms of Distributions, Springer-Verlag,
Berlin, 1980.
Oberhettinger, F., Tables of Mellin Transforms, Springer-Verlag, New York, 1974.
Oberhettinger, F and Badii, L., Tables of Laplace Transforms, Springer-Verlag, New York, 1973.
Prudnikov, A P., Brychkov, Yu A., and Marichev, O I., Integrals and Series, Vol 4, Direct Laplace
Transforms, Gordon & Breach, New York, 1992.
Prudnikov, A P., Brychkov, Yu A., and Marichev, O I., Integrals and Series, Vol 5, Inverse Laplace
Transforms, Gordon & Breach, New York, 1992.
Trang 3Chapter T4
Orthogonal Curvilinear
Systems of Coordinate
T4.1 Arbitrary Curvilinear Coordinate Systems
T4.1.1 General Nonorthogonal Curvilinear Coordinates
T4.1.1-1 Metric tensor Arc length and volume elements in curvilinear coordinates
The curvilinear coordinates x1, x2, x3are defined as functions of the rectangular Cartesian
coordinates x, y, z:
x1= x1(x, y, z), x2 = x2(x, y, z), x3= x3(x, y, z).
Using these formulas, one can express x, y, z in terms of the curvilinear coordinates
x1, x2, x3as follows:
x = x(x1, x2, x3), y = y(x1, x2, x3), z = z(x1, x2, x3)
The metric tensor components g ij are determined by the formulas
g ij (x1, x2, x3) = ∂x ∂x i ∂x ∂x j + ∂x ∂y i ∂x ∂y j + ∂x ∂z i ∂x ∂z j;
g ij (x1, x2, x3) = g ji (x1, x2, x3); i , j =1, 2, 3
The arc length dl between close points (x, y, z)≡(x1, x2, x3) and (x + dx, y + dy, z + dz)≡
(x1+ dx1, x2+ dx2, x3+ dx3) is expressed as
(dl)2= (dx)2+ (dy)2+ (dz)2=
3
i=1
3
j=1
g ij (x1, x2, x3) dx i dx j.
The volume of the elementary parallelepiped with vertices at the eight points (x1, x2, x3),
(x1+dx1, x2, x3), (x1, x2+dx2, x3), (x1, x2, x3+dx3), (x1+dx1, x2+dx2, x3), (x1+dx1,
x2, x3+ dx3), (x1, x2+ dx2, x3+ dx3), (x1+ dx1, x2+ dx2, x3+ dx3) is given by
dV = ∂ (x, y, z)
∂ (x1, x2, x3) dx
det|g ij|dx1dx2dx3.
Here, the plus sign corresponds to the standard situation where the tangent vectors to
the coordinate lines x1, x2, x3, pointing toward the direction of growth of the respective
coordinate, form a right-handed triple, just as unit vectorsi,j, k of a right-handed rectangular
Cartesian coordinate system
1195
Trang 41196 ORTHOGONALCURVILINEARSYSTEMS OFCOORDINATE
T4.1.1-2 Vector components in Cartesian and curvilinear coordinate systems
The unit vectors i, j, k of a rectangular Cartesian coordinate system* x, y, z and the unit vectors i1, i2, i3 of a curvilinear coordinate system x1, x2, x3 are connected by the linear relations
√
g nn
∂x
∂x n k
, n=1,2,3;
i = √g11 ∂x1
2
3
∂x i3;
j = √g11 ∂x1
∂y i1+√
2
∂y i2+√
3
∂y i3;
k = √g11 ∂x1
∂z i1+√
2
∂z i2+√
3
∂z i3
In the general case, the vectorsi1,i2,i3are not orthogonal and change their direction from point to point
The components v x , v y , v z of a vector v in a rectangular Cartesian coordinate system
x , y, z and the components v1, v2, v3of the same vector in a curvilinear coordinate system
x1, x2, x3are related by
v = v x i + v y j + v z k = v1i1+ v2i2+ v3i3,
g nn
∂x n
n
n
, n=1,2,3;
v x= ∂x ∂x1 √ v g1
∂x
∂x2
v2
∂x
∂x3
v3
33;
v y = ∂x ∂y1 √ v g1
∂y
∂x2
v2
∂y
∂x3
v3
33;
v z = ∂x ∂z1 √ v g1
∂z
∂x2
v2
∂z
∂x3
v3
33.
T4.1.2 General Orthogonal Curvilinear Coordinates
T4.1.2-1 Orthogonal coordinates Length, area, and volume elements
A system of coordinates is orthogonal if
g ij (x1, x2, x3) =0 for i≠j
In this case the third invariant of the metric tensor is given by
g= det|g ij|= g11g22g33.
The Lam´e coefficients L kof orthogonal curvilinear coordinates are expressed in terms
of the components of the metric tensor as
g ii= ∂x ∂x i
2 +
∂y
∂x i
2 +
∂z
∂x i
2 , i=1,2,3
* Here and henceforth the coordinate axes and the respective coordinates of points in space are denoted by the same letters.
Trang 5T4.1 A RBITRARY C URVILINEAR C OORDINATE S YSTEMS 1197
Arc length element:
(L1dx1)2+ (L2dx2)2+ (L3dx3)2=
g11(dx1)2+ g22(dx2)2+ g33(dx3)2.
The area elements ds i of the respective coordinate surfaces x i= const are given by
ds1= dl2dl3= L2L3dx2dx3=√
g22g33dx2dx3,
ds2= dl1dl3= L1L3dx1dx3=√
g11g33dx1dx3,
ds3= dl1dl2= L1L2dx1dx2=√
g11g22dx1dx2.
Volume element:
dV = L1L2L3dx1dx2dx3 =√
g11g22g33 dx1dx2dx3.
T4.1.2-2 Basic differential relations in orthogonal curvilinear coordinates
In what follows, we present the basic differential operators in the orthogonal curvilinear
coordinates x1, x2, x3 The corresponding unit vectors are denoted byi1, i2, i3
The gradient of a scalar f is expressed as
grad f ≡ ∇f = 1
11
∂f
∂x1i1+ 1
22
∂f
∂x2i2+ 1
33
∂f
∂x3i3
Divergence of a vector v = i1v1+i2v2+i3v3:
div v≡ ∇ ⋅v= 1
∂
∂x1
11
+ ∂
∂x2
22
+ ∂
∂x3
33
Gradient of a scalar f along a vector v:
(v⋅ ∇)f = √ v g1
11
∂f
v2
22
∂f
v3
33
∂f
∂x3.
Gradient of a vector w along a vector v:
(v⋅ ∇) w = i1(v⋅ ∇)w1+i2(v⋅ ∇)w2+i3(v⋅ ∇)w3
Curl of a vector v:
curl v ≡ ∇ ×v = i1
√
g11
√ g
∂
g33 – ∂
g22
+i2
22
∂
g11 – ∂
g33
+i3
33
∂
g22 – ∂
g11 Remark. Sometimes curl v is denoted by rot v.
Laplace operator of a scalar f :
Δf ≡ ∇2f = 1
√ g
∂
∂x1
√
g
g11
∂f
∂x1
+ ∂
∂x2
√
g
g22
∂f
∂x2
+ ∂
∂x3
√
g
g33
∂f
∂x3
Trang 6
1198 ORTHOGONALCURVILINEARSYSTEMS OFCOORDINATE
T4.2 Special Curvilinear Coordinate Systems
T4.2.1 Cylindrical Coordinates
T4.2.1-1 Transformations of coordinates and vectors The metric tensor components The Cartesian coordinates are expressed in terms of the cylindrical ones as
x = ρ cos ϕ, y = ρ sin ϕ, z = z
(0 ≤ρ<∞, 0 ≤ϕ<2π, –∞ < z < ∞).
The cylindrical coordinates are expressed in terms of the cylindrical ones as
x2+ y2, tan ϕ = y/x, z = z (sin ϕ = y/ρ).
Coordinate surfaces:
x2+ y2 = ρ2 (right circular cylinders with their axis coincident with the z-axis),
y = x tan ϕ (half-planes through the z-axis),
z = z (planes perpendicular to the z-axis).
Direct and inverse transformations of the components of a vector v = v x i + v y j + v z k=
v ρ i ρ + v ϕ i ϕ + v z i z:
v ρ = v x cos ϕ + v y sin ϕ,
v ϕ = –v x sin ϕ + v y cos ϕ,
v z = v z;
v x = v ρ cos ϕ – v ϕ sin ϕ,
v y = v ρ sin ϕ + v ϕ cos ϕ,
v z = v z.
Metric tensor components:
g = ρ.
T4.2.1-2 Basic differential relations
Gradient of a scalar f :
∇f = ∂f
∂ρ i ρ+ 1
ρ
∂f
∂ϕ i ϕ+ ∂f
∂z i z
Divergence of a vector v:
∇ ⋅v = 1
ρ
∂ (ρv ρ)
ρ
∂v ϕ
Gradient of a scalar f along a vector v:
(v⋅ ∇)f = v ρ ∂f
ρ
∂f
∂z
Gradient of a vector w along a vector v:
(v⋅ ∇) w = (v⋅ ∇)w ρ i ρ + (v⋅ ∇)w ϕ i ϕ + (v⋅ ∇)w z i z
Curl of a vector v:
∇ ×v=
1
ρ
∂v z
∂z
i ρ+
∂v ρ
∂ρ
i ϕ+ 1
ρ
∂ (ρv ϕ)
∂ϕ
i z
Laplacian of a scalar f :
Δf = 1
ρ
∂
∂ρ
∂ρ
+ 1
ρ2
∂2f
∂2f
∂z2.
Remark. The cylindrical coordinates ρ, ϕ are also used as polar coordinates on the plane xy.
Trang 7T4.2 S PECIAL C URVILINEAR C OORDINATE S YSTEMS 1199 T4.2.2 Spherical Coordinates
T4.2.2-1 Transformations of coordinates and vectors The metric tensor components Cartesian coordinates via spherical ones:
x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ
(0 ≤r<∞, 0 ≤θ≤π, 0 ≤ϕ<2π)
Spherical coordinates via Cartesian ones:
x2+ y2+ z2, θ= arccos z
r, tan ϕ = y
x
sin ϕ = y
x2+ y2
Coordinate surfaces:
x2+ y2+ z2= r2 (spheres),
x2+ y2– z2tan2θ=0 (circular cones),
y = x tan ϕ (half-planes trough the z-axis).
Direct and inverse transformations of the components of a vector v = v x i + v y j + v z k=
v r i r + v θ i θ + v ϕ i ϕ:
v r = v x sin θ cos ϕ + v y sin θ sin ϕ + v z cos θ,
v θ = v x cos θ cos ϕ + v y cos θ sin ϕ – v z sin θ,
v ϕ = –v x sin ϕ + v y cos ϕ;
v x = v r sin θ cos ϕ + v θ cos θ cos ϕ – v ϕ sin ϕ,
v y = v r sin θ sin ϕ + v θ cos θ sin ϕ + v ϕ cos ϕ,
v z = v r cos θ – v θ sin θ.
The metric tensor components are
g rr =1, g θθ = r2, g ϕϕ = r2sin2θ, √
g = r2sin θ.
T4.2.2-2 Basic differential relations
Gradient of a scalar f :
∇f = ∂f
∂r i r+ 1
r
∂f
r sin θ
∂f
∂ϕ i ϕ
Divergence of a vector v:
∇ ⋅v= 1
r2
∂
2v
r
r sin θ
∂
∂θ sin θ v θ
r sin ϕ
∂v ϕ
Gradient of a scalar f along a vector v:
(v⋅ ∇)f = v r ∂f
r
∂f
r sin θ
∂f
∂ϕ
Gradient of a vector w along a vector v:
(v⋅ ∇) w = (v⋅ ∇)w r i r + (v⋅ ∇)w θ i θ + (v⋅ ∇)w ϕ i ϕ
Curl of a vector v:
∇×v= 1
r sin θ
∂ (sin θ v ϕ)
∂ϕ
i r+1
r
1
sin θ
∂v r
∂r
i θ+1
r
∂ (rv θ)
∂θ
i ϕ
Laplacian of a scalar f :
Δf = 1
r2
∂
∂r
r2∂f
∂r
r2sin θ
∂
∂θ
sin θ ∂f
∂θ
r2sin2θ
∂2f
∂ϕ2.
...z = z (planes perpendicular to the z-axis).
Direct and inverse transformations of the components of a vector v = v x i + v y j + v z ...
y = x tan ϕ (half-planes trough the z-axis).
Direct and inverse transformations of the components of a vector v = v x i + v y j + v z ... OORDINATE S YSTEMS 1199 T4.2.2 Spherical Coordinates
T4.2.2-1 Transformations of coordinates and vectors The metric tensor components Cartesian coordinates via spherical ones: