List of Figures
-
1 Algebraic and Analytic Methods
-
1.4.1 Piecewise continuous function on
.
-
1.4.2 Convex function
.
-
1.6.1 Right-hand rule for cross products.
-
1.10.1 Domain
.
-
2 Asymptotic Approximations
-
2.10.1 Contour
.
-
2.11.1
.
-
3 Numerical Methods
-
3.1.1 Representation of data in the binary interchange formats
for binary32, binary64 and binary128.
-
3.11.1 Error of the minimax rational approximation to
-
4 Elementary Functions
-
4.2.1 Branch cut for
and 
.
-
4.3.1
and 
.
-
4.3.2 Conformal mapping of exponential and logarithm.
-
4.3.3
.
-
4.3.4
.
-
4.13.1 Branches
, 
of the Lambert 
-function.
-
4.13.2 The
function on the first 5 Riemann sheets.
-
4.15.1
and 
.
-
4.15.2
and 
.
-
4.15.3
and 
.
-
4.15.4
and 
.
-
4.15.5
and 
.
-
4.15.6
and 
.
-
4.15.7 Conformal mapping of sine and inverse sine.
-
4.15.8
.
-
4.15.9
.
-
4.15.10
.
-
4.15.11
.
-
4.15.12
.
-
4.15.13
.
-
4.16.1 Quadrants for the angle
.
-
4.23.1 Branch cuts for the inverse trigonometric functions.
-
4.29.1
and 
.
-
4.29.2
and 
.
-
4.29.3
and 
.
-
4.29.4
and 
.
-
4.29.5
and 
.
-
4.29.6
and 
.
-
4.37.1 Branch cuts for the inverse hyperbolic functions.
-
4.42.1 Planar right triangle.
-
4.42.2 Planar triangle.
-
4.42.3 Spherical triangle.
-
5 Gamma Function
-
5.3.1
, 
.
-
5.3.2
.
-
5.3.3
.
-
5.3.4
.
-
5.3.5
.
-
5.3.6
.
-
5.9.1 Contour for Hankelβs loop integral.
-
5.12.1 Contour for first loop integral for the beta function.
-
5.12.2 Contour for second loop integral for the beta function.
-
5.12.3 Contour for Pochhammerβs integral.
-
6 Exponential, Logarithmic, Sine, and Cosine Integrals
-
6.3.1
, 
, 
.
-
6.3.2
, 
.
-
6.3.3
, 
, 
.
-
6.16.1 Gibbs phenomenon.
-
6.16.2
, 
, 
.
-
7 Error Functions, Dawsonβs and Fresnel Integrals
-
7.3.1
, 
, 
.
-
7.3.2
, 
.
-
7.3.3
, 
, 
.
-
7.3.4
, 
.
-
7.3.5
, , 
.
-
7.3.6
, , .
-
7.18.1 Repeated integrals of the scaled complementary error function.
-
7.19.1 Voigt function
,
, 2.5, 5, 10.
-
7.19.2 Voigt function
,
, 2.5, 5, 10.
-
7.20.1 Cornuβs spiral.
-
8 Incomplete Gamma and Related Functions
-
8.3.1
, 
= 0.25, 1, 2, 2.5, 3.
-
8.3.2
,
= 0.25, 0.5, 0.75, 1.
-
8.3.3 ,
= 1, 2, 2.5, 3.
-
8.3.4
(= 
),
= 0.25, 0.5, 0.75, 1, 2.
-
8.3.5
(= 
),
= 0.25, 0.5, 1, 2.
-
8.3.6 (= ),
,
.
-
8.3.7 (= ),
,
.
-
8.3.8
,
, .
-
8.3.9
,
, .
-
8.3.10
,
, .
-
8.3.11
,
, .
-
8.3.12
,
, .
-
8.3.13
,
, .
-
8.3.14
,
, .
-
8.3.15
,
, .
-
8.3.16
,
, .
-
8.19.1
,
, 
.
-
8.19.2
,
, .
-
8.19.3
,
, .
-
8.19.4
,
, .
-
8.19.5
,
, .
-
9 Airy and Related Functions
-
9.3.1
, 
, 
.
-
9.3.2
, 
, 
.
-
9.3.3
.
-
9.3.4
.
-
9.3.5
.
-
9.3.6
.
-
9.12.1
, 
.
-
9.12.2
, 
.
-
9.13.1 Paths
, 
,
, 
.
-
9.13.2 Paths
, 
,
.
-
10 Bessel Functions
-
10.3.1 ,
, 
, 
, 
.
-
10.3.2
, 
, 
, .
-
10.3.3
, 
, 
, .
-
10.3.4
, 
, .
-
10.3.5
, , 
.
-
10.3.6
, 
, .
-
10.3.7
, , .
-
10.3.8
, 
, .
-
10.3.9
, 
, .
-
10.3.10
, 
, 
.
-
10.3.11
, , .
-
10.3.12
, , .
-
10.3.13
, , .
-
10.3.14
, 
, .
-
10.3.15
, , .
-
10.3.16
, , .
-
10.3.17
, 
, 
.
-
10.3.18
, 
, .
-
10.3.19
, 
, .
-
10.20.1
-plane.
-
10.20.2
-plane.
-
10.20.3 Domain
.
-
10.21.1 Zeros of
in 
.
-
10.21.2 Zeros of
in .
-
10.21.3 Zeros of in .
-
10.21.4 Zeros of in .
-
10.21.5 Zeros of in .
-
10.21.6 Zeros of in .
-
10.23.1 Grafβs and Gegenbauerβs addition theorems.
-
10.26.1
, 
, 
, 
, .
-
10.26.2
, 
, 
, 
, .
-
10.26.3
, 
, 
.
-
10.26.4
, 
, .
-
10.26.5
, , .
-
10.26.6
, 
, .
-
10.26.7
, 
, 
.
-
10.26.8
, 
, .
-
10.26.9
, 
, .
-
10.26.10 , .
-
10.41.1 -plane.
-
10.41.2
-plane.
-
10.48.1
, 
, 
.
-
10.48.2
, , 
.
-
10.48.3
, 
,
,
.
-
10.48.4
, 
,
,
.
-
10.48.5
, 
, 
,
.
-
10.48.6
, .
-
10.48.7
, 
, 
,
.
-
10.62.1
, 
, 
,
, .
-
10.62.2
, 
, 
,
, .
-
10.62.3
,
,
, .
-
10.62.4
, 
,
, .
-
11 Struve and Related Functions
-
11.3.1
, , 
.
-
11.3.2
, 
, .
-
11.3.3 , ,
.
-
11.3.4 , ,
.
-
11.3.5 , ,
.
-
11.3.6 , , .
-
11.3.7
, , .
-
11.3.8
, , .
-
11.3.9
, , .
-
11.3.10
, , .
-
11.3.11
, 
, .
-
11.3.12
, , .
-
11.3.13
, 
, .
-
11.3.14
, 
, .
-
11.3.15 ,
, .
-
11.3.16 , , .
-
11.3.17 ,
, .
-
11.3.18 , , .
-
11.3.19
, , .
-
11.3.20
, , .
-
11.10.1
, , 
.
-
11.10.2
, , .
-
11.10.3 , , .
-
11.10.4 , , .
-
12 Parabolic Cylinder Functions
-
12.3.1
, = 0.5, 2, 3.5, 5, 8.
-
12.3.2
, = 0.5, 2, 3.5, 5, 8.
-
12.3.3 , = β0.5, β2, β3.5, β5.
-
12.3.4 , = β0.5, β2, β3.5, β5.
-
12.3.5
, 
, 
, 
.
-
12.3.6
, 
, 
, .
-
12.3.7 ,
, 
.
-
12.3.8 , , .
-
12.3.9
, 
, 
.
-
12.3.10
, 
, 
.
-
12.14.1
, 
, 
, .
-
12.14.2
, 
, 
, .
-
12.14.3
, 
, 
, .
-
12.14.4
, 
, .
-
13 Confluent Hypergeometric Functions
-
13.4.1 Contour of integration in (13.4.11).
-
13.7.1 Regions for error bounds of
.
-
14 Legendre and Related Functions
-
14.4.1
, 
.
-
14.4.2
, .
-
14.4.3
, .
-
14.4.4
, .
-
14.4.5
, .
-
14.4.6
, .
-
14.4.7
, 
.
-
14.4.8
, .
-
14.4.9
, .
-
14.4.10
, .
-
14.4.11
, .
-
14.4.12
, .
-
14.4.13 ,
.
-
14.4.14 , .
-
14.4.15 ,
.
-
14.4.16 ,
.
-
14.4.17
, .
-
14.4.18
, .
-
14.4.19
, .
-
14.4.20
, .
-
14.4.21
, .
-
14.4.22
, .
-
14.4.23
, .
-
14.4.24
, 
.
-
14.4.25
, .
-
14.4.26
, .
-
14.4.27
, .
-
14.4.28
, .
-
14.4.29 ,
, 
.
-
14.4.30 , , .
-
14.4.31 ,
, .
-
14.4.32 , , .
-
14.20.1
, 
.
-
14.20.2
, 
.
-
14.20.3
, .
-
14.20.4
, 
.
-
14.20.5
, .
-
14.20.6
, .
-
14.20.7
.
-
14.20.8
, .
-
14.22.1
, , .
-
14.22.2
,
, .
-
14.22.3
,
, .
-
14.22.4
,
, .
-
15 Hypergeometric Function
-
15.3.1
.
-
15.3.2
.
-
15.3.3
.
-
15.3.4
.
-
15.3.5
.
-
15.3.6
.
-
15.3.7
.
-
15.6.1 Contour of integration in (15.6.5).
-
16 Generalized Hypergeometric Functions & Meijer G-Function
-
16.17.1 Integration path
for the Meijer 
-function.
-
18 Orthogonal Polynomials
-
18.4.1 Jacobi polynomials
, 
.
-
18.4.2 Jacobi polynomials
, 
.
-
18.4.3 Chebyshev polynomials
, .
-
18.4.4 Legendre polynomials
, .
-
18.4.5 Laguerre polynomials
, .
-
18.4.6 Laguerre polynomials
,
.
-
18.4.7 Monic Hermite polynomials
, .
-
18.4.8 Laguerre polynomials ,
, .
-
18.4.9 Laguerre polynomials
,
, .
-
18.21.1 Askey scheme.
-
18.39.1 Graphs of the first and fourth excited state eigenfunctions of the harmonic oscillator, for
, of (18.39.13),
in 
, 
and those of the rational potential of (18.39.19), in 
, 
.
Both sets satisfy the Sturm oscillation theorem.
-
18.39.2 CoulombβPollaczek weight functions,
![x\in[-1,1]](.././lof/m836.png)
, (18.39.50) for 
, 
, and 
. For 
the weight function, red curve,
has an essential singularity at 
, as all derivatives vanish as 
; the green curve is 
,
to be compared with its histogram approximation in Β§18.40(ii). For 
the weight function, blue curve, is non-zero at ,
but this point is also an essential singularity as the discrete parts of the weight function of (18.39.51) accumulate as 
, 
.
-
18.40.1 Histogram approximations to the Repulsive CoulombβPollaczek, RCP,
weight function integrated over
, see Figure 18.39.2 for an exact result,
for , shown for 
and 
.
-
18.40.2 Derivative Rule inversions for
carried out via Lagrange
and PWCF interpolations. Shown are the absolute errors of approximation (18.40.8) at the points
, 
for 
. For the derivative rule
Lagrange interpolation (red points) gives
digits in the central region, while PWCF interpolation (blue points) gives 
.
-
19 Elliptic Integrals
-
19.3.1
, 
, 
.
-
19.3.2
, 
, .
-
19.3.3
, 
, 
.
-
19.3.4
, , .
-
19.3.5
, 
, 
.
-
19.3.6
, 
, 
.
-
19.3.7 ,
,
.
-
19.3.8 , ,
.
-
19.3.9
, ,
.
-
19.3.10
, ,
.
-
19.3.11
, ,
.
-
19.3.12
, ,
.
-
19.17.1
, 
, 
.
-
19.17.2
, , .
-
19.17.3
, 
, 
.
-
19.17.4
, , .
-
19.17.5
, , .
-
19.17.6
, , .
-
19.17.7
, 
, 
.
-
19.17.8
, 
, 
.
-
20 Theta Functions
-
20.2.1 Fundamental parallelogram.
-
20.3.1
,
,
.
-
20.3.2
,
,
= 0.05, 0.5, 0.7, 0.9.
-
20.3.3
,
,
= 0.05, 0.5, 0.7, 0.9.
-
20.3.4
,
,
= 0.05, 0.5, 0.7, 0.9.
-
20.3.5
,
,
= 0.05, 0.5, 0.7, 0.9.
-
20.3.6
,
,
= 0, 0.4, 5, 10, 40.
-
20.3.7
,
,
= 0, 0.4, 5, 10, 40.
-
20.3.8
,
,
= 0, 0.4, 5, 10, 40.
-
20.3.9
,
,
= 0, 0.4, 5, 10, 40.
-
20.3.10 ,
,
.
-
20.3.11 ,
,
.
-
20.3.12 ,
,
.
-
20.3.13 ,
,
.
-
20.3.14
,
,
.
-
20.3.15
,
,
.
-
20.3.16
,
,
.
-
20.3.17
,
,
.
-
20.3.18
,
,
.
-
20.3.19
,
,
.
-
20.3.20
,
,
.
-
20.3.21
,
,
.
-
21 Multidimensional Theta Functions
-
21.4.1
parametrized by (21.4.1).
-
21.4.2
,
, 
.
-
21.4.3
,
, 
.
-
21.4.4
,
, 
.
-
21.4.5
,
, 
.
-
21.7.1 A basis of cycles for a genus 2 surface.
-
21.9.1 Two-dimensional periodic waves in a shallow water wave tank.
-
21.9.2 Contour plot of a two-phase solution of Equation (21.9.3).
-
22 Jacobian Elliptic Functions
-
22.3.1
, 
, 
,
,
,
.
-
22.3.2 , , ,
,
,
.
-
22.3.3 , , ,
,
,
.
-
22.3.4 , , ,
,
,
.
-
22.3.5
, 
, 
,
,
,
.
-
22.3.6 , , ,
,
,
.
-
22.3.7 , , ,
,
,
.
-
22.3.8 , , ,
,
,
.
-
22.3.9
, 
, 
,
,
,
.
-
22.3.10 , , ,
,
,
.
-
22.3.11 , , ,
,
,
.
-
22.3.12 , , ,
,
,
.
-
22.3.13
for
,
to 20,
.
-
22.3.14
for ,
to 20,
.
-
22.3.15
for ,
to 20,
.
-
22.3.16
, , , 
.
-
22.3.17
, , , .
-
22.3.18
, , , .
-
22.3.19
, , , .
-
22.3.20
, , , .
-
22.3.21
, , .
-
22.3.22
.
-
22.3.23
.
-
22.3.24 .
-
22.3.25
.
-
22.3.26 Density plot of
.
-
22.3.27 Density plot of
.
-
22.3.28 Density plot of
.
-
22.3.29 Density plot of
.
-
22.4.1 Poles, zeros of the principal Jacobian elliptic functions.
-
22.4.2 Fundamental unit cell.
-
22.16.1
, 
, 
.
-
22.16.2
, , .
-
22.16.3
, , .
-
22.19.1 , ,
.
-
23 Weierstrass Elliptic and Modular Functions
-
23.4.1
, 
,
= 0.1, 0.2, 0.5, 0.8.
-
23.4.2
, ,
= 0.1, 0.2, 0.5, 0.8.
-
23.4.3
, ,
= 0.1, 0.2, 0.5, 0.8.
-
23.4.4
, ,
= 0.1, 0.2, 0.5, 0.8.
-
23.4.5
, ,
= 0.1, 0.2, 0.5, 0.8.
-
23.4.6
, ,
= 0.1, 0.2, 0.5, 0.8.
-
23.4.7
, , 
= 0.2, 0.8, 0.95, 0.99.
-
23.4.8
,
,
,
.
-
23.4.9
, 
, 
-
23.4.10
, , 
-
23.4.11
, , 
-
23.4.12
, 
, 
.
-
23.5.1 Rhombic lattice.
.
-
23.5.2 Equianharmonic lattice.
,
.
-
23.16.1
, 
, 
,
.
-
23.16.2
, 
, 
.
-
23.16.3
, 
, 
.
-
24 Bernoulli and Euler Polynomials
-
24.3.1 Bernoulli polynomials
, 
.
-
24.3.2 Euler polynomials
, .
-
25 Zeta and Related Functions
-
25.3.1
, 
, .
-
25.3.2 , ,
.
-
25.3.3
, , 
.
-
25.3.4
, 
.
-
25.3.5 ,
.
-
25.3.6 ,
.
-
25.11.1
, = 0.3, 0.5, 0.8, 1, .
-
25.11.2 ,
, 
.
-
25.12.1
, 
-
25.12.2
,
,
.
-
26 Combinatorial Analysis
- 26.9.1 Ferrers graph of the partition
.
-
26.9.2 The partition
represented as a lattice path.
-
26.12.1 A plane partition of 75.
-
28 Mathieu Functions and Hillβs Equation
-
28.2.1 Eigenvalues
,
of Mathieuβs equation.
-
28.3.1
, 
, 
.
-
28.3.2
, , .
-
28.3.3
, , .
-
28.3.4
, , .
-
28.3.5
, , .
-
28.3.6
, , .
-
28.3.7
, , 
.
-
28.3.8
, , .
-
28.3.9
, 
, 
.
-
28.3.10
, , .
-
28.3.11
, , .
-
28.3.12
, , .
-
28.3.13
, , .
-
28.5.1
, 
, .
-
28.5.2
, 
, .
-
28.5.3
, 
, .
-
28.5.4
, 
, .
-
28.5.5
, 
, .
-
28.5.6
, 
, .
-
28.7.1 Branch point of the eigenvalues
and
:
.
-
28.13.1
, 
;
,
(βs), (
βs).
-
28.13.2 ,
, .
-
28.13.3
, 
, .
-
28.13.4
, 
, .
-
28.13.5
, 
, 
.
-
28.17.1 Stability chart for eigenvalues of Mathieuβs equation (28.2.1).
-
28.21.1
, 
, .
-
28.21.2
, , .
-
28.21.3
, .
-
28.21.4
, 
, .
-
28.21.5
, , .
-
28.21.6
, , .
-
29 LamΓ© Functions
-
29.2.1 Singularities of LamΓ©βs equation.
-
29.4.1
, 
, 
.
-
29.4.2
.
-
29.4.3
, 
.
-
29.4.4
, 
, .
-
29.4.5
, 
, .
-
29.4.6
.
-
29.4.7
-
29.4.8
, 
.
-
29.4.9
.
-
29.4.10
.
-
29.4.11
.
-
29.4.12
.
-
29.4.13
,
, 
.
-
29.4.14
,
, 
.
-
29.4.15
,
, .
-
29.4.16
,
, .
-
29.4.17
,
, .
-
29.4.18
,
, .
-
29.4.19
,
, .
-
29.4.20
,
, .
-
29.4.21
,
, .
-
29.4.22
,
, .
-
29.4.23
,
, .
-
29.4.24
,
, .
-
29.4.25
.
-
29.4.26
.
-
29.4.27
.
-
29.4.28
.
-
29.4.29
.
-
29.4.30
.
-
29.4.31
.
-
29.4.32
.
-
29.13.1
, 
.
-
29.13.2
, 
.
-
29.13.3
, 
.
-
29.13.4
, 
.
-
29.13.5
for
, .
.
-
29.13.6
for
, .
.
-
29.13.7
for
, .
.
-
29.13.8
for
, .
.
-
29.13.9
for
, .
.
-
29.13.10
for
, .
.
-
29.13.11
for
, .
.
-
29.13.12
for
, .
.
-
29.13.13
for
, 
.
.
-
29.13.14
for
, .
.
-
29.13.15
for
, .
.
-
29.13.16
for
, .
.
-
29.13.17
for
, .
.
-
29.13.18
for
, .
.
-
29.13.19
for
, .
.
-
29.13.20
for
, .
.
-
29.13.21
for ,
. ,
.
-
29.13.22
for ,
. 
.
-
29.13.23
for ,
. ,
.
-
30 Spheroidal Wave Functions
-
30.7.1
,
, 
.
-
30.7.2
,
, .
-
30.7.3
,
, 
.
-
30.7.4
,
, 
.
-
30.7.5
, , .
-
30.7.6
, , .
-
30.7.7
, , .
-
30.7.8
, , .
-
30.7.9
,
, 
.
-
30.7.10
,
, .
-
30.7.11
, , 
.
-
30.7.12
, , .
-
30.7.13
, for ,
.
-
30.7.14
, , .
-
30.7.15
.
-
30.7.16
,
, 
.
-
30.7.17
,
, .
-
30.7.18
,
, .
-
30.7.19
,
, .
-
30.7.20
,
, .
-
30.7.21
,
, .
-
30.11.1
,
, 
.
-
30.11.2
,
, 
.
-
30.11.3
,
, .
-
30.11.4
,
, .
-
32 PainlevΓ© Transcendents
-
32.3.1
, 
, 
, 0.75, 1, 1.25.
-
32.3.2 ,
, 
, β0.25, 0, 1, 2.
-
32.3.3 ,
, 
, 1.85185β5.
-
32.3.4 ,
, 
, β0.45142β8.
-
32.3.5 ,
, 
, .
-
32.3.6 , ,
, 1.001.
-
32.3.7
, 
, 
, 0.33554β692.
-
32.3.8
, , 
.
-
32.3.9
, , 
, 0.38737.
-
32.3.10
, , 
, 0.24499β3.
-
33 Coulomb Functions
-
33.3.1
, 
,
, 
.
-
33.3.2 , ,
,
.
-
33.3.3 ,
,
,
.
-
33.3.4 ,
,
, .
-
33.3.5 , ,
,
, 
.
-
33.3.6 , ,
,
, .
-
33.3.7
,
, 
.
-
33.3.8
,
, 
.
-
33.15.1
,
.
-
33.15.2 ,
.
-
33.15.3 ,
.
-
33.15.4 ,
.
-
33.15.5 ,
.
-
33.15.6
, 
.
-
33.15.7
, .
-
33.15.8 ,
.
-
33.15.9 , .
-
33.15.10
, 
.
-
33.15.11
, .
-
34 3j, 6j, 9j Symbols
-
34.2.1 Angular momenta
and projective quantum numbers.
-
34.4.1 Tetrahedron corresponding to
symbol.
-
36 Integrals with Coalescing Saddles
-
36.3.1 Modulus of Pearcey integral
.
-
36.3.2 Modulus of swallowtail canonical integral function
.
-
36.3.3 Modulus of swallowtail canonical integral function
.
-
36.3.4 Modulus of swallowtail canonical integral function
.
-
36.3.5 Modulus of swallowtail canonical integral function
.
-
36.3.6 Modulus of elliptic umbilic canonical integral function
.
-
36.3.7 Modulus of elliptic umbilic canonical integral function
.
-
36.3.8 Modulus of elliptic umbilic canonical integral function
.
-
36.3.9 Modulus of hyperbolic umbilic canonical integral function
.
-
36.3.10 Modulus of hyperbolic umbilic canonical integral function
.
-
36.3.11 Modulus of hyperbolic umbilic canonical integral function
.
-
36.3.12 Modulus of hyperbolic umbilic canonical integral function
.
-
36.3.13 Phase of Pearcey integral
.
-
36.3.14 Density plots of phase of swallowtail canonical integrals.
-
36.3.15 Phase of elliptic umbilic canonical integral
.
-
36.3.16 Phase of elliptic umbilic canonical integral
.
-
36.3.17 Phase of elliptic umbilic canonical integral
.
-
36.3.18 Phase of hyperbolic umbilic canonical integral
.
-
36.3.19 Phase of hyperbolic umbilic canonical integral
.
-
36.3.20 Phase of hyperbolic umbilic canonical integral
.
-
36.3.21 Phase of hyperbolic umbilic canonical integral
.
-
36.4.1 Bifurcation set of cusp catastrophe.
-
36.4.2 Bifurcation set of swallowtail catastrophe.
-
36.4.3 Bifurcation set of elliptic umbilic catastrophe.
-
36.4.4 Bifurcation set of hyperbolic umbilic catastrophe.
-
36.5.1 Cusp catastrophe.
-
36.5.2 Swallowtail catastrophe with
.
-
36.5.3 Swallowtail catastrophe with
.
-
36.5.4 Swallowtail catastrophe with
.
-
36.5.5 Elliptic
umbilic catastrophe with
.
-
36.5.6 Hyperbolic
umbilic catastrophe with .
-
36.5.7 Sheets of the Stokes surface for the swallowtail catastrophe
(colored and with mesh) and the bifurcation set (gray).
-
36.5.8 Sheets of the Stokes surface for the elliptic umbilic catastrophe.
-
36.5.9 Sheets of the Stokes surface for the hyperbolic umbilic catastrophe
-
36.13.1 Kelvinβs ship wave pattern.
(i)
-plane
A B C
D
E
F
βββ (i)
D
E
F
β1
(i) 
and 
(ii) 
(iii) 
and 
(iv) 
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
Case (i)
Case (ii)
Case (iii)
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(a1) (b1) (c1)
(a2) (b2) (c2)
(a3) (b3) (c3)
(a) 
(b) 
(c) 
(a) Density plot. (b) 3D plot.
(a) Density plot. (b) 3D plot.
(a) Density plot. (b) 3D plot.
(a) Density plot. (b) 3D plot.
(a) Density plot. (b) 3D plot.
(a) Density plot. (b) 3D plot.
(a) Density plot. (b) 3D plot.
(a) Density plot. (b) 3D plot.
(a) Density plot. (b) 3D plot.
(a) Density plot. (b) 3D plot.
(a) Density plot. (b) 3D plot.
(a) Density plot. (b) 3D plot.
(a) Contour plot, at intervals of 
.
(b) Density plot.
(a) 
.
(b) 
.
(c) 
.
(d) 
.
(a) Contour plot.
(b) Density plot.
(a) Contour plot.
(b) Density plot.
(a) Contour plot.
(b) Density plot.
(a) Contour plot.
(b) Density plot.
(a) Contour plot.
(b) Density plot.
(a) Contour plot.
(b) Density plot.
(a) Contour plot.
(b) Density plot.
