List of Figures 1 Algebraic and Analytic Methods 1.4.1 Piecewise continuous function on [a,b). 1.4.2 Convex function f(x). 1.6.1 Right-hand rule for cross products. 1.10.1 Domain D. 2 Asymptotic Approximations 2.10.1 Contour 𝒞. 2.11.1 |erfc(50c(θ))|. 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 J0(x) 4 Elementary Functions 4.2.1 Branch cut for lnz and zα. 4.3.1 lnx and ex. (i) z-plane (ii) w-plane A B C C¯ D D¯ E E¯ F z 0 r r+iπ r−iπ iπ −iπ −r+iπ −r−iπ −r w 1 er −er+i0 −er−i0 −1+i0 −1−i0 −e−r+i0 −e−r−i0 e−r 4.3.2 Conformal mapping of exponential and logarithm. 4.3.3 ln(x+iy). 4.3.4 ex+iy. 4.13.1 Branches W0(x), W±1(x∓0i) of the Lambert W-function. 4.13.2 The W(z) function on the first 5 Riemann sheets. 4.15.1 sinx and cosx. 4.15.2 Arcsinx and Arccosx. 4.15.3 tanx and cotx. 4.15.4 arctanx and arccotx. 4.15.5 cscx and secx. 4.15.6 arccscx and arcsecx. (i) z-plane (ii) w-plane A B C C¯ D D¯ E E¯ F z 0 12π 12π+ir 12π−ir ir −ir −12π+ir −12π−ir −12π w 0 1 coshr+i0 coshr−i0 isinhr −isinhr −coshr+i0 −coshr−i0 −1 4.15.7 Conformal mapping of sine and inverse sine. 4.15.8 sin(x+iy). 4.15.9 arcsin(x+iy). 4.15.10 tan(x+iy). 4.15.11 arctan(x+iy). 4.15.12 csc(x+iy). 4.15.13 arccsc(x+iy). 4.16.1 Quadrants for the angle θ. (i) arcsinz and arccosz (ii) arctanz (iii) arccscz and arcsecz (iv) arccotz 4.23.1 Branch cuts for the inverse trigonometric functions. 4.29.1 sinhx and coshx. 4.29.2 arcsinhx and arccoshx. 4.29.3 tanhx and cothx. 4.29.4 arctanhx and arccothx. 4.29.5 cschx and sechx. 4.29.6 arccschx and arcsechx. (i) arcsinhz (ii) arccoshz (iii) arctanhz (iv) arccschz (v) arcsechz (vi) arccothz 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 Γ(x), 1/Γ(x). 5.3.2 lnΓ(x). 5.3.3 ψ(x). 5.3.4 |Γ(x+iy)|. 5.3.5 1/|Γ(x+iy)|. 5.3.6 |ψ(x+iy)|. 5.9.1 Contour for Hankel’s 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 Pochhammer’s integral. 6 Exponential, Logarithmic, Sine, and Cosine Integrals 6.3.1 E1(x), Ei(x), 0<x≤2. 6.3.2 Si(x),Ci(x), 0≤x≤15. 6.3.3 |E1(x+iy)|, −4≤x≤4, −4≤y≤4. 6.16.1 Gibbs phenomenon. 6.16.2 li(x), π(x), x=10,20,…,1000. 7 Error Functions, Dawson’s and Fresnel Integrals 7.3.1 erfcx, erfc(10x), −3≤x≤3. 7.3.2 F(x), −3.5≤x≤3.5. 7.3.3 C(x), S(x), 0≤x≤4. 7.3.4 |ℱ(x)|2, −8≤x≤8. 7.3.5 |erf(x+iy)|, −3≤x≤3, −3≤y≤3. 7.3.6 |erfc(x+iy)|, −3≤x≤3, −3≤y≤3. 7.18.1 Repeated integrals of the scaled complementary error function. 7.19.1 Voigt function 𝖴(x,t), t=0.1, 2.5, 5, 10. 7.19.2 Voigt function 𝖵(x,t), t=0.1, 2.5, 5, 10. 7.20.1 Cornu’s spiral. 8 Incomplete Gamma and Related Functions 8.3.1 Γ(a,x), a = 0.25, 1, 2, 2.5, 3. 8.3.2 γ(a,x), a = 0.25, 0.5, 0.75, 1. 8.3.3 γ(a,x), a = 1, 2, 2.5, 3. 8.3.4 γ∗(a,x) (= x−aP(a,x)), a = 0.25, 0.5, 0.75, 1, 2. 8.3.5 x−a−γ∗(a,x) (= x−aQ(a,x)), a = 0.25, 0.5, 1, 2. 8.3.6 γ∗(a,x) (= x−aP(a,x)), −4≤x≤4, −5≤a≤4. 8.3.7 x−a−γ∗(a,x) (= x−aQ(a,x)), 0≤x≤4, −5≤a≤5. 8.3.8 Γ(0.25,x+iy), −3≤x≤3, −3≤y≤3. 8.3.9 γ(0.25,x+iy), −3≤x≤3, −3≤y≤3. 8.3.10 γ∗(0.25,x+iy), −3≤x≤3, −3≤y≤3. 8.3.11 Γ(1,x+iy), −3≤x≤3, −3≤y≤3. 8.3.12 γ(1,x+iy), −3≤x≤3, −3≤y≤3. 8.3.13 γ∗(1,x+iy), −3≤x≤3, −3≤y≤3. 8.3.14 Γ(2.5,x+iy), −2.2≤x≤3, −3≤y≤3. 8.3.15 γ(2.5,x+iy), −2.2≤x≤3, −3≤y≤3. 8.3.16 γ∗(2.5,x+iy), −3≤x≤3, −3≤y≤3. 8.19.1 Ep(x), 0≤x≤3, 0≤p≤8. 8.19.2 E12(x+iy), −4≤x≤4, −4≤y≤4. 8.19.3 E1(x+iy), −4≤x≤4, −4≤y≤4. 8.19.4 E32(x+iy), −3≤x≤3, −3≤y≤3. 8.19.5 E2(x+iy), −3≤x≤3, −3≤y≤3. 9 Airy and Related Functions 9.3.1 Ai(x), Bi(x), M(x). 9.3.2 Ai′(x), Bi′(x), N(x). 9.3.3 Ai(x+iy). 9.3.4 Bi(x+iy). 9.3.5 Ai′(x+iy). 9.3.6 Bi′(x+iy). 9.12.1 Gi(x), Gi′(x). 9.12.2 Hi(x), Hi′(x). 9.13.1 Paths ℒ0, ℒ1, ℒ2, ℒ3. 9.13.2 Paths ℐ1, ℐ2, ℐ3. 10 Bessel Functions 10.3.1 J0(x), Y0(x), J1(x), Y1(x), 0≤x≤10. 10.3.2 J5(x), Y5(x), M5(x), 0≤x≤15. 10.3.3 J5′(x), Y5′(x), N5(x), 0≤x≤15. 10.3.4 θ5(x), ϕ5(x), 0≤x≤15. 10.3.5 Jν(x), 0≤x≤10, 0≤ν≤5. 10.3.6 Yν(x), 0<x≤10, 0≤ν≤5. 10.3.7 Jν′(x), 0≤x≤10, 0≤ν≤5. 10.3.8 Yν′(x), 0.2≤x≤10, 0≤ν≤5. 10.3.9 J0(x+iy), −10≤x≤10, −4≤y≤4. 10.3.10 H0(1)(x+iy), −10≤x≤5, −2.8≤y≤4. 10.3.11 J1(x+iy), −10≤x≤10, −4≤y≤4. 10.3.12 H1(1)(x+iy), −10≤x≤5, −2.8≤y≤4. 10.3.13 J5(x+iy), −10≤x≤10, −4≤y≤4. 10.3.14 H5(1)(x+iy), −20≤x≤10, −4≤y≤4. 10.3.15 J5.5(x+iy), −10≤x≤10, −4≤y≤4. 10.3.16 H5.5(1)(x+iy), −20≤x≤10, −4≤y≤4. 10.3.17 J~1/2(x), Y~1/2(x), 0.01≤x≤10. 10.3.18 J~1(x), Y~1(x), 0.01≤x≤10. 10.3.19 J~5(x), Y~5(x), 0.01≤x≤10. 10.20.1 z-plane. 10.20.2 ζ-plane. 10.20.3 Domain 𝐊. 10.21.1 Zeros of Yn(nz) in |phz|≤π. 10.21.2 Zeros of Hn(1)(nz) in |phz|≤π. 10.21.3 Zeros of Yn(nz) in |phz|≤π. 10.21.4 Zeros of Hn(1)(nz) in |phz|≤π. 10.21.5 Zeros of Yn(nz) in |phz|≤π. 10.21.6 Zeros of Hn(1)(nz) in |phz|≤π. 10.23.1 Graf’s and Gegenbauer’s addition theorems. 10.26.1 I0(x), I1(x), K0(x), K1(x), 0≤x≤3. 10.26.2 e−xI0(x), e−xI1(x), exK0(x), exK1(x), 0≤x≤10. 10.26.3 Iν(x), 0≤x≤5, 0≤ν≤4. 10.26.4 Kν(x), 0.1≤x≤5, 0≤ν≤4. 10.26.5 Iν′(x), 0≤x≤5, 0≤ν≤4. 10.26.6 Kν′(x), 0.3≤x≤5, 0≤ν≤4. 10.26.7 I~1/2(x), K~1/2(x), 0.01≤x≤3. 10.26.8 I~1(x), K~1(x), 0.01≤x≤3. 10.26.9 I~5(x), K~5(x), 0.01≤x≤3. 10.26.10 K~5(x), 0.01≤x≤3. 10.41.1 z-plane. 10.41.2 η-plane. 10.48.1 𝗃n(x), n=0(1)4, 0≤x≤12. 10.48.2 𝗒n(x), n=0(1)4, 0<x≤12. 10.48.3 𝗃5(x), 𝗒5(x), 𝗃52(x)+𝗒52(x), 0≤x≤12. 10.48.4 𝗃5′(x), 𝗒5′(x), 𝗃5′2(x)+𝗒5′2(x), 0≤x≤12. 10.48.5 𝗂0(1)(x), 𝗂0(2)(x), 𝗄0(x), 0≤x≤4. 10.48.6 𝗂1(1)(x),𝗂1(2)(x),𝗄1(x), 0≤x≤4. 10.48.7 𝗂5(1)(x), 𝗂5(2)(x), 𝗄5(x), 0≤x≤8. 10.62.1 berx, beix, ber′x, bei′x, 0≤x≤8. 10.62.2 kerx, keix, ker′x, kei′x, 0≤x≤8. 10.62.3 e−x/2berx, e−x/2beix, e−x/2M(x), 0≤x≤8. 10.62.4 ex/2kerx, ex/2keix, ex/2N(x), 0≤x≤8. 11 Struve and Related Functions 11.3.1 𝐇ν(x), 0≤x≤12, ν=0,12,1,32,2,3. 11.3.2 𝐊ν(x), 0<x≤16, ν=0,12,1,32,2,3. 11.3.3 𝐇ν(x), 0≤x≤12, ν=−3,−2,−32,−1,−12. 11.3.4 𝐊ν(x), 0<x≤16, ν=−4,−3,−2,−1,0. 11.3.5 𝐇ν(x), 0≤x≤8, −4≤ν≤4. 11.3.6 𝐊ν(x), 0≤x≤8, −4≤ν≤4. 11.3.7 |𝐇0(x+iy)|, −8≤x≤8, −3≤y≤3. 11.3.8 |𝐊0(x+iy)|, −8≤x≤8, −3≤y≤3. 11.3.9 |𝐇12(x+iy)|, −8≤x≤8, −3≤y≤3. 11.3.10 |𝐊12(x+iy)|, −8≤x≤8, −3≤y≤3. 11.3.11 |𝐇1(x+iy)|, −8.5≤x≤8.5, −3≤y≤3. 11.3.12 |𝐊1(x+iy)|, −8≤x≤8, −3≤y≤3. 11.3.13 𝐋ν(x), 0≤x<4.38, ν=0,12,1,32,2,3. 11.3.14 𝐌ν(x), 0≤x≤16, ν=0,12,1,32,2,3. 11.3.15 𝐋ν(x), 0≤x<4.25, ν=−3,−2,−32,−1,−12. 11.3.16 𝐌ν(x), 0<x≤16, ν=−3,−2,−32,−1,−12. 11.3.17 𝐋ν(x), 0≤x≤5.6, −4≤ν≤4. 11.3.18 𝐌ν(x), 0≤x≤8, −4≤ν≤4. 11.3.19 |𝐌−12(x+iy)|, −3≤x≤3, −3≤y≤3. 11.3.20 |𝐌12(x+iy)|, −3≤x≤3, −3≤y≤3. 11.10.1 𝐉ν(x), −8≤x≤8, ν=0,12,1,32. 11.10.2 𝐄ν(x), −8≤x≤8, ν=0,12,1,32. 11.10.3 𝐉ν(x), −10≤x≤10, 0≤ν≤5. 11.10.4 𝐄ν(x), −10≤x≤10, 0≤ν≤5. 12 Parabolic Cylinder Functions 12.3.1 U(a,x), a = 0.5, 2, 3.5, 5, 8. 12.3.2 V(a,x), a = 0.5, 2, 3.5, 5, 8. 12.3.3 U(a,x), a = −0.5, −2, −3.5, −5. 12.3.4 V(a,x), a = −0.5, −2, −3.5, −5. 12.3.5 U(−8,x), U¯(−8,x), F(−8,x), −42≤x≤42. 12.3.6 U′(−8,x), U¯′(−8,x), G(−8,x), −42≤x≤42. 12.3.7 U(a,x), −2.5≤a≤2.5, −2.5≤x≤2.5. 12.3.8 V(a,x), −2.5≤a≤2.5, −2.5≤x≤2.5. 12.3.9 U(3.5,x+iy), −3.6≤x≤5, −5≤y≤5. 12.3.10 U(−3.5,x+iy), −5≤x≤5, −3.5≤y≤3.5. 12.14.1 k−1/2W(3,x), k1/2W(3,−x), F~(3,x), 0≤x≤8. 12.14.2 k−1/2W′(3,x), k1/2W′(3,−x), G~(3,x), 0≤x≤8. 12.14.3 k−1/2W(−3,x), k1/2W(−3,−x), F~(−3,x), 0≤x≤8. 12.14.4 k−1/2W′(−3,x),k1/2W′(−3,−x), G~(−3,x), 0≤x≤8. 13 Confluent Hypergeometric Functions 13.4.1 Contour of integration in (13.4.11). 13.7.1 Regions for error bounds of U. 14 Legendre and Related Functions 14.4.1 𝖯ν0(x), ν=0,12,1,2,4. 14.4.2 𝖰ν0(x), ν=0,12,1,2,4. 14.4.3 𝖯ν−1/2(x), ν=0,12,1,2,4. 14.4.4 𝖰ν1/2(x), ν=0,12,1,2,4. 14.4.5 𝖯ν−1(x), ν=0,12,1,2,4. 14.4.6 𝖰ν1(x), ν=0,12,1,2,4. 14.4.7 𝖯0−μ(x), μ=0,12,1,2,4. 14.4.8 𝖰0μ(x), μ=0,12,1,2,4. 14.4.9 𝖯1/2−μ(x), μ=0,12,1,2,4. 14.4.10 𝖰1/2μ(x), μ=0,12,1,2,4. 14.4.11 𝖯1−μ(x), μ=0,12,1,2,4. 14.4.12 𝖰1μ(x), μ=0,12,1,2,4. 14.4.13 𝖯ν0(x), 0≤ν≤10,−1<x<1. 14.4.14 𝖰ν0(x), 0≤ν≤10,−1<x<1. 14.4.15 𝖯0−μ(x), 0≤μ≤10,−1<x<1. 14.4.16 𝖰0μ(x), 0≤μ≤6.2,−1<x<1. 14.4.17 Pν0(x), ν=0,12,1,2,4. 14.4.18 𝑸ν0(x), ν=0,12,1,2,4. 14.4.19 Pν−1/2(x), ν=0,12,1,2,4. 14.4.20 𝑸ν1/2(x), ν=0,12,1,2,4. 14.4.21 Pν−1(x), ν=0,12,1,2,4. 14.4.22 𝑸ν1(x), ν=0,12,1,2,4. 14.4.23 P0−μ(x), μ=0,12,1,2,4. 14.4.24 𝑸0μ(x), μ=0,2,4,8. 14.4.25 P1/2−μ(x), μ=0,12,1,2,4. 14.4.26 𝑸1/2μ(x), μ=0,2,4,8. 14.4.27 P1−μ(x), μ=0,12,1,2,4. 14.4.28 𝑸1μ(x), μ=0,2,4,8. 14.4.29 Pν0(x), 0≤ν≤10, 1<x<10. 14.4.30 𝑸ν0(x), 0≤ν≤10, 1<x<10. 14.4.31 P0−μ(x), 0≤μ≤10, 1<x<10. 14.4.32 𝑸0μ(x), 0≤μ≤10, 1<x<10. 14.20.1 𝖯−12+iτ0(x), τ=0,1,2,4,8. 14.20.2 𝖰^−12+iτ0(x), τ=0,12,1,2,4. 14.20.3 𝖯−12+iτ−1/2(x), τ=0,1,2,4,8. 14.20.4 𝖰^−12+iτ−1/2(x), τ=12,1,2,4. 14.20.5 𝖯−12+iτ−1(x), τ=0,1,2,4,8. 14.20.6 𝖰^−12+iτ−1(x), τ=0,12,1,2,4. 14.20.7 𝖯−12+iτ−2(x),τ=0,1,2,4,8. 14.20.8 𝖰^−12+iτ−2(x), τ=0,12,1,2,4. 14.22.1 P1/20(x+iy), −5≤x≤5, −5≤y≤5. 14.22.2 P1/2−1/2(x+iy), −5≤x≤5, −5≤y≤5. 14.22.3 P1/2−1(x+iy), −5≤x≤5, −5≤y≤5. 14.22.4 𝑸00(x+iy), −5≤x≤5, −5≤y≤5. 15 Hypergeometric Function 15.3.1 F(43,916;145;x),−100≤x≤1. 15.3.2 F(5,−10;1;x),−0.023≤x≤1. 15.3.3 F(1,−10;10;x),−3≤x≤1. 15.3.4 F(5,10;1;x),−1≤x≤0.022. 15.3.5 F(43,916;145;x+iy),0≤x≤2,−0.5≤y≤0.5. 15.3.6 F(−3,35;u+iv;12),−6≤u≤2,−2≤v≤2. 15.3.7 |𝐅(−3,35;u+iv;12)|,−6≤u≤2,−2≤v≤2. 15.6.1 Contour of integration in (15.6.5). 16 Generalized Hypergeometric Functions & Meijer G-Function Case (i) Case (ii) Case (iii) 16.17.1 Integration path L for the Meijer G-function. 18 Orthogonal Polynomials 18.4.1 Jacobi polynomials Pn(1.5,−0.5)(x), n=1,2,3,4,5. 18.4.2 Jacobi polynomials Pn(1.25,0.75)(x), n=7,8. 18.4.3 Chebyshev polynomials Tn(x), n=1,2,3,4,5. 18.4.4 Legendre polynomials Pn(x), n=1,2,3,4,5. 18.4.5 Laguerre polynomials Ln(x), n=1,2,3,4,5. 18.4.6 Laguerre polynomials L3(α)(x), α=0,1,2,3,4. 18.4.7 Monic Hermite polynomials hn(x)=2−nHn(x), n=1,2,3,4,5. 18.4.8 Laguerre polynomials L3(α)(x), 0≤α≤3, 0≤x≤10. 18.4.9 Laguerre polynomials L4(α)(x), 0≤α≤3, 0≤x≤10. 18.21.1 Askey scheme. 18.39.1 Graphs of the first and fourth excited state eigenfunctions of the harmonic oscillator, for ℏ=k=m=1, of (18.39.13), in ψ1(x), ψ4(x) and those of the rational potential of (18.39.19), in ψ^3(x), ψ^6(x). Both sets satisfy the Sturm oscillation theorem. 18.39.2 Coulomb–Pollaczek weight functions, x∈[−1,1], (18.39.50) for s=10, l=0, and Z=±1. For Z=+1 the weight function, red curve, has an essential singularity at x=−1, as all derivatives vanish as x→−1+; the green curve is ∫−1xwCP(y)dy, to be compared with its histogram approximation in §18.40(ii). For Z=−1 the weight function, blue curve, is non-zero at x=−1, but this point is also an essential singularity as the discrete parts of the weight function of (18.39.51) accumulate as k→∞, xk→−1−. 18.40.1 Histogram approximations to the Repulsive Coulomb–Pollaczek, RCP, weight function integrated over [−1,x), see Figure 18.39.2 for an exact result, for Z=+1, shown for N=12 and N=120. 18.40.2 Derivative Rule inversions for wRCP(x) carried out via Lagrange and PWCF interpolations. Shown are the absolute errors of approximation (18.40.8) at the points xi,N, i=1,2,…,N for N=40. For the derivative rule Lagrange interpolation (red points) gives ∼15 digits in the central region, while PWCF interpolation (blue points) gives ∼25. 19 Elliptic Integrals 19.3.1 K(k), E(k), −2≤k2≤1. 19.3.2 RC(x,1), RC(x,−1), 0≤x≤5. 19.3.3 F(ϕ,k), −1≤k2≤2, 0≤sin2ϕ≤1. 19.3.4 E(ϕ,k), −1≤k2≤2, 0≤sin2ϕ≤1. 19.3.5 Π(α2,k), −2≤k2<1, −2≤α2≤2. 19.3.6 Π(ϕ,2,k), −1≤k2≤3, 0≤sin2ϕ<1. 19.3.7 K(k), −2≤ℜ(k2)≤2, −2≤ℑ(k2)≤2. 19.3.8 E(k), −2≤ℜ(k2)≤2, −2≤ℑ(k2)≤2. 19.3.9 ℜ(K(k)), −2≤ℜ(k2)≤2, −2≤ℑ(k2)≤2. 19.3.10 ℑ(K(k)), −2≤ℜ(k2)≤2, −2≤ℑ(k2)≤2. 19.3.11 ℜ(E(k)), −2≤ℜ(k2)≤2, −2≤ℑ(k2)≤2. 19.3.12 ℑ(E(k)), −2≤ℜ(k2)≤2, −2≤ℑ(k2)≤2. 19.17.1 RF(x,y,1), 0≤x≤1, y=0, 0.1, 0.5, 1. 19.17.2 RG(x,y,1), 0≤x≤1, y=0, 0.1, 0.5, 1. 19.17.3 RD(x,y,1), 0≤x≤2, y=0, 0.1, 1, 5, 25. 19.17.4 RJ(x,y,1,2), 0≤x≤1, y=0, 0.1, 0.5, 1. 19.17.5 RJ(x,y,1,0.5), 0≤x≤1, y=0, 0.1, 0.5, 1. 19.17.6 RJ(x,y,1,−0.5), 0≤x≤1, y=0, 0.1, 0.5, 1. 19.17.7 RJ(0.5,y,1,p), y=0, 0.01, 0.05, 0.2, 1, −1≤p<0. 19.17.8 RJ(0,y,1,p), 0≤y≤1, −1≤p≤2. 20 Theta Functions 20.2.1 Fundamental parallelogram. 20.3.1 θj(πx,0.15), 0≤x≤2, j=1,2,3,4. 20.3.2 θ1(πx,q), 0≤x≤2, q = 0.05, 0.5, 0.7, 0.9. 20.3.3 θ2(πx,q), 0≤x≤2, q = 0.05, 0.5, 0.7, 0.9. 20.3.4 θ3(πx,q), 0≤x≤2, q = 0.05, 0.5, 0.7, 0.9. 20.3.5 θ4(πx,q), 0≤x≤2, q = 0.05, 0.5, 0.7, 0.9. 20.3.6 θ1(x,q), 0≤q≤1, x = 0, 0.4, 5, 10, 40. 20.3.7 θ2(x,q), 0≤q≤1, x = 0, 0.4, 5, 10, 40. 20.3.8 θ3(x,q), 0≤q≤1, x = 0, 0.4, 5, 10, 40. 20.3.9 θ4(x,q), 0≤q≤1, x = 0, 0.4, 5, 10, 40. 20.3.10 θ1(πx,q), 0≤x≤2, 0≤q≤0.99. 20.3.11 θ2(πx,q), 0≤x≤2, 0≤q≤0.99. 20.3.12 θ3(πx,q), 0≤x≤2, 0≤q≤0.99. 20.3.13 θ4(πx,q), 0≤x≤2, 0≤q≤0.99. 20.3.14 θ1(πx+iy,0.12), −1≤x≤1, −1≤y≤2.3. 20.3.15 θ2(πx+iy,0.12), −1≤x≤1, −1≤y≤2.3. 20.3.16 θ3(πx+iy,0.12), −1≤x≤1, −1≤y≤1.5. 20.3.17 θ4(πx+iy,0.12), −1≤x≤1, −1≤y≤1.5. 20.3.18 θ1(0.1|u+iv), −1≤u≤1, 0.005≤v≤0.5. 20.3.19 θ2(0|u+iv), −1≤u≤1, 0.005≤v≤0.1. 20.3.20 θ3(0|u+iv), −1≤u≤1, 0.005≤v≤0.1. 20.3.21 θ4(0|u+iv), −1≤u≤1, 0.005≤v≤0.1. 21 Multidimensional Theta Functions (a1) (b1) (c1) (a2) (b2) (c2) (a3) (b3) (c3) 21.4.1 θ^(𝐳|𝛀) parametrized by (21.4.1). 21.4.2 ℜθ^(x+iy,0|𝛀1), 0≤x≤1, 0≤y≤5. 21.4.3 |θ^(x+iy,0|𝛀1)|, 0≤x≤1, 0≤y≤2. 21.4.4 θ^(ix,iy|𝛀1), 0≤x≤4, 0≤y≤4. 21.4.5 ℜθ^(x+iy,0,0|𝛀2), 0≤x≤1, 0≤y≤3. 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 sn(x,k), cn(x,k), dn(x,k), k=0.4, −3K≤x≤3K, K=1.6399…. 22.3.2 sn(x,k), cn(x,k), dn(x,k), k=0.7, −3K≤x≤3K, K=1.8456…. 22.3.3 sn(x,k), cn(x,k), dn(x,k), k=0.99, −3K≤x≤3K, K=3.3566…. 22.3.4 sn(x,k), cn(x,k), dn(x,k), k=0.999999, −3K≤x≤3K, K=7.9474…. 22.3.5 ds(x,k), sd(x,k), dc(x,k), k=0.4, −2K≤x≤2K, K=1.6399…. 22.3.6 ds(x,k), sd(x,k), dc(x,k), k=0.7, −2K≤x≤2K, K=1.8456…. 22.3.7 ds(x,k), sd(x,k), dc(x,k), k=0.99, −2K≤x≤2K, K=3.3566…. 22.3.8 ds(x,k), sd(x,k), dc(x,k), k=0.999999, −2K≤x≤2K, K=7.9474…. 22.3.9 cs(x,k), ns(x,k), sc(x,k), k=0.4, −2K≤x≤2K, K=1.6399…. 22.3.10 cs(x,k), ns(x,k), sc(x,k), k=0.7, −2K≤x≤2K, K=1.8456…. 22.3.11 cs(x,k), ns(x,k), sc(x,k), k=0.99, −2K≤x≤2K, K=3.3566…. 22.3.12 cs(x,k), ns(x,k), sc(x,k), k=0.999999, −2K≤x≤2K, K=7.9474…. 22.3.13 sn(x,k) for k=1−e−n, n=0 to 20, −5π≤x≤5π. 22.3.14 cn(x,k) for k=1−e−n, n=0 to 20, −5π≤x≤5π. 22.3.15 dn(x,k) for k=1−e−n, n=0 to 20, −5π≤x≤5π. 22.3.16 sn(x+iy,k), k=0.99, −3K≤x≤3K, 0≤y≤4K′. 22.3.17 cn(x+iy,k), k=0.99, −3K≤x≤3K, 0≤y≤4K′. 22.3.18 dn(x+iy,k), k=0.99, −3K≤x≤3K, 0≤y≤4K′. 22.3.19 cd(x+iy,k), k=0.99, −3K≤x≤3K, 0≤y≤4K′. 22.3.20 dc(x+iy,k), k=0.99, −3K≤x≤3K, 0≤y≤4K′. 22.3.21 ns(x+iy,k), k=0.99, −3K≤x≤3K. 22.3.22 ℜsn(x,k). 22.3.23 ℑsn(x,k). 22.3.24 sn(x+iy,k). 22.3.25 sn(5,k). 22.3.26 Density plot of |sn(5,k)|. 22.3.27 Density plot of |sn(10,k)|. 22.3.28 Density plot of |sn(20,k)|. 22.3.29 Density plot of |sn(30,k)|. (a) sn(z,k) (b) cn(z,k) (c) dn(z,k) 22.4.1 Poles, zeros of the principal Jacobian elliptic functions. 22.4.2 Fundamental unit cell. 22.16.1 am(x,k), 0≤x≤10π, k=0.4,0.7,0.99,0.999999. 22.16.2 ℰ(x,k), 0≤x≤10π, k=0.4,0.7,0.99,0.999999. 22.16.3 Z(x|k), 0≤x≤10π, k=0.4,0.7,0.99,0.999999. 22.19.1 am(x,k), 0≤x≤10π, k=0.5,0.9999,1.0001,2. 23 Weierstrass Elliptic and Modular Functions 23.4.1 ℘(x;g2,0), 0≤x≤9, g2 = 0.1, 0.2, 0.5, 0.8. 23.4.2 ℘(x;0,g3), 0≤x≤9, g3 = 0.1, 0.2, 0.5, 0.8. 23.4.3 ζ(x;g2,0), 0≤x≤8, g2 = 0.1, 0.2, 0.5, 0.8. 23.4.4 ζ(x;0,g3), 0≤x≤8, g3 = 0.1, 0.2, 0.5, 0.8. 23.4.5 σ(x;g2,0), −5≤x≤5, g2 = 0.1, 0.2, 0.5, 0.8. 23.4.6 σ(x;0,g3), −5≤x≤5, g3 = 0.1, 0.2, 0.5, 0.8. 23.4.7 ℘(x), 0≤x≤9, k2 = 0.2, 0.8, 0.95, 0.99. 23.4.8 ℘(x+iy), −2K(k)≤x≤2K(k), 0≤y≤6K′(k), k2=0.9. 23.4.9 ℘(x+iy;1,4i), −3.8≤x≤3.8, −3.8≤y≤3.8. 23.4.10 ζ(x+iy;1,0), −5≤x≤5, −5≤y≤5. 23.4.11 σ(x+iy;1,i), −2.5≤x≤2.5, −2.5≤y≤2.5. 23.4.12 ℘(3.7;a+ib,0), −5≤a≤3, −4≤b≤4. 23.5.1 Rhombic lattice. ℜ(2ω3)=ω1. 23.5.2 Equianharmonic lattice. 2ω3=eπi/32ω1, 2ω1−2ω3=e−πi/32ω1. 23.16.1 λ(iy), J(iy), η(iy), 0≤y≤3. 23.16.2 λ(x+iy), −0.25≤x≤0.25, 0.005≤y≤0.1. 23.16.3 η(x+iy), −0.0625≤x≤0.0625, 0.0001≤y≤0.07. 24 Bernoulli and Euler Polynomials 24.3.1 Bernoulli polynomials Bn(x), n=2,3,…,6. 24.3.2 Euler polynomials En(x), n=2,3,…,6. 25 Zeta and Related Functions 25.3.1 ζ(x), ζ′(x), −20≤x≤10. 25.3.2 ζ(x), ζ′(x), −12≤x≤−2. 25.3.3 |ζ(x+iy)|, −4≤x≤4, −10≤y≤40. 25.3.4 Z(t), 0≤t≤50. 25.3.5 Z(t), 1000≤t≤1050. 25.3.6 Z(t), 10000≤t≤10050. 25.11.1 ζ(x,a), a = 0.3, 0.5, 0.8, 1, −20≤x≤10. 25.11.2 ζ(x,a), −19.5≤x≤10, 0.02≤a≤1. 25.12.1 Li2(x), −20≤x<1. 25.12.2 |Li2(x+iy)|, −20≤x≤20, −20≤y≤20. 26 Combinatorial Analysis 26.9.1 Ferrers graph of the partition 7+4+3+3+2+1. 26.9.2 The partition 5+5+3+2 represented as a lattice path. 26.12.1 A plane partition of 75. 28 Mathieu Functions and Hill’s Equation 28.2.1 Eigenvalues an(q), bn(q) of Mathieu’s equation. 28.3.1 ce2n(x,1), 0≤x≤π/2, n=0,1,2,3. 28.3.2 ce2n(x,10), 0≤x≤π/2, n=0,1,2,3. 28.3.3 ce2n+1(x,1), 0≤x≤π/2, n=0,1,2,3. 28.3.4 ce2n+1(x,10), 0≤x≤π/2, n=0,1,2,3. 28.3.5 se2n+1(x,1), 0≤x≤π/2, n=0,1,2,3. 28.3.6 se2n+1(x,10), 0≤x≤π/2, n=0,1,2,3. 28.3.7 se2n(x,1), 0≤x≤π/2, n=1,2,3,4. 28.3.8 se2n(x,10), 0≤x≤π/2, n=1,2,3,4. 28.3.9 ce0(x,q), 0≤x≤2π, 0≤q≤10. 28.3.10 se1(x,q), 0≤x≤2π, 0≤q≤10. 28.3.11 ce1(x,q), 0≤x≤2π, 0≤q≤10. 28.3.12 se2(x,q), 0≤x≤2π, 0≤q≤10. 28.3.13 ce2(x,q), 0≤x≤2π, 0≤q≤10. 28.5.1 fe0(x,0.5), ce0(x,0.5), 0≤x≤2π. 28.5.2 fe0(x,1), ce0(x,1), 0≤x≤2π. 28.5.3 fe1(x,0.5), ce1(x,0.5), 0≤x≤2π. 28.5.4 fe1(x,1), ce1(x,1), 0≤x≤2π. 28.5.5 ge1(x,0.5), se1(x,0.5), 0≤x≤2π. 28.5.6 ge1(x,1), se1(x,1), 0≤x≤2π. 28.7.1 Branch point of the eigenvalues a0(iq^) and a2(iq^): 0≤q^≤2.5. 28.13.1 λν(q), ν=0.5(1)3.5; an(q),bn(q), n=0,1,2,3,4 (a’s), n=1,2,3,4 (b’s). 28.13.2 λν(q), −2<ν<2, 0≤q≤10. 28.13.3 ceν(x,1), −1<ν<1, 0≤x≤2π. 28.13.4 seν(x,1), 0<ν<1, 0≤x≤2π. 28.13.5 meiμ(x,1), 0.1≤μ≤0.4, −π≤x≤π. 28.17.1 Stability chart for eigenvalues of Mathieu’s equation (28.2.1). 28.21.1 Mc0(1)(x,h), 0≤h≤3, 0≤x≤2. 28.21.2 Mc1(1)(x,h), 0≤h≤3, 0≤x≤2. 28.21.3 Mc0(2)(x,h) 0.1≤h≤2, 0≤x≤2. 28.21.4 Mc1(2)(x,h), 0.2≤h≤2, 0≤x≤2. 28.21.5 Ms1(1)(x,h), 0≤h≤3, 0≤x≤2. 28.21.6 Ms1(2)(x,h), 0.2≤h≤2, 0≤x≤2. 29 Lamé Functions 29.2.1 Singularities of Lamé’s equation. 29.4.1 aνm(0.5), bνm+1(0.5), m=0,1,2,3. 29.4.2 aν3(0.5)−bν3(0.5). 29.4.3 a1.5m(k2), b1.5m+1(k2). 29.4.4 aνm(0.1), bνm+1(0.1), m=0,1,2,3. 29.4.5 aνm(0.9), bνm+1(0.9), m=0,1,2,3. 29.4.6 aν2(0.5)−bν2(0.5). 29.4.7 aν4(0.5)−bν4(0.5) 29.4.8 a2.5m(k2), b2.5m+1(k2). 29.4.9 aν0(k2). 29.4.10 bν1(k2). 29.4.11 aν1(k2). 29.4.12 bν2(k2). 29.4.13 𝐸𝑐1.5m(x,0.5), −2K≤x≤2K, m=0,1,2. 29.4.14 𝐸𝑠1.5m(x,0.5), −2K≤x≤2K, m=1,2,3. 29.4.15 𝐸𝑐1.5m(x,0.1), −2K≤x≤2K, m=0,1,2. 29.4.16 𝐸𝑠1.5m(x,0.1), −2K≤x≤2K, m=1,2,3. 29.4.17 𝐸𝑐1.5m(x,0.9), −2K≤x≤2K, m=0,1,2. 29.4.18 𝐸𝑠1.5m(x,0.9), −2K≤x≤2K, m=1,2,3. 29.4.19 𝐸𝑐2.5m(x,0.1), −2K≤x≤2K, m=0,1,2. 29.4.20 𝐸𝑠2.5m(x,0.1), −2K≤x≤2K, m=1,2,3. 29.4.21 𝐸𝑐2.5m(x,0.5), −2K≤x≤2K, m=0,1,2. 29.4.22 𝐸𝑠2.5m(x,0.5), −2K≤x≤2K, m=1,2,3. 29.4.23 𝐸𝑐2.5m(x,0.9), −2K≤x≤2K, m=0,1,2. 29.4.24 𝐸𝑠2.5m(x,0.9), −2K≤x≤2K, m=1,2,3. 29.4.25 𝐸𝑐1.50(x,k2). 29.4.26 𝐸𝑠1.51(x,k2). 29.4.27 𝐸𝑐1.51(x,k2). 29.4.28 𝐸𝑠1.52(x,k2). 29.4.29 𝐸𝑐2.50(x,k2). 29.4.30 𝐸𝑠2.51(x,k2). 29.4.31 𝐸𝑐2.51(x,k2). 29.4.32 𝐸𝑠2.52(x,k2). 29.13.1 a2m(k2), b2m(k2). 29.13.2 a1m(k2), b1m(k2). 29.13.3 a3m(k2), b3m(k2). 29.13.4 a4m(k2), b4m(k2). 29.13.5 𝑢𝐸4m(x,0.1) for −2K≤x≤2K, m=0,1,2. K=1.61244…. 29.13.6 𝑢𝐸4m(x,0.9) for −2K≤x≤2K, m=0,1,2. K=2.57809…. 29.13.7 𝑠𝐸5m(x,0.1) for −2K≤x≤2K, m=0,1,2. K=1.61244…. 29.13.8 𝑠𝐸5m(x,0.9) for −2K≤x≤2K, m=0,1,2. K=2.57809…. 29.13.9 𝑐𝐸5m(x,0.1) for −2K≤x≤2K, m=0,1,2. K=1.61244…. 29.13.10 𝑐𝐸5m(x,0.9) for −2K≤x≤2K, m=0,1,2. K=2.57809…. 29.13.11 𝑑𝐸5m(x,0.1) for −2K≤x≤2K, m=0,1,2. K=1.61244…. 29.13.12 𝑑𝐸5m(x,0.9) for −2K≤x≤2K, m=0,1,2. K=2.57809…. 29.13.13 𝑠𝑐𝐸4m(x,0.1) for −2K≤x≤2K, m=0,1. K=1.61244…. 29.13.14 𝑠𝑐𝐸4m(x,0.9) for −2K≤x≤2K, m=0,1. K=2.57809…. 29.13.15 𝑠𝑑𝐸4m(x,0.1) for −2K≤x≤2K, m=0,1. K=1.61244…. 29.13.16 𝑠𝑑𝐸4m(x,0.9) for −2K≤x≤2K, m=0,1. K=2.57809…. 29.13.17 𝑐𝑑𝐸4m(x,0.1) for −2K≤x≤2K, m=0,1. K=1.61244…. 29.13.18 𝑐𝑑𝐸4m(x,0.9) for −2K≤x≤2K, m=0,1. K=2.57809…. 29.13.19 𝑠𝑐𝑑𝐸5m(x,0.1) for −2K≤x≤2K, m=0,1. K=1.61244…. 29.13.20 𝑠𝑐𝑑𝐸5m(x,0.9) for −2K≤x≤2K, m=0,1. K=2.57809…. 29.13.21 |𝑢𝐸41(x+iy,0.1)| for −3K≤x≤3K, 0≤y≤2K′. K=1.61244…, K′=2.57809…. 29.13.22 |𝑢𝐸41(x+iy,0.5)| for −3K≤x≤3K, 0≤y≤2K′. K=K′=1.85407…. 29.13.23 |𝑢𝐸41(x+iy,0.9)| for −3K≤x≤3K, 0≤y≤2K′. K=2.57809…, K′=1.61244…. 30 Spheroidal Wave Functions 30.7.1 λn0(γ2), n=0,1,2,3, −10≤γ2≤10. 30.7.2 λn1(γ2), n=1,2,3,4, −10≤γ2≤10. 30.7.3 λn5(γ2), n=5,6,7,8, −40≤γ2≤40. 30.7.4 λn10(γ2), n=10,11,12,13, −50≤γ2≤150. 30.7.5 𝖯𝗌n0(x,4), n=0,1,2,3, −1≤x≤1. 30.7.6 𝖯𝗌n0(x,−4), n=0,1,2,3, −1≤x≤1. 30.7.7 𝖯𝗌n1(x,30), n=1,2,3,4, −1≤x≤1. 30.7.8 𝖯𝗌n1(x,−30), n=1,2,3,4, −1≤x≤1. 30.7.9 𝖯𝗌20(x,γ2), −1≤x≤1, −50≤γ2≤50. 30.7.10 𝖯𝗌31(x,γ2), −1≤x≤1, −50≤γ2≤50. 30.7.11 𝖰𝗌n0(x,4), n=0,1,2,3, −1<x<1. 30.7.12 𝖰𝗌n0(x,−4), n=0,1,2,3, −1<x<1. 30.7.13 𝖰𝗌n1(x,4), for n=1,2,3,4, −1<x<1. 30.7.14 𝖰𝗌n1(x,−4), n=1,2,3,4, −1<x<1. 30.7.15 𝖰𝗌10(x,γ2),−1<x<1,−10≤γ2≤10. 30.7.16 |𝑃𝑠00(x+iy,4)|, −2≤x≤2, −2≤y≤2. 30.7.17 |𝑃𝑠00(x+iy,−4)|, −2≤x≤2, −2≤y≤2. 30.7.18 |𝑃𝑠11(x+iy,4)|, −2≤x≤2, −2≤y≤2. 30.7.19 |𝑃𝑠11(x+iy,−4)|, −2≤x≤2, −2≤y≤2. 30.7.20 |𝑄𝑠00(x+iy,4)|, −2≤x≤2, −2≤y≤2. 30.7.21 |𝑄𝑠00(x+iy,−4)|, −1.8≤x≤1.8, −2≤y≤2. 30.11.1 Sn0(1)(x,2), n=0,1, 1≤x≤10. 30.11.2 Sn0(1)(iy,2i), n=0,1, 0≤y≤10. 30.11.3 Sn1(1)(x,2), n=1,2, 1≤x≤10. 30.11.4 Sn1(1)(iy,2i), n=1,2, 0≤y≤10. 32 Painlevé Transcendents 32.3.1 wk(x), −12≤x≤1.33, k=0.5, 0.75, 1, 1.25. 32.3.2 wk(x), −12≤x≤2.43, k=−0.5, −0.25, 0, 1, 2. 32.3.3 wk(x), −12≤x≤0.73, k=1.85185 3, 1.85185 5. 32.3.4 wk(x), −12≤x≤2.3, k=−0.45142 7, −0.45142 8. 32.3.5 wk(x), kAi(x), −10≤x≤4, k=0.5. 32.3.6 wk(x), −10≤x≤4, k=0.999, 1.001. 32.3.7 uk(x;−12), −12≤x≤4, k=0.33554 691, 0.33554 692. 32.3.8 uk(x;12), −12≤x≤4, k=0.47443. 32.3.9 uk(x;32), −12≤x≤4, k=0.38736, 0.38737. 32.3.10 uk(x;52), −12≤x≤4, k=0.24499 2, 0.24499 3. 33 Coulomb Functions 33.3.1 Fℓ(η,ρ), Gℓ(η,ρ), ℓ=0, η=−2. 33.3.2 Fℓ(η,ρ), Gℓ(η,ρ), ℓ=0, η=0. 33.3.3 Fℓ(η,ρ), Gℓ(η,ρ), ℓ=0, η=2. 33.3.4 Fℓ(η,ρ), Gℓ(η,ρ), ℓ=0, η=2. 33.3.5 Fℓ(η,ρ), Gℓ(η,ρ), Mℓ(η,ρ), ℓ=0, η=15/2. 33.3.6 Fℓ(η,ρ), Gℓ(η,ρ), Mℓ(η,ρ), ℓ=5, η=0. 33.3.7 F0(η,ρ), −2≤η≤2, 0≤ρ≤5. 33.3.8 G0(η,ρ), −2≤η≤2, 0<ρ≤5. 33.15.1 f(ϵ,ℓ;r),h(ϵ,ℓ;r), ℓ=0,ϵ=4. 33.15.2 f(ϵ,ℓ;r),h(ϵ,ℓ;r), ℓ=1,ϵ=4. 33.15.3 f(ϵ,ℓ;r),h(ϵ,ℓ;r), ℓ=0,ϵ=−1/ν2,ν=1.5. 33.15.4 f(ϵ,ℓ;r),h(ϵ,ℓ;r), ℓ=0,ϵ=−1/ν2,ν=2. 33.15.5 f(ϵ,ℓ;r),h(ϵ,ℓ;r), ℓ=0,ϵ=−1/ν2,ν=2.5. 33.15.6 f(ϵ,ℓ;r), ℓ=0,−2<ϵ<2,−15<r<15. 33.15.7 h(ϵ,ℓ;r), ℓ=0,−2<ϵ<2,−15<r<15. 33.15.8 f(ϵ,ℓ;r), ℓ=1,−2<ϵ<2,−15<r<15. 33.15.9 h(ϵ,ℓ;r), ℓ=1,−2<ϵ<2,−15<r<15. 33.15.10 s(ϵ,ℓ;r), ℓ=0,−0.15<ϵ<0.10,0<r<65. 33.15.11 c(ϵ,ℓ;r), ℓ=0,−0.15<ϵ<0.10,0<r<65. 34 3j, 6j, 9j Symbols 34.2.1 Angular momenta jr and projective quantum numbers. 34.4.1 Tetrahedron corresponding to 6j symbol. 36 Integrals with Coalescing Saddles (a) Density plot. (b) 3D plot. 36.3.1 Modulus of Pearcey integral |Ψ2(x,y)|. (a) Density plot. (b) 3D plot. 36.3.2 Modulus of swallowtail canonical integral function |Ψ3(x,y,3)|. (a) Density plot. (b) 3D plot. 36.3.3 Modulus of swallowtail canonical integral function |Ψ3(x,y,0)|. (a) Density plot. (b) 3D plot. 36.3.4 Modulus of swallowtail canonical integral function |Ψ3(x,y,−3)|. (a) Density plot. (b) 3D plot. 36.3.5 Modulus of swallowtail canonical integral function |Ψ3(x,y,−7.5)|. (a) Density plot. (b) 3D plot. 36.3.6 Modulus of elliptic umbilic canonical integral function |Ψ(E)(x,y,0)|. (a) Density plot. (b) 3D plot. 36.3.7 Modulus of elliptic umbilic canonical integral function |Ψ(E)(x,y,2)|. (a) Density plot. (b) 3D plot. 36.3.8 Modulus of elliptic umbilic canonical integral function |Ψ(E)(x,y,4)|. (a) Density plot. (b) 3D plot. 36.3.9 Modulus of hyperbolic umbilic canonical integral function |Ψ(H)(x,y,0)|. (a) Density plot. (b) 3D plot. 36.3.10 Modulus of hyperbolic umbilic canonical integral function |Ψ(H)(x,y,1)|. (a) Density plot. (b) 3D plot. 36.3.11 Modulus of hyperbolic umbilic canonical integral function |Ψ(H)(x,y,2)|. (a) Density plot. (b) 3D plot. 36.3.12 Modulus of hyperbolic umbilic canonical integral function |Ψ(H)(x,y,3)|. (a) Contour plot, at intervals of π/4. (b) Density plot. 36.3.13 Phase of Pearcey integral phΨ2(x,y). (a) phΨ3(x,y,3). (b) phΨ3(x,y,0). (c) phΨ3(x,y,−3). (d) phΨ3(x,y,−7.5). 36.3.14 Density plots of phase of swallowtail canonical integrals. (a) Contour plot. (b) Density plot. 36.3.15 Phase of elliptic umbilic canonical integral phΨ(E)(x,y,0). (a) Contour plot. (b) Density plot. 36.3.16 Phase of elliptic umbilic canonical integral phΨ(E)(x,y,2). (a) Contour plot. (b) Density plot. 36.3.17 Phase of elliptic umbilic canonical integral phΨ(E)(x,y,4). (a) Contour plot. (b) Density plot. 36.3.18 Phase of hyperbolic umbilic canonical integral phΨ(H)(x,y,0). (a) Contour plot. (b) Density plot. 36.3.19 Phase of hyperbolic umbilic canonical integral phΨ(H)(x,y,1). (a) Contour plot. (b) Density plot. 36.3.20 Phase of hyperbolic umbilic canonical integral phΨ(H)(x,y,2). (a) Contour plot. (b) Density plot. 36.3.21 Phase of hyperbolic umbilic canonical integral phΨ(H)(x,y,3). 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 z<0. 36.5.3 Swallowtail catastrophe with z=0. 36.5.4 Swallowtail catastrophe with z>0. 36.5.5 Elliptic umbilic catastrophe with z=constant. 36.5.6 Hyperbolic umbilic catastrophe with z=constant. 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 Kelvin’s ship wave pattern.
