Results of Lamé Functions: Difference between revisions

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Latest revision as of 16:27, 19 January 2020

DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
29.2.E1 d 2 w d z 2 + ( h - ν ⁢ ( ν + 1 ) ⁢ k 2 ⁢ sn 2 ⁡ ( z , k ) ) ⁢ w = 0 derivative 𝑤 𝑧 2 ℎ 𝜈 𝜈 1 superscript 𝑘 2 Jacobi-elliptic-sn 2 𝑧 𝑘 𝑤 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+(h-\nu% (\nu+1)k^{2}{\operatorname{sn}^{2}}\left(z,k\right))w=0}} diff(w, [z$(2)])+(h - nu*(nu + 1)*(k)^(2)* (JacobiSN(z, k))^(2))* w = 0 D[w, {z, 2}]+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (JacobiSN[z, (k)^2])^(2))* w = 0 Failure Failure
Fail
7.948739768-7.545139452*I <- {h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), k = 1}
51.95468921+27.97388965*I <- {h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), k = 2}
-3.910915794-11.82960931*I <- {h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), k = 3}
6.183103152-8.579425434*I <- {h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
 Skip
29.2.E2 d 2 w d ξ 2 + 1 2 ⁢ ( 1 ξ + 1 ξ - 1 + 1 ξ - k - 2 ) ⁢ d w d ξ + h ⁢ k - 2 - ν ⁢ ( ν + 1 ) ⁢ ξ 4 ⁢ ξ ⁢ ( ξ - 1 ) ⁢ ( ξ - k - 2 ) ⁢ w = 0 derivative 𝑤 𝜉 2 1 2 1 𝜉 1 𝜉 1 1 𝜉 superscript 𝑘 2 derivative 𝑤 𝜉 ℎ superscript 𝑘 2 𝜈 𝜈 1 𝜉 4 𝜉 𝜉 1 𝜉 superscript 𝑘 2 𝑤 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\xi}^{2}}+% \frac{1}{2}\left(\frac{1}{\xi}+\frac{1}{\xi-1}+\frac{1}{\xi-k^{-2}}\right)% \frac{\mathrm{d}w}{\mathrm{d}\xi}+\frac{hk^{-2}-\nu(\nu+1)\xi}{4\xi(\xi-1)(\xi% -k^{-2})}w=0}} diff(w, [xi$(2)])+(1)/(2)*((1)/(xi)+(1)/(xi - 1)+(1)/(xi - (k)^(- 2)))* diff(w, xi)+(h*(k)^(- 2)- nu*(nu + 1)* xi)/(4*xi*(xi - 1)*(xi - (k)^(- 2)))*w = 0 D[w, {\[Xi], 2}]+Divide[1,2]*(Divide[1,\[Xi]]+Divide[1,\[Xi]- 1]+Divide[1,\[Xi]- (k)^(- 2)])* D[w, \[Xi]]+Divide[h*(k)^(- 2)- \[Nu]*(\[Nu]+ 1)* \[Xi],4*\[Xi]*(\[Xi]- 1)*(\[Xi]- (k)^(- 2))]*w = 0 Failure Failure
Fail
-1.197364586+.3597957455*I <- {h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), k = 1}
-1.025513349+.2414101023e-1*I <- {h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), k = 2}
-.9825277432-.7730059408e-2*I <- {h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), k = 3}
.1006999677+1.062488515*I <- {h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
 Skip
29.2.E4 ( 1 - k 2 ⁢ cos 2 ⁡ ϕ ) ⁢ d 2 w d ϕ 2 + k 2 ⁢ cos ⁡ ϕ ⁢ sin ⁡ ϕ ⁢ d w d ϕ + ( h - ν ⁢ ( ν + 1 ) ⁢ k 2 ⁢ cos 2 ⁡ ϕ ) ⁢ w = 0 1 superscript 𝑘 2 2 italic-ϕ derivative 𝑤 italic-ϕ 2 superscript 𝑘 2 italic-ϕ italic-ϕ derivative 𝑤 italic-ϕ ℎ 𝜈 𝜈 1 superscript 𝑘 2 2 italic-ϕ 𝑤 0 {\displaystyle{\displaystyle(1-k^{2}{\cos^{2}}\phi)\frac{{\mathrm{d}}^{2}w}{{% \mathrm{d}\phi}^{2}}+k^{2}\cos\phi\sin\phi\frac{\mathrm{d}w}{\mathrm{d}\phi}+(% h-\nu(\nu+1)k^{2}{\cos^{2}}\phi)w=0}} (1 - (k)^(2)* (cos(phi))^(2))* diff(w, [phi$(2)])+ (k)^(2)* cos(phi)*sin(phi)*diff(w, phi)+(h - nu*(nu + 1)*(k)^(2)* (cos(phi))^(2))* w = 0 (1 - (k)^(2)* (Cos[\[Phi]])^(2))* D[w, {\[Phi], 2}]+ (k)^(2)* Cos[\[Phi]]*Sin[\[Phi]]*D[w, \[Phi]]+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (Cos[\[Phi]])^(2))* w = 0 Failure Failure
Fail
-32.55364142+30.82095554*I <- {h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), k = 1}
-130.2145657+111.2838222*I <- {h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), k = 2}
-292.9827729+245.3885999*I <- {h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), k = 3}
30.82095554+32.55364142*I <- {h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
 Skip
29.2.E10 d 2 w d ζ 2 + 1 2 ⁢ ( 1 ζ - e 1 + 1 ζ - e 2 + 1 ζ - e 3 ) ⁢ d w d ζ + g - ν ⁢ ( ν + 1 ) ⁢ ζ 4 ⁢ ( ζ - e 1 ) ⁢ ( ζ - e 2 ) ⁢ ( ζ - e 3 ) ⁢ w = 0 derivative 𝑤 𝜁 2 1 2 1 𝜁 subscript 𝑒 1 1 𝜁 subscript 𝑒 2 1 𝜁 subscript 𝑒 3 derivative 𝑤 𝜁 𝑔 𝜈 𝜈 1 𝜁 4 𝜁 subscript 𝑒 1 𝜁 subscript 𝑒 2 𝜁 subscript 𝑒 3 𝑤 0 {\displaystyle{\displaystyle{\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+% \frac{1}{2}\left(\frac{1}{\zeta-e_{1}}+\frac{1}{\zeta-e_{2}}+\frac{1}{\zeta-e_% {3}}\right)\frac{\mathrm{d}w}{\mathrm{d}\zeta}}+\frac{g-\nu(\nu+1)\zeta}{4(% \zeta-e_{1})(\zeta-e_{2})(\zeta-e_{3})}w=0}} diff(w, [zeta$(2)])+(1)/(2)*((1)/(zeta - e[1])+(1)/(zeta - e[2])+(1)/(zeta - e[3]))* diff(w, zeta)+(g - nu*(nu + 1)* zeta)/(4*(zeta - e[1])*(zeta - e[2])*(zeta - e[3]))*w = 0 D[w, {\[zeta], 2}]+Divide[1,2]*(Divide[1,\[zeta]- Subscript[e, 1]]+Divide[1,\[zeta]- Subscript[e, 2]]+Divide[1,\[zeta]- Subscript[e, 3]])* D[w, \[zeta]]+Divide[g - \[Nu]*(\[Nu]+ 1)* \[zeta],4*(\[zeta]- Subscript[e, 1])*(\[zeta]- Subscript[e, 2])*(\[zeta]- Subscript[e, 3])]*w = 0 Failure Failure Skip Error
29.8.E1 x = k 2 ⁢ sn ⁡ ( z , k ) ⁢ sn ⁡ ( z 1 , k ) ⁢ sn ⁡ ( z 2 , k ) ⁢ sn ⁡ ( z 3 , k ) - k 2 k ′ 2 ⁢ cn ⁡ ( z , k ) ⁢ cn ⁡ ( z 1 , k ) ⁢ cn ⁡ ( z 2 , k ) ⁢ cn ⁡ ( z 3 , k ) + 1 k ′ 2 ⁢ dn ⁡ ( z , k ) ⁢ dn ⁡ ( z 1 , k ) ⁢ dn ⁡ ( z 2 , k ) ⁢ dn ⁡ ( z 3 , k ) 𝑥 superscript 𝑘 2 Jacobi-elliptic-sn 𝑧 𝑘 Jacobi-elliptic-sn subscript 𝑧 1 𝑘 Jacobi-elliptic-sn subscript 𝑧 2 𝑘 Jacobi-elliptic-sn subscript 𝑧 3 𝑘 superscript 𝑘 2 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-cn 𝑧 𝑘 Jacobi-elliptic-cn subscript 𝑧 1 𝑘 Jacobi-elliptic-cn subscript 𝑧 2 𝑘 Jacobi-elliptic-cn subscript 𝑧 3 𝑘 1 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-dn 𝑧 𝑘 Jacobi-elliptic-dn subscript 𝑧 1 𝑘 Jacobi-elliptic-dn subscript 𝑧 2 𝑘 Jacobi-elliptic-dn subscript 𝑧 3 𝑘 {\displaystyle{\displaystyle x=k^{2}\operatorname{sn}\left(z,k\right)% \operatorname{sn}\left(z_{1},k\right)\operatorname{sn}\left(z_{2},k\right)% \operatorname{sn}\left(z_{3},k\right)-\frac{k^{2}}{{k^{\prime}}^{2}}% \operatorname{cn}\left(z,k\right)\operatorname{cn}\left(z_{1},k\right)% \operatorname{cn}\left(z_{2},k\right)\operatorname{cn}\left(z_{3},k\right)+% \frac{1}{{k^{\prime}}^{2}}\operatorname{dn}\left(z,k\right)\operatorname{dn}% \left(z_{1},k\right)\operatorname{dn}\left(z_{2},k\right)\operatorname{dn}% \left(z_{3},k\right)}} x = (k)^(2)* JacobiSN(z, k)*JacobiSN(z[1], k)*JacobiSN(z[2], k)*JacobiSN(z[3], k)-((k)^(2))/(1 - (k)^(2))*JacobiCN(z, k)*JacobiCN(z[1], k)*JacobiCN(z[2], k)*JacobiCN(z[3], k)+(1)/(1 - (k)^(2))*JacobiDN(z, k)*JacobiDN(z[1], k)*JacobiDN(z[2], k)*JacobiDN(z[3], k) x = (k)^(2)* JacobiSN[z, (k)^2]*JacobiSN[Subscript[z, 1], (k)^2]*JacobiSN[Subscript[z, 2], (k)^2]*JacobiSN[Subscript[z, 3], (k)^2]-Divide[(k)^(2),1 - (k)^(2)]*JacobiCN[z, (k)^2]*JacobiCN[Subscript[z, 1], (k)^2]*JacobiCN[Subscript[z, 2], (k)^2]*JacobiCN[Subscript[z, 3], (k)^2]+Divide[1,1 - (k)^(2)]*JacobiDN[z, (k)^2]*JacobiDN[Subscript[z, 1], (k)^2]*JacobiDN[Subscript[z, 2], (k)^2]*JacobiDN[Subscript[z, 3], (k)^2] Failure Failure Skip Successful
29.8.E2 μ ⁢ w ⁢ ( z 1 ) ⁢ w ⁢ ( z 2 ) ⁢ w ⁢ ( z 3 ) = ∫ - 2 ⁢ K ⁡ 2 ⁢ K ⁡ 𝖯 ν ⁡ ( x ) ⁢ w ⁢ ( z ) ⁢ d z 𝜇 𝑤 subscript 𝑧 1 𝑤 subscript 𝑧 2 𝑤 subscript 𝑧 3 superscript subscript 2 complete-elliptic-integral-first-kind-K 𝑘 2 complete-elliptic-integral-first-kind-K 𝑘 shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 𝑤 𝑧 𝑧 {\displaystyle{\displaystyle\mu w(z_{1})w(z_{2})w(z_{3})=\int_{-2\!K\!}^{2\!K% \!}\mathsf{P}_{\nu}\left(x\right)w(z)\mathrm{d}z}} mu*w*(z[1])* w*(z[2])* w*(z[3])= int(LegendreP(nu, x)*w*(z), z = - 2*EllipticK(k)..2*EllipticK(k)) \[Mu]*w*(Subscript[z, 1])* w*(Subscript[z, 2])* w*(Subscript[z, 3])= Integrate[LegendreP[\[Nu], x]*w*(z), {z, - 2*EllipticK[(k)^2], 2*EllipticK[(k)^2]}] Failure Failure - Skip
29.8#Ex1 w ⁢ ( z + 2 ⁢ K ⁡ ) = σ ⁢ w ⁢ ( z ) 𝑤 𝑧 2 complete-elliptic-integral-first-kind-K 𝑘 𝜎 𝑤 𝑧 {\displaystyle{\displaystyle w(z+2\!K\!)=\sigma w(z)}} w*(z + 2*EllipticK(k))= sigma*w*(z) w*(z + 2*EllipticK[(k)^2])= \[Sigma]*w*(z) Failure Failure Error
Fail
DirectedInfinity[] <- {Rule[k, 1], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[11.090638940207405, -2.3226169111049417] <- {Rule[k, 2], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[9.565752102125508, -2.5159784259617144] <- {Rule[k, 3], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 1], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[σ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
29.8#Ex2 w 2 ⁢ ( z + 2 ⁢ K ⁡ ) = τ ⁢ w ⁢ ( z ) + σ ⁢ w 2 ⁢ ( z ) subscript 𝑤 2 𝑧 2 complete-elliptic-integral-first-kind-K 𝑘 𝜏 𝑤 𝑧 𝜎 subscript 𝑤 2 𝑧 {\displaystyle{\displaystyle w_{2}(z+2\!K\!)=\tau w(z)+\sigma w_{2}(z)}} w[2]*(z + 2*EllipticK(k))= tau*w*(z)+ sigma*w[2]*(z) Subscript[w, 2]*(z + 2*EllipticK[(k)^2])= \[Tau]*w*(z)+ \[Sigma]*Subscript[w, 2]*(z) Failure Failure Skip Skip
29.8.E6 y = 1 k ′ ⁢ dn ⁡ ( z , k ) ⁢ dn ⁡ ( z 1 , k ) 𝑦 1 superscript 𝑘 ′ Jacobi-elliptic-dn 𝑧 𝑘 Jacobi-elliptic-dn subscript 𝑧 1 𝑘 {\displaystyle{\displaystyle y=\frac{1}{k^{\prime}}\operatorname{dn}\left(z,k% \right)\operatorname{dn}\left(z_{1},k\right)}} y =(1)/(sqrt(1 - (k)^(2)))*JacobiDN(z, k)*JacobiDN(z[1], k) y =Divide[1,Sqrt[1 - (k)^(2)]]*JacobiDN[z, (k)^2]*JacobiDN[Subscript[z, 1], (k)^2] Failure Failure
Fail
Float(infinity)+Float(infinity)*I <- {z = 2^(1/2)+I*2^(1/2), z[1] = 2^(1/2)+I*2^(1/2), k = 1, y = 1}
Float(infinity)+Float(infinity)*I <- {z = 2^(1/2)+I*2^(1/2), z[1] = 2^(1/2)+I*2^(1/2), k = 1, y = 2}
Float(infinity)+Float(infinity)*I <- {z = 2^(1/2)+I*2^(1/2), z[1] = 2^(1/2)+I*2^(1/2), k = 1, y = 3}
3.937738242+.2897798728*I <- {z = 2^(1/2)+I*2^(1/2), z[1] = 2^(1/2)+I*2^(1/2), k = 2, y = 1}
... skip entries to safe data
Fail
DirectedInfinity[] <- {Rule[k, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 1], Rule[y, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
29.11.E1 d 2 w d z 2 + ( h - ν ⁢ ( ν + 1 ) ⁢ k 2 ⁢ sn 2 ⁡ ( z , k ) + k 2 ⁢ ω 2 ⁢ sn 4 ⁡ ( z , k ) ) ⁢ w = 0 derivative 𝑤 𝑧 2 ℎ 𝜈 𝜈 1 superscript 𝑘 2 Jacobi-elliptic-sn 2 𝑧 𝑘 superscript 𝑘 2 superscript 𝜔 2 Jacobi-elliptic-sn 4 𝑧 𝑘 𝑤 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+(h-\nu% (\nu+1)k^{2}{\operatorname{sn}^{2}}\left(z,k\right)+k^{2}\omega^{2}{% \operatorname{sn}^{4}}\left(z,k\right))w=0}} diff(w, [z$(2)])+(h - nu*(nu + 1)*(k)^(2)* (JacobiSN(z, k))^(2)+ (k)^(2)* (omega)^(2)* (JacobiSN(z, k))^(4))* w = 0 D[w, {z, 2}]+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (JacobiSN[z, (k)^2])^(2)+ (k)^(2)* (\[Omega])^(2)* (JacobiSN[z, (k)^2])^(4))* w = 0 Failure Failure Skip Skip
29.14.E3 w ⁢ ( s , t ) = sn 2 ⁡ ( K ⁡ + i ⁢ t , k ) - sn 2 ⁡ ( s , k ) 𝑤 𝑠 𝑡 Jacobi-elliptic-sn 2 complete-elliptic-integral-first-kind-K 𝑘 imaginary-unit 𝑡 𝑘 Jacobi-elliptic-sn 2 𝑠 𝑘 {\displaystyle{\displaystyle w(s,t)={\operatorname{sn}^{2}}\left(\!K\!+\mathrm% {i}t,k\right)-{\operatorname{sn}^{2}}\left(s,k\right)}} w*(s , t)= (JacobiSN(EllipticK(k)+ I*t, k))^(2)- (JacobiSN(s, k))^(2) w*(s , t)= (JacobiSN[EllipticK[(k)^2]+ I*t, (k)^2])^(2)- (JacobiSN[s, (k)^2])^(2) Failure Failure Error Error
29.17.E1 F ⁢ ( z ) = E ⁢ ( z ) ⁢ ∫ i ⁢ K ′ ⁡ z d u ( E ⁢ ( u ) ) 2 𝐹 𝑧 𝐸 𝑧 superscript subscript imaginary-unit complementary-complete-elliptic-integral-first-kind-K 𝑘 𝑧 𝑢 superscript 𝐸 𝑢 2 {\displaystyle{\displaystyle F(z)=E(z)\int_{\mathrm{i}\!{K^{\prime}}\!}^{z}% \frac{\mathrm{d}u}{(E(u))^{2}}}} F*(z)= E*(z)* int((1)/((E*(u))^(2)), u = I*EllipticCK(k)..z) F*(z)= E*(z)* Integrate[Divide[1,(E*(u))^(2)], {u, I*EllipticK[1-(k)^2], z}] Failure Failure Skip
Fail
Complex[0.03667157790603448, 4.331207863265408] <- {Rule[F, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.11462120344581403, 4.482500644617256] <- {Rule[F, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.2493650915526525, 4.6172445327240945] <- {Rule[F, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[4.69908730443685, 0.33120786326540785] <- {Rule[F, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
29.18#Ex1 x = k ⁢ r ⁢ sn ⁡ ( β , k ) ⁢ sn ⁡ ( γ , k ) 𝑥 𝑘 𝑟 Jacobi-elliptic-sn 𝛽 𝑘 Jacobi-elliptic-sn 𝛾 𝑘 {\displaystyle{\displaystyle x=kr\operatorname{sn}\left(\beta,k\right)% \operatorname{sn}\left(\gamma,k\right)}} x = k*r*JacobiSN(beta, k)*JacobiSN(gamma, k) x = k*r*JacobiSN[\[Beta], (k)^2]*JacobiSN[\[Gamma], (k)^2] Failure Failure
Fail
.2066406870-.8535459454*I <- {beta = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), k = 1, x = 1}
1.206640687-.8535459454*I <- {beta = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), k = 1, x = 2}
2.206640687-.8535459454*I <- {beta = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), k = 1, x = 3}
.9014522664-2.018283670*I <- {beta = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), k = 2, x = 1}
... skip entries to safe data
 Skip
29.18#Ex2 y = i ⁢ k k ′ ⁢ r ⁢ cn ⁡ ( β , k ) ⁢ cn ⁡ ( γ , k ) 𝑦 imaginary-unit 𝑘 superscript 𝑘 ′ 𝑟 Jacobi-elliptic-cn 𝛽 𝑘 Jacobi-elliptic-cn 𝛾 𝑘 {\displaystyle{\displaystyle y=\mathrm{i}\frac{k}{k^{\prime}}r\operatorname{cn% }\left(\beta,k\right)\operatorname{cn}\left(\gamma,k\right)}} y = I*(k)/(sqrt(1 - (k)^(2)))*r*JacobiCN(beta, k)*JacobiCN(gamma, k) y = I*Divide[k,Sqrt[1 - (k)^(2)]]*r*JacobiCN[\[Beta], (k)^2]*JacobiCN[\[Gamma], (k)^2] Failure Failure
Fail
Float(infinity)+Float(infinity)*I <- {beta = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), k = 1, y = 1}
Float(infinity)+Float(infinity)*I <- {beta = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), k = 1, y = 2}
Float(infinity)+Float(infinity)*I <- {beta = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), k = 1, y = 3}
3.451665915+.7621257589*I <- {beta = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), k = 2, y = 1}
... skip entries to safe data
 Skip
29.18#Ex3 z = 1 k ′ ⁢ r ⁢ dn ⁡ ( β , k ) ⁢ dn ⁡ ( γ , k ) 𝑧 1 superscript 𝑘 ′ 𝑟 Jacobi-elliptic-dn 𝛽 𝑘 Jacobi-elliptic-dn 𝛾 𝑘 {\displaystyle{\displaystyle z=\frac{1}{k^{\prime}}r\operatorname{dn}\left(% \beta,k\right)\operatorname{dn}\left(\gamma,k\right)}} z =(1)/(sqrt(1 - (k)^(2)))*r*JacobiDN(beta, k)*JacobiDN(gamma, k) z =Divide[1,Sqrt[1 - (k)^(2)]]*r*JacobiDN[\[Beta], (k)^2]*JacobiDN[\[Gamma], (k)^2] Failure Failure
Fail
Float(infinity)+Float(infinity)*I <- {beta = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), k = 1}
1.356638323+2.584421950*I <- {beta = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), k = 2}
1.490898086+1.415864964*I <- {beta = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), k = 3}
Float(infinity)+Float(infinity)*I <- {beta = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
 Skip
29.18#Ex7 0 ≤ γ 0 𝛾 {\displaystyle{\displaystyle 0<=\gamma}} 0 < = gamma 0 < = \[Gamma] Failure Failure Successful Successful
29.18#Ex7 γ ≤ 4 ⁢ K ⁡ 𝛾 4 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle\gamma<=4\!K\!}} gamma < = 4*EllipticK(k) \[Gamma]< = 4*EllipticK[(k)^2] Failure Failure Error Successful
29.18.E5 d d r ⁡ ( r 2 ⁢ d u 1 d r ) + ( ω 2 ⁢ r 2 - ν ⁢ ( ν + 1 ) ) ⁢ u 1 = 0 derivative 𝑟 superscript 𝑟 2 derivative subscript 𝑢 1 𝑟 superscript 𝜔 2 superscript 𝑟 2 𝜈 𝜈 1 subscript 𝑢 1 0 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}r}\left(r^{2}\frac{% \mathrm{d}u_{1}}{\mathrm{d}r}\right)+(\omega^{2}r^{2}-\nu(\nu+1))u_{1}=0}} diff(((r)^(2)* diff(u[1], r))+((omega)^(2)* (r)^(2)- nu*(nu + 1))* u[1], r)= 0 D[((r)^(2)* D[Subscript[u, 1], r])+((\[Omega])^(2)* (r)^(2)- \[Nu]*(\[Nu]+ 1))* Subscript[u, 1], r]= 0 Failure Failure
Fail
-31.99999996+0.*I <- {omega = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), u[1] = 2^(1/2)+I*2^(1/2)}
0.+31.99999996*I <- {omega = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), u[1] = 2^(1/2)-I*2^(1/2)}
31.99999996+0.*I <- {omega = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), u[1] = -2^(1/2)-I*2^(1/2)}
0.-31.99999996*I <- {omega = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), u[1] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
-32.0 <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ω, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[u, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 32.0] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ω, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[u, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
32.0 <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ω, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[u, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -32.0] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ω, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[u, 1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
29.18.E6 d 2 u 2 d β 2 + ( h - ν ⁢ ( ν + 1 ) ⁢ k 2 ⁢ sn 2 ⁡ ( β , k ) ) ⁢ u 2 = 0 derivative subscript 𝑢 2 𝛽 2 ℎ 𝜈 𝜈 1 superscript 𝑘 2 Jacobi-elliptic-sn 2 𝛽 𝑘 subscript 𝑢 2 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}u_{2}}{{\mathrm{d}\beta}^{2}% }+(h-\nu(\nu+1)k^{2}{\operatorname{sn}^{2}}\left(\beta,k\right))u_{2}=0}} diff(u[2], [beta$(2)])+(h - nu*(nu + 1)*(k)^(2)* (JacobiSN(beta, k))^(2))* u[2]= 0 D[Subscript[u, 2], {\[Beta], 2}]+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (JacobiSN[\[Beta], (k)^2])^(2))* Subscript[u, 2]= 0 Failure Failure
Fail
7.948739768-7.545139452*I <- {beta = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), u[2] = 2^(1/2)+I*2^(1/2), k = 1}
51.95468921+27.97388965*I <- {beta = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), u[2] = 2^(1/2)+I*2^(1/2), k = 2}
-3.910915794-11.82960931*I <- {beta = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), u[2] = 2^(1/2)+I*2^(1/2), k = 3}
-7.545139452-7.948739768*I <- {beta = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), u[2] = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
 Skip
29.18.E7 d 2 u 3 d γ 2 + ( h - ν ⁢ ( ν + 1 ) ⁢ k 2 ⁢ sn 2 ⁡ ( γ , k ) ) ⁢ u 3 = 0 derivative subscript 𝑢 3 𝛾 2 ℎ 𝜈 𝜈 1 superscript 𝑘 2 Jacobi-elliptic-sn 2 𝛾 𝑘 subscript 𝑢 3 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}u_{3}}{{\mathrm{d}\gamma}^{2% }}+(h-\nu(\nu+1)k^{2}{\operatorname{sn}^{2}}\left(\gamma,k\right))u_{3}=0}} diff(u[3], [gamma$(2)])+(h - nu*(nu + 1)*(k)^(2)* (JacobiSN(gamma, k))^(2))* u[3]= 0 D[Subscript[u, 3], {\[Gamma], 2}]+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (JacobiSN[\[Gamma], (k)^2])^(2))* Subscript[u, 3]= 0 Error Failure - Skip
29.18#Ex8 x = k ⁢ sn ⁡ ( α , k ) ⁢ sn ⁡ ( β , k ) ⁢ sn ⁡ ( γ , k ) 𝑥 𝑘 Jacobi-elliptic-sn 𝛼 𝑘 Jacobi-elliptic-sn 𝛽 𝑘 Jacobi-elliptic-sn 𝛾 𝑘 {\displaystyle{\displaystyle x=k\operatorname{sn}\left(\alpha,k\right)% \operatorname{sn}\left(\beta,k\right)\operatorname{sn}\left(\gamma,k\right)}} x = k*JacobiSN(alpha, k)*JacobiSN(beta, k)*JacobiSN(gamma, k) x = k*JacobiSN[\[Alpha], (k)^2]*JacobiSN[\[Beta], (k)^2]*JacobiSN[\[Gamma], (k)^2] Failure Failure
Fail
.3496751872-.4759618703e-1*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), k = 1, x = 1}
1.349675187-.4759618703e-1*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), k = 1, x = 2}
2.349675187-.4759618703e-1*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), k = 1, x = 3}
.8887187920-1.136817507*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), k = 2, x = 1}
... skip entries to safe data
 Skip
29.18#Ex9 y = - k k ′ ⁢ cn ⁡ ( α , k ) ⁢ cn ⁡ ( β , k ) ⁢ cn ⁡ ( γ , k ) 𝑦 𝑘 superscript 𝑘 ′ Jacobi-elliptic-cn 𝛼 𝑘 Jacobi-elliptic-cn 𝛽 𝑘 Jacobi-elliptic-cn 𝛾 𝑘 {\displaystyle{\displaystyle y=-\frac{k}{k^{\prime}}\operatorname{cn}\left(% \alpha,k\right)\operatorname{cn}\left(\beta,k\right)\operatorname{cn}\left(% \gamma,k\right)}} y = -(k)/(sqrt(1 - (k)^(2)))*JacobiCN(alpha, k)*JacobiCN(beta, k)*JacobiCN(gamma, k) y = -Divide[k,Sqrt[1 - (k)^(2)]]*JacobiCN[\[Alpha], (k)^2]*JacobiCN[\[Beta], (k)^2]*JacobiCN[\[Gamma], (k)^2] Failure Failure
Fail
Float(infinity)+Float(infinity)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), k = 1, y = 1}
Float(infinity)+Float(infinity)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), k = 1, y = 2}
Float(infinity)+Float(infinity)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), k = 1, y = 3}
-.314074508-.9043815133*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), k = 2, y = 1}
... skip entries to safe data
 Skip
29.18#Ex10 z = i k ⁢ k ′ ⁢ dn ⁡ ( α , k ) ⁢ dn ⁡ ( β , k ) ⁢ dn ⁡ ( γ , k ) 𝑧 imaginary-unit 𝑘 superscript 𝑘 ′ Jacobi-elliptic-dn 𝛼 𝑘 Jacobi-elliptic-dn 𝛽 𝑘 Jacobi-elliptic-dn 𝛾 𝑘 {\displaystyle{\displaystyle z=\frac{\mathrm{i}}{kk^{\prime}}\operatorname{dn}% \left(\alpha,k\right)\operatorname{dn}\left(\beta,k\right)\operatorname{dn}% \left(\gamma,k\right)}} z =(I)/(k*sqrt(1 - (k)^(2)))*JacobiDN(alpha, k)*JacobiDN(beta, k)*JacobiDN(gamma, k) z =Divide[I,k*Sqrt[1 - (k)^(2)]]*JacobiDN[\[Alpha], (k)^2]*JacobiDN[\[Beta], (k)^2]*JacobiDN[\[Gamma], (k)^2] Failure Failure
Fail
Float(infinity)+Float(infinity)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), k = 1}
1.349197762+2.073332495*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), k = 2}
1.413658462+1.427095902*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), k = 3}
Float(infinity)+Float(infinity)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
 Skip
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