Parabolic Cylinder Functions - 12.10 Uniform Asymptotic Expansions for Large Parameter

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
12.10#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle a = +\tfrac{1}{2}\mu^{2}}
a = +\tfrac{1}{2}\mu^{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
a = +(1)/(2)*(mu)^(2)
 
a == +Divide[1,2]*\[Mu]^(2)
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = \mu t\sqrt{2}}
x = \mu t\sqrt{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
x = mu*t*sqrt(2)
 
x == \[Mu]*t*Sqrt[2]
 Skipped - no semantic math Skipped - no semantic math - -
12.10.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv[2]{w}{t} = \mu^{4}(t^{2}+ 1)w}
\deriv[2]{w}{t} = \mu^{4}(t^{2}+ 1)w		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff(w, [t$(2)]) = (mu)^(4)*((t)^(2)+ 1)*w

D[w, {t, 2}] == \[Mu]^(4)*((t)^(2)+ 1)*w
Failure Failure
Failed [300 / 300]
Result: 2.814582564-1.625000003*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = 1/2*3^(1/2)+1/2*I}


Result: 1.625000003+2.814582564*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[2.814582562299425, -1.6250000000000009]
Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.814582562299425, -1.6250000000000009]
Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
12.10.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv[2]{w}{t} = \mu^{4}(t^{2}- 1)w}
\deriv[2]{w}{t} = \mu^{4}(t^{2}- 1)w		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff(w, [t$(2)]) = (mu)^(4)*((t)^(2)- 1)*w

D[w, {t, 2}] == \[Mu]^(4)*((t)^(2)- 1)*w
Failure Failure
Failed [300 / 300]
Result: 1.082531755-.6250000011*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = 1/2*3^(1/2)+1/2*I}


Result: .6250000011+1.082531755*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.0825317547305482, -0.6250000000000002]
Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.0825317547305482, -0.6250000000000002]
Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
12.10.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle v_{s}(t) = u_{s}(t)+\tfrac{1}{2}tu_{s-1}(t)-r_{s-2}(t)}
v_{s}(t) = u_{s}(t)+\tfrac{1}{2}tu_{s-1}(t)-r_{s-2}(t)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
v[s](t) = u[s](t)+(1)/(2)*tu[s - 1](t)- r[s - 2](t)
 
Subscript[v, s][t] == Subscript[u, s][t]+Divide[1,2]*Subscript[tu, s - 1][t]- Subscript[r, s - 2][t]
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \gamma_{0} = 1}
\gamma_{0} = 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
gamma[0] = 1
 
Subscript[\[Gamma], 0] == 1
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \gamma_{1} = -\tfrac{1}{24}}
\gamma_{1} = -\tfrac{1}{24}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
gamma[1] = -(1)/(24)
 
Subscript[\[Gamma], 1] == -Divide[1,24]
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \gamma_{2} = \tfrac{1}{1152}}
\gamma_{2} = \tfrac{1}{1152}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
gamma[2] = (1)/(1152)
 
Subscript[\[Gamma], 2] == Divide[1,1152]
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \gamma_{3} = \tfrac{1003}{4\;14720}}
\gamma_{3} = \tfrac{1003}{4\;14720}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
gamma[3] = (1003)/(414720)
 
Subscript[\[Gamma], 3] == Divide[1003,414720]
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \gamma_{4} = -\tfrac{4027}{398\;13120}}
\gamma_{4} = -\tfrac{4027}{398\;13120}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
gamma[4] = -(4027)/(39813120)
 
Subscript[\[Gamma], 4] == -Divide[4027,39813120]
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex18 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mathsf{A}_{1}(\tau) = -\tfrac{1}{12}\tau(20\tau^{2}+30\tau+9)}
\mathsf{A}_{1}(\tau) = -\tfrac{1}{12}\tau(20\tau^{2}+30\tau+9)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A[1]*(((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))) = -(1)/(12)*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))*(20*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(2)+ 30*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))+ 9)
 
Subscript[A, 1]*((Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))) == -Divide[1,12]*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))*(20*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(2)+ 30*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))+ 9)
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex19 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mathsf{A}_{2}(\tau) = \tfrac{1}{288}\tau^{2}(6160\tau^{4}+18480\tau^{3}+19404\tau^{2}+8028\tau+945)}
\mathsf{A}_{2}(\tau) = \tfrac{1}{288}\tau^{2}(6160\tau^{4}+18480\tau^{3}+19404\tau^{2}+8028\tau+945)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A[2]*(((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))) = (1)/(288)*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(2)*(6160*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(4)+ 18480*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(3)+ 19404*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(2)+ 8028*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))+ 945)
 
Subscript[A, 2]*((Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))) == Divide[1,288]*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(2)*(6160*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(4)+ 18480*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(3)+ 19404*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(2)+ 8028*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))+ 945)
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex22 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{s}(\zeta) = \zeta^{-3s}\sum_{m=0}^{2s}\beta_{m}(\phi(\zeta))^{6(2s-m)}u_{2s-m}(t)}
A_{s}(\zeta) = \zeta^{-3s}\sum_{m=0}^{2s}\beta_{m}(\phi(\zeta))^{6(2s-m)}u_{2s-m}(t)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A[s](zeta) = (zeta)^(- 3*s)* sum((-(6*m + 1)/(6*m - 1)*alpha[m])*((((zeta)/((t)^(2)- 1))^((1)/(4))))^(6*(2*s - m))* u[2*s - m](t), m = 0..2*s)
 
Subscript[A, s][\[Zeta]] == \[Zeta]^(- 3*s)* Sum[(-Divide[6*m + 1,6*m - 1]*Subscript[\[Alpha], m])*(((Divide[\[Zeta],(t)^(2)- 1])^(Divide[1,4])))^(6*(2*s - m))* Subscript[u, 2*s - m][t], {m, 0, 2*s}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex23 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \zeta^{2}B_{s}(\zeta) = -\zeta^{-3s}\sum_{m=0}^{2s+1}\alpha_{m}(\phi(\zeta))^{6(2s-m+1)}u_{2s-m+1}(t)}
\zeta^{2}B_{s}(\zeta) = -\zeta^{-3s}\sum_{m=0}^{2s+1}\alpha_{m}(\phi(\zeta))^{6(2s-m+1)}u_{2s-m+1}(t)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(zeta)^(2)* B[s](zeta) = - (zeta)^(- 3*s)* sum((alpha[m]((((zeta)/((t)^(2)- 1))^((1)/(4)))))^(6*(2*s - m + 1))* u[2*s - m + 1](t), m = 0..2*s + 1)
 
\[Zeta]^(2)* Subscript[B, s][\[Zeta]] == - \[Zeta]^(- 3*s)* Sum[(Subscript[\[Alpha], m][((Divide[\[Zeta],(t)^(2)- 1])^(Divide[1,4]))])^(6*(2*s - m + 1))* Subscript[u, 2*s - m + 1][t], {m, 0, 2*s + 1}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex28 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \zeta C_{s}(\zeta) = -\zeta^{-3s}\sum_{m=0}^{2s+1}\beta_{m}(\phi(\zeta))^{6(2s-m+1)}v_{2s-m+1}(t)}
\zeta C_{s}(\zeta) = -\zeta^{-3s}\sum_{m=0}^{2s+1}\beta_{m}(\phi(\zeta))^{6(2s-m+1)}v_{2s-m+1}(t)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
zeta*C[s](zeta) = - (zeta)^(- 3*s)* sum((-(6*m + 1)/(6*m - 1)*alpha[m])*((((zeta)/((t)^(2)- 1))^((1)/(4))))^(6*(2*s - m + 1))* v[2*s - m + 1](t), m = 0..2*s + 1)
 
\[Zeta]*Subscript[C, s][\[Zeta]] == - \[Zeta]^(- 3*s)* Sum[(-Divide[6*m + 1,6*m - 1]*Subscript[\[Alpha], m])*(((Divide[\[Zeta],(t)^(2)- 1])^(Divide[1,4])))^(6*(2*s - m + 1))* Subscript[v, 2*s - m + 1][t], {m, 0, 2*s + 1}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex29 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle D_{s}(\zeta) = \zeta^{-3s}\sum_{m=0}^{2s}\alpha_{m}(\phi(\zeta))^{6(2s-m)}v_{2s-m}(t)}
D_{s}(\zeta) = \zeta^{-3s}\sum_{m=0}^{2s}\alpha_{m}(\phi(\zeta))^{6(2s-m)}v_{2s-m}(t)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
D[s](zeta) = (zeta)^(- 3*s)* sum((alpha[m]((((zeta)/((t)^(2)- 1))^((1)/(4)))))^(6*(2*s - m))* v[2*s - m](t), m = 0..2*s)
 
Subscript[D, s][\[Zeta]] == \[Zeta]^(- 3*s)* Sum[(Subscript[\[Alpha], m][((Divide[\[Zeta],(t)^(2)- 1])^(Divide[1,4]))])^(6*(2*s - m))* Subscript[v, 2*s - m][t], {m, 0, 2*s}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
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