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 a = + 1 2 ⁒ ΞΌ 2 π‘Ž 1 2 superscript πœ‡ 2 {\displaystyle{\displaystyle a=+\tfrac{1}{2}\mu^{2}}}
a = +\tfrac{1}{2}\mu^{2}		 

a = +(1)/(2)*(mu)^(2)
 
a == +Divide[1,2]*\[Mu]^(2)
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex2 x = ΞΌ ⁒ t ⁒ 2 π‘₯ πœ‡ 𝑑 2 {\displaystyle{\displaystyle x=\mu t\sqrt{2}}}
x = \mu t\sqrt{2}		 

x = mu*t*sqrt(2)
 
x == \[Mu]*t*Sqrt[2]
 Skipped - no semantic math Skipped - no semantic math - -
12.10.E2 d 2 w d t 2 = ΞΌ 4 ⁒ ( t 2 + 1 ) ⁒ w derivative 𝑀 𝑑 2 superscript πœ‡ 4 superscript 𝑑 2 1 𝑀 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=\mu^{4% }(t^{2}+1)w}}
\deriv[2]{w}{t} = \mu^{4}(t^{2}+ 1)w		 

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[ΞΌ, 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[ΞΌ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
12.10.E2 d 2 w d t 2 = ΞΌ 4 ⁒ ( t 2 - 1 ) ⁒ w derivative 𝑀 𝑑 2 superscript πœ‡ 4 superscript 𝑑 2 1 𝑀 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=\mu^{4% }(t^{2}-1)w}}
\deriv[2]{w}{t} = \mu^{4}(t^{2}- 1)w		 

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[ΞΌ, 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[ΞΌ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
12.10.E13 v s ⁒ ( t ) = u s ⁒ ( t ) + 1 2 ⁒ t ⁒ u s - 1 ⁒ ( t ) - r s - 2 ⁒ ( t ) subscript 𝑣 𝑠 𝑑 subscript 𝑒 𝑠 𝑑 1 2 𝑑 subscript 𝑒 𝑠 1 𝑑 subscript π‘Ÿ 𝑠 2 𝑑 {\displaystyle{\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)		 

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 Ξ³ 0 = 1 subscript 𝛾 0 1 {\displaystyle{\displaystyle\gamma_{0}=1}}
\gamma_{0} = 1		 

gamma[0] = 1
 
Subscript[\[Gamma], 0] == 1
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex10 Ξ³ 1 = - 1 24 subscript 𝛾 1 1 24 {\displaystyle{\displaystyle\gamma_{1}=-\tfrac{1}{24}}}
\gamma_{1} = -\tfrac{1}{24}		 

gamma[1] = -(1)/(24)
 
Subscript[\[Gamma], 1] == -Divide[1,24]
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex11 Ξ³ 2 = 1 1152 subscript 𝛾 2 1 1152 {\displaystyle{\displaystyle\gamma_{2}=\tfrac{1}{1152}}}
\gamma_{2} = \tfrac{1}{1152}		 

gamma[2] = (1)/(1152)
 
Subscript[\[Gamma], 2] == Divide[1,1152]
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex12 Ξ³ 3 = 1003 4 14720 subscript 𝛾 3 1003 4 14720 {\displaystyle{\displaystyle\gamma_{3}=\tfrac{1003}{4\;14720}}}
\gamma_{3} = \tfrac{1003}{4\;14720}		 

gamma[3] = (1003)/(414720)
 
Subscript[\[Gamma], 3] == Divide[1003,414720]
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex13 Ξ³ 4 = - 4027 398 13120 subscript 𝛾 4 4027 398 13120 {\displaystyle{\displaystyle\gamma_{4}=-\tfrac{4027}{398\;13120}}}
\gamma_{4} = -\tfrac{4027}{398\;13120}		 

gamma[4] = -(4027)/(39813120)
 
Subscript[\[Gamma], 4] == -Divide[4027,39813120]
 Skipped - no semantic math Skipped - no semantic math - -
12.10#Ex18 𝖠 1 ⁒ ( Ο„ ) = - 1 12 ⁒ Ο„ ⁒ ( 20 ⁒ Ο„ 2 + 30 ⁒ Ο„ + 9 ) subscript 𝖠 1 𝜏 1 12 𝜏 20 superscript 𝜏 2 30 𝜏 9 {\displaystyle{\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)		 

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 𝖠 2 ⁒ ( Ο„ ) = 1 288 ⁒ Ο„ 2 ⁒ ( 6160 ⁒ Ο„ 4 + 18480 ⁒ Ο„ 3 + 19404 ⁒ Ο„ 2 + 8028 ⁒ Ο„ + 945 ) subscript 𝖠 2 𝜏 1 288 superscript 𝜏 2 6160 superscript 𝜏 4 18480 superscript 𝜏 3 19404 superscript 𝜏 2 8028 𝜏 945 {\displaystyle{\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)		 

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 A s ⁒ ( ΞΆ ) = ΞΆ - 3 ⁒ s ⁒ βˆ‘ m = 0 2 ⁒ s Ξ² m ⁒ ( Ο• ⁒ ( ΞΆ ) ) 6 ⁒ ( 2 ⁒ s - m ) ⁒ u 2 ⁒ s - m ⁒ ( t ) subscript 𝐴 𝑠 𝜁 superscript 𝜁 3 𝑠 superscript subscript π‘š 0 2 𝑠 subscript 𝛽 π‘š superscript italic-Ο• 𝜁 6 2 𝑠 π‘š subscript 𝑒 2 𝑠 π‘š 𝑑 {\displaystyle{\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)		 

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 ΞΆ 2 ⁒ B s ⁒ ( ΞΆ ) = - ΞΆ - 3 ⁒ s ⁒ βˆ‘ m = 0 2 ⁒ s + 1 Ξ± m ⁒ ( Ο• ⁒ ( ΞΆ ) ) 6 ⁒ ( 2 ⁒ s - m + 1 ) ⁒ u 2 ⁒ s - m + 1 ⁒ ( t ) superscript 𝜁 2 subscript 𝐡 𝑠 𝜁 superscript 𝜁 3 𝑠 superscript subscript π‘š 0 2 𝑠 1 subscript 𝛼 π‘š superscript italic-Ο• 𝜁 6 2 𝑠 π‘š 1 subscript 𝑒 2 𝑠 π‘š 1 𝑑 {\displaystyle{\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)		 

(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 ΞΆ ⁒ C s ⁒ ( ΞΆ ) = - ΞΆ - 3 ⁒ s ⁒ βˆ‘ m = 0 2 ⁒ s + 1 Ξ² m ⁒ ( Ο• ⁒ ( ΞΆ ) ) 6 ⁒ ( 2 ⁒ s - m + 1 ) ⁒ v 2 ⁒ s - m + 1 ⁒ ( t ) 𝜁 subscript 𝐢 𝑠 𝜁 superscript 𝜁 3 𝑠 superscript subscript π‘š 0 2 𝑠 1 subscript 𝛽 π‘š superscript italic-Ο• 𝜁 6 2 𝑠 π‘š 1 subscript 𝑣 2 𝑠 π‘š 1 𝑑 {\displaystyle{\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)		 

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 D s ⁒ ( ΞΆ ) = ΞΆ - 3 ⁒ s ⁒ βˆ‘ m = 0 2 ⁒ s Ξ± m ⁒ ( Ο• ⁒ ( ΞΆ ) ) 6 ⁒ ( 2 ⁒ s - m ) ⁒ v 2 ⁒ s - m ⁒ ( t ) subscript 𝐷 𝑠 𝜁 superscript 𝜁 3 𝑠 superscript subscript π‘š 0 2 𝑠 subscript 𝛼 π‘š superscript italic-Ο• 𝜁 6 2 𝑠 π‘š subscript 𝑣 2 𝑠 π‘š 𝑑 {\displaystyle{\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)		 

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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