Airy and Related Functions - 9.7 Asymptotic Expansions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
9.7#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle v_{k} = \frac{6k+1}{1-6k}u_{k}}
v_{k} = \frac{6k+1}{1-6k}u_{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
v[k] = (6*k + 1)/(1 - 6*k)*u[k]
 
Subscript[v, k] == Divide[6*k + 1,1 - 6*k]*Subscript[u, k]
 Skipped - no semantic math Skipped - no semantic math - -
9.7#Ex3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \AiryAi@{x} \leq \frac{e^{-\xi}}{2\sqrt{\pi}x^{1/4}}}
\AiryAi@{x} \leq \frac{e^{-\xi}}{2\sqrt{\pi}x^{1/4}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
AiryAi(x) <= (exp(- xi))/(2*sqrt(Pi)*(x)^(1/4))

AiryAi[x] <= Divide[Exp[- \[Xi]],2*Sqrt[Pi]*(x)^(1/4)]
Failure Failure
Failed [6 / 30]
Result: .7174949701e-1 <= .5687619445e-1
Test Values: {x = 1.5, xi = 1.5}


Result: .7174949701e-1 <= .3449715573e-1
Test Values: {x = 1.5, xi = 2}

... skip entries to safe data
Failed [18 / 30]
Result: LessEqual[0.07174949700810541, Complex[0.09409145494595897, -0.05140239611000222]]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: LessEqual[0.07174949700810541, Complex[0.2722702697816324, -0.3201383097997893]]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
9.7#Ex4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\AiryAi'@{x}| \leq \frac{x^{1/4}e^{-\xi}}{2\sqrt{\pi}}\left(1+\frac{7}{72\xi}\right)}
|\AiryAi'@{x}| \leq \frac{x^{1/4}e^{-\xi}}{2\sqrt{\pi}}\left(1+\frac{7}{72\xi}\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
abs(diff( AiryAi(x), x$(1) )) <= ((x)^(1/4)* exp(- xi))/(2*sqrt(Pi))*(1 +(7)/(72*xi))

Abs[D[AiryAi[x], {x, 1}]] <= Divide[(x)^(1/4)* Exp[- \[Xi]],2*Sqrt[Pi]]*(1 +Divide[7,72*\[Xi]])
Failure Failure
Failed [6 / 30]
Result: .9738201284e-1 <= .7417375145e-1
Test Values: {x = 1.5, xi = 1.5}


Result: .9738201284e-1 <= .4430404444e-1
Test Values: {x = 1.5, xi = 2}

... skip entries to safe data
Failed [18 / 30]
Result: LessEqual[0.0973820128423013, Complex[0.12188040935651077, -0.07385727120906739]]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: LessEqual[0.0973820128423013, Complex[0.2842390979283623, -0.4011043708701865]]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
9.7#Ex5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \AiryBi@{x} \leq \frac{\expe^{\xi}}{\sqrt{\cpi}x^{1/4}}\left(1+\left(\chi(\tfrac{7}{6})+1\right)\frac{5}{72\xi}\right)}
\AiryBi@{x} \leq \frac{\expe^{\xi}}{\sqrt{\cpi}x^{1/4}}\left(1+\left(\chi(\tfrac{7}{6})+1\right)\frac{5}{72\xi}\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
AiryBi(x) <= (exp(xi))/(sqrt(Pi)*(x)^(1/4))*(1 +(chi((7)/(6))+ 1)*(5)/(72*xi))

AiryBi[x] <= Divide[Exp[\[Xi]],Sqrt[Pi]*(x)^(1/4)]*(1 +(\[Chi][Divide[7,6]]+ 1)*Divide[5,72*\[Xi]])
Failure Failure
Failed [72 / 300]
Result: 1.878941504 <= .1177021246
Test Values: {chi = -1.5, x = 1.5, xi = -1.5}


Result: 1.878941504 <= .3414205364
Test Values: {chi = -1.5, x = 1.5, xi = -.5}

... skip entries to safe data
Failed [264 / 300]
Result: LessEqual[1.8789415037478943, Complex[1.2339745412017042, 0.6261685922908564]]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: LessEqual[1.8789415037478943, Complex[1.1007209136525415, 0.6652662717134934]]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[χ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
9.7#Ex6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \AiryBi'@{x} \leq \frac{x^{1/4}e^{\xi}}{\sqrt{\pi}}\left(1+\left(\frac{\cpi}{2}+1\right)\frac{7}{72\xi}\right)}
\AiryBi'@{x} \leq \frac{x^{1/4}e^{\xi}}{\sqrt{\pi}}\left(1+\left(\frac{\cpi}{2}+1\right)\frac{7}{72\xi}\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff( AiryBi(x), x$(1) ) <= ((x)^(1/4)* exp(xi))/(sqrt(Pi))*(1 +((Pi)/(2)+ 1)*(7)/(72*xi))

D[AiryBi[x], {x, 1}] <= Divide[(x)^(1/4)* Exp[\[Xi]],Sqrt[Pi]]*(1 +(Divide[Pi,2]+ 1)*Divide[7,72*\[Xi]])
Failure Failure
Failed [12 / 30]
Result: 1.886212255 <= .1161037548
Test Values: {x = 1.5, xi = -1.5}


Result: 1.886212255 <= .1893988815
Test Values: {x = 1.5, xi = -.5}

... skip entries to safe data
Failed [24 / 30]
Result: LessEqual[1.8862122548481652, Complex[1.673615932813246, 0.7029160379352533]]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: LessEqual[1.8862122548481652, Complex[0.2771293070670663, 0.19932445926872913]]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
9.7.E18 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \AiryAi@{z} = \frac{e^{-\zeta}}{2\sqrt{\pi}z^{1/4}}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{u_{k}}{\zeta^{k}}+R_{n}(z)\right)}
\AiryAi@{z} = \frac{e^{-\zeta}}{2\sqrt{\pi}z^{1/4}}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{u_{k}}{\zeta^{k}}+R_{n}(z)\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
AiryAi(z) = (exp(-(2)/(3)*(z)^((3)/(2))))/(2*sqrt(Pi)*(z)^(1/4))*(sum((- 1)^(k)*(u[k])/((2)/(3)*((z)^((3)/(2)))^(k)), k = 0..n - 1)+ R[n](z))

AiryAi[z] == Divide[Exp[-Divide[2,3]*(z)^(Divide[3,2])],2*Sqrt[Pi]*(z)^(1/4)]*(Sum[(- 1)^(k)*Divide[Subscript[u, k],Divide[2,3]*((z)^(Divide[3,2]))^(k)], {k, 0, n - 1}, GenerateConditions->None]+ Subscript[R, n][z])
Failure Failure
Failed [300 / 300]
Result: -.2820623089-.1436891013*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, u[k] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -.1105804716-.3445358858*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, u[k] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
 Skipped - Because timed out
9.7.E19 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \AiryAi'@{z} = -\frac{z^{1/4}e^{-\zeta}}{2\sqrt{\pi}}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{v_{k}}{\zeta^{k}}+S_{n}(z)\right)}
\AiryAi'@{z} = -\frac{z^{1/4}e^{-\zeta}}{2\sqrt{\pi}}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{v_{k}}{\zeta^{k}}+S_{n}(z)\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff( AiryAi(z), z$(1) ) = -((z)^(1/4)* exp(-(2)/(3)*(z)^((3)/(2))))/(2*sqrt(Pi))*(sum((- 1)^(k)*(v[k])/((2)/(3)*((z)^((3)/(2)))^(k)), k = 0..n - 1)+ S[n](z))

D[AiryAi[z], {z, 1}] == -Divide[(z)^(1/4)* Exp[-Divide[2,3]*(z)^(Divide[3,2])],2*Sqrt[Pi]]*(Sum[(- 1)^(k)*Divide[Subscript[v, k],Divide[2,3]*((z)^(Divide[3,2]))^(k)], {k, 0, n - 1}, GenerateConditions->None]+ Subscript[S, n][z])
Failure Failure
Failed [300 / 300]
Result: .2167391543+.2356234249*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, S[n] = 1/2*3^(1/2)+1/2*I, v[k] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -.8825541e-3+.3852437556*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, S[n] = 1/2*3^(1/2)+1/2*I, v[k] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
 Skipped - Because timed out
9.7.E22 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \scterminant{p}@{z} = \frac{e^{z}}{2\pi}\EulerGamma@{p}\incGamma@{1-p}{z}}
\scterminant{p}@{z} = \frac{e^{z}}{2\pi}\EulerGamma@{p}\incGamma@{1-p}{z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{p} > 0} 
(exp(z)/(2*Pi))*GAMMA(p)*GAMMA(1-p,z) = (exp(z))/(2*Pi)*GAMMA(p)*GAMMA(1 - p, z)

Error
Successful Missing Macro Error - -
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