Airy and Related Functions - 9.12 Scorer Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
9.12.E1 d 2 w d z 2 - z ⁒ w = 1 Ο€ derivative 𝑀 𝑧 2 𝑧 𝑀 1 πœ‹ {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-zw=% \frac{1}{\pi}}}
\deriv[2]{w}{z}-zw = \frac{1}{\pi}		 

diff(w, [z$(2)])- z*w = (1)/(Pi)

D[w, {z, 2}]- z*w == Divide[1,Pi]
Failure Failure
Failed [70 / 70]
Result: -.8183098865-.8660254040*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: .5477155179-.5000000004*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [70 / 70]
Result: Complex[-0.8183098861837907, -0.8660254037844386]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.5477155176006481, -0.49999999999999994]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
9.12.E4 Gi ⁑ ( z ) = Bi ⁑ ( z ) ⁒ ∫ z ∞ Ai ⁑ ( t ) ⁒ d t + Ai ⁑ ( z ) ⁒ ∫ 0 z Bi ⁑ ( t ) ⁒ d t Scorer-Gi 𝑧 Airy-Bi 𝑧 superscript subscript 𝑧 Airy-Ai 𝑑 𝑑 Airy-Ai 𝑧 superscript subscript 0 𝑧 Airy-Bi 𝑑 𝑑 {\displaystyle{\displaystyle\mathrm{Gi}\left(z\right)=\mathrm{Bi}\left(z\right% )\int_{z}^{\infty}\mathrm{Ai}\left(t\right)\mathrm{d}t+\mathrm{Ai}\left(z% \right)\int_{0}^{z}\mathrm{Bi}\left(t\right)\mathrm{d}t}}
\ScorerGi@{z} = \AiryBi@{z}\int_{z}^{\infty}\AiryAi@{t}\diff{t}+\AiryAi@{z}\int_{0}^{z}\AiryBi@{t}\diff{t}		 

AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))) = AiryBi(z)*int(AiryAi(t), t = z..infinity)+ AiryAi(z)*int(AiryBi(t), t = 0..z)

ScorerGi[z] == AiryBi[z]*Integrate[AiryAi[t], {t, z, Infinity}, GenerateConditions->None]+ AiryAi[z]*Integrate[AiryBi[t], {t, 0, z}, GenerateConditions->None]
Successful Failure - Successful [Tested: 7]
9.12.E5 Hi ⁑ ( z ) = Bi ⁑ ( z ) ⁒ ∫ - ∞ z Ai ⁑ ( t ) ⁒ d t - Ai ⁑ ( z ) ⁒ ∫ - ∞ z Bi ⁑ ( t ) ⁒ d t Scorer-Hi 𝑧 Airy-Bi 𝑧 superscript subscript 𝑧 Airy-Ai 𝑑 𝑑 Airy-Ai 𝑧 superscript subscript 𝑧 Airy-Bi 𝑑 𝑑 {\displaystyle{\displaystyle\mathrm{Hi}\left(z\right)=\mathrm{Bi}\left(z\right% )\int_{-\infty}^{z}\mathrm{Ai}\left(t\right)\mathrm{d}t-\mathrm{Ai}\left(z% \right)\int_{-\infty}^{z}\mathrm{Bi}\left(t\right)\mathrm{d}t}}
\ScorerHi@{z} = \AiryBi@{z}\int_{-\infty}^{z}\AiryAi@{t}\diff{t}-\AiryAi@{z}\int_{-\infty}^{z}\AiryBi@{t}\diff{t}		 

AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = AiryBi(z)*int(AiryAi(t), t = - infinity..z)- AiryAi(z)*int(AiryBi(t), t = - infinity..z)

ScorerHi[z] == AiryBi[z]*Integrate[AiryAi[t], {t, - Infinity, z}, GenerateConditions->None]- AiryAi[z]*Integrate[AiryBi[t], {t, - Infinity, z}, GenerateConditions->None]
Successful Failure - Successful [Tested: 7]
9.12.E6 Gi ⁑ ( 0 ) = 1 2 ⁒ Hi ⁑ ( 0 ) Scorer-Gi 0 1 2 Scorer-Hi 0 {\displaystyle{\displaystyle\mathrm{Gi}\left(0\right)=\tfrac{1}{2}\mathrm{Hi}% \left(0\right)}}
\ScorerGi@{0} = \tfrac{1}{2}\ScorerHi@{0}		 

AiryBi(0)*(int(AiryAi(t), t = (0) .. infinity))+AiryAi(0)*(int(AiryBi(t), t = 0 .. (0))) = (1)/(2)*AiryBi(0)*(int(AiryAi(t), t = -infinity .. (0)))-AiryAi(0)*(int(AiryBi(t), t = -infinity .. (0)))

ScorerGi[0] == Divide[1,2]*ScorerHi[0]
Failure Successful Skip - No test values generated Successful [Tested: 1]
9.12.E6 1 2 ⁒ Hi ⁑ ( 0 ) = 1 3 ⁒ Bi ⁑ ( 0 ) 1 2 Scorer-Hi 0 1 3 Airy-Bi 0 {\displaystyle{\displaystyle\tfrac{1}{2}\mathrm{Hi}\left(0\right)=\tfrac{1}{3}% \mathrm{Bi}\left(0\right)}}
\tfrac{1}{2}\ScorerHi@{0} = \tfrac{1}{3}\AiryBi@{0}		 

(1)/(2)*AiryBi(0)*(int(AiryAi(t), t = -infinity .. (0)))-AiryAi(0)*(int(AiryBi(t), t = -infinity .. (0))) = (1)/(3)*AiryBi(0)

Divide[1,2]*ScorerHi[0] == Divide[1,3]*AiryBi[0]
Failure Successful Skip - No test values generated Successful [Tested: 1]
9.12.E6 1 3 ⁒ Bi ⁑ ( 0 ) = 1 / ( 3 7 / 6 ⁒ Ξ“ ⁑ ( 2 3 ) ) = 0.20497 55424 ⁒ … , 1 3 Airy-Bi 0 1 superscript 3 7 6 Euler-Gamma 2 3 0.20497 55424 … {\displaystyle{\displaystyle\tfrac{1}{3}\mathrm{Bi}\left(0\right)={1\Big{/}\!% \left(3^{7/6}\Gamma\left(\tfrac{2}{3}\right)\right)=0.20497\;55424\ldots,}}}
\tfrac{1}{3}\AiryBi@{0} = {1\Big{/}\!\left(3^{7/6}\EulerGamma@{\tfrac{2}{3}}\right)=0.20497\;55424\ldots,}		 

(1)/(3)*AiryBi(0) = 1/((3)^(7/6)* GAMMA((2)/(3))) = 0.2049755424

Divide[1,3]*AiryBi[0] == 1/((3)^(7/6)* Gamma[Divide[2,3]]) == 0.2049755424
Error Failure Skip - symbolical successful subtest Error
9.12.E7 Gi β€² ⁑ ( 0 ) = 1 2 ⁒ Hi β€² ⁑ ( 0 ) diffop Scorer-Gi 1 0 1 2 diffop Scorer-Hi 1 0 {\displaystyle{\displaystyle\mathrm{Gi}'\left(0\right)=\tfrac{1}{2}\mathrm{Hi}% '\left(0\right)}}
\ScorerGi'@{0} = \tfrac{1}{2}\ScorerHi'@{0}		 

subs( temp=0, diff( AiryBi(temp)*(int(AiryAi(t), t = (temp) .. infinity))+AiryAi(temp)*(int(AiryBi(t), t = 0 .. (temp))), temp$(1) ) ) = (1)/(2)*subs( temp=0, diff( AiryBi(temp)*(int(AiryAi(t), t = -infinity .. (temp)))-AiryAi(temp)*(int(AiryBi(t), t = -infinity .. (temp))), temp$(1) ) )

(D[ScorerGi[temp], {temp, 1}]/.temp-> 0) == Divide[1,2]*(D[ScorerHi[temp], {temp, 1}]/.temp-> 0)
Successful Successful - Successful [Tested: 1]
9.12.E7 1 2 ⁒ Hi β€² ⁑ ( 0 ) = 1 3 ⁒ Bi β€² ⁑ ( 0 ) 1 2 diffop Scorer-Hi 1 0 1 3 diffop Airy-Bi 1 0 {\displaystyle{\displaystyle\tfrac{1}{2}\mathrm{Hi}'\left(0\right)=\tfrac{1}{3% }\mathrm{Bi}'\left(0\right)}}
\tfrac{1}{2}\ScorerHi'@{0} = \tfrac{1}{3}\AiryBi'@{0}		 

(1)/(2)*subs( temp=0, diff( AiryBi(temp)*(int(AiryAi(t), t = -infinity .. (temp)))-AiryAi(temp)*(int(AiryBi(t), t = -infinity .. (temp))), temp$(1) ) ) = (1)/(3)*subs( temp=0, diff( AiryBi(temp), temp$(1) ) )

Divide[1,2]*(D[ScorerHi[temp], {temp, 1}]/.temp-> 0) == Divide[1,3]*(D[AiryBi[temp], {temp, 1}]/.temp-> 0)
Successful Successful - Successful [Tested: 1]
9.12.E7 1 3 ⁒ Bi β€² ⁑ ( 0 ) = 1 / ( 3 5 / 6 ⁒ Ξ“ ⁑ ( 1 3 ) ) 1 3 diffop Airy-Bi 1 0 1 superscript 3 5 6 Euler-Gamma 1 3 {\displaystyle{\displaystyle\tfrac{1}{3}\mathrm{Bi}'\left(0\right)=1\Big{/}% \left(3^{5/6}\Gamma\left(\tfrac{1}{3}\right)\right)}}
\tfrac{1}{3}\AiryBi'@{0} = 1\Big{/}\left(3^{5/6}\EulerGamma@{\tfrac{1}{3}}\right)		 

(1)/(3)*subs( temp=0, diff( AiryBi(temp), temp$(1) ) ) = 1/((3)^(5/6)* GAMMA((1)/(3)))

Divide[1,3]*(D[AiryBi[temp], {temp, 1}]/.temp-> 0) == 1/((3)^(5/6)* Gamma[Divide[1,3]])
Successful Successful - Successful [Tested: 1]
9.12.E7 1 / ( 3 5 / 6 ⁒ Ξ“ ⁑ ( 1 3 ) ) = 0.14942 94524 ⁒ … 1 superscript 3 5 6 Euler-Gamma 1 3 0.14942 94524 … {\displaystyle{\displaystyle 1\Big{/}\left(3^{5/6}\Gamma\left(\tfrac{1}{3}% \right)\right)=0.14942\;94524\ldots}}
1\Big{/}\left(3^{5/6}\EulerGamma@{\tfrac{1}{3}}\right) = 0.14942\;94524\ldots		 

1/((3)^(5/6)* GAMMA((1)/(3))) = 0.1494294524

1/((3)^(5/6)* Gamma[Divide[1,3]]) == 0.1494294524
Successful Failure - Successful [Tested: 1]
9.12.E11 Gi ⁑ ( z ) + Hi ⁑ ( z ) = Bi ⁑ ( z ) Scorer-Gi 𝑧 Scorer-Hi 𝑧 Airy-Bi 𝑧 {\displaystyle{\displaystyle\mathrm{Gi}\left(z\right)+\mathrm{Hi}\left(z\right% )=\mathrm{Bi}\left(z\right)}}
\ScorerGi@{z}+\ScorerHi@{z} = \AiryBi@{z}		 

AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z)))+ AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = AiryBi(z)

ScorerGi[z]+ ScorerHi[z] == AiryBi[z]
Successful Successful - Successful [Tested: 7]
9.12.E12 Gi ⁑ ( z ) = 1 2 ⁒ e Ο€ ⁒ i / 3 ⁒ Hi ⁑ ( z ⁒ e - 2 ⁒ Ο€ ⁒ i / 3 ) + 1 2 ⁒ e - Ο€ ⁒ i / 3 ⁒ Hi ⁑ ( z ⁒ e 2 ⁒ Ο€ ⁒ i / 3 ) Scorer-Gi 𝑧 1 2 superscript 𝑒 πœ‹ 𝑖 3 Scorer-Hi 𝑧 superscript 𝑒 2 πœ‹ 𝑖 3 1 2 superscript 𝑒 πœ‹ 𝑖 3 Scorer-Hi 𝑧 superscript 𝑒 2 πœ‹ 𝑖 3 {\displaystyle{\displaystyle\mathrm{Gi}\left(z\right)=\tfrac{1}{2}e^{\pi i/3}% \mathrm{Hi}\left(ze^{-2\pi i/3}\right)+\tfrac{1}{2}e^{-\pi i/3}\mathrm{Hi}% \left(ze^{2\pi i/3}\right)}}
\ScorerGi@{z} = \tfrac{1}{2}e^{\pi i/3}\ScorerHi@{ze^{-2\pi i/3}}+\tfrac{1}{2}e^{-\pi i/3}\ScorerHi@{ze^{2\pi i/3}}		 

AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))) = (1)/(2)*exp(Pi*I/3)*AiryBi(z*exp(- 2*Pi*I/3))*(int(AiryAi(t), t = -infinity .. (z*exp(- 2*Pi*I/3))))-AiryAi(z*exp(- 2*Pi*I/3))*(int(AiryBi(t), t = -infinity .. (z*exp(- 2*Pi*I/3))))+(1)/(2)*exp(- Pi*I/3)*AiryBi(z*exp(2*Pi*I/3))*(int(AiryAi(t), t = -infinity .. (z*exp(2*Pi*I/3))))-AiryAi(z*exp(2*Pi*I/3))*(int(AiryBi(t), t = -infinity .. (z*exp(2*Pi*I/3))))

ScorerGi[z] == Divide[1,2]*Exp[Pi*I/3]*ScorerHi[z*Exp[- 2*Pi*I/3]]+Divide[1,2]*Exp[- Pi*I/3]*ScorerHi[z*Exp[2*Pi*I/3]]
Failure Successful
Failed [7 / 7]
Result: -.2356545741-.3070803572*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: .1024598659-.3465846956*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Successful [Tested: 7]
9.12.E13 Gi ⁑ ( z ) = e - Ο€ ⁒ i / 3 ⁒ Hi ⁑ ( z ⁒ e + 2 ⁒ Ο€ ⁒ i / 3 ) + i ⁒ Ai ⁑ ( z ) Scorer-Gi 𝑧 superscript 𝑒 πœ‹ 𝑖 3 Scorer-Hi 𝑧 superscript 𝑒 2 πœ‹ 𝑖 3 𝑖 Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Gi}\left(z\right)=e^{-\pi i/3}\mathrm{Hi}% \left(ze^{+2\pi i/3}\right)+i\mathrm{Ai}\left(z\right)}}
\ScorerGi@{z} = e^{-\pi i/3}\ScorerHi@{ze^{+ 2\pi i/3}}+ i\AiryAi@{z}		 

AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))) = exp(- Pi*I/3)*AiryBi(z*exp(+ 2*Pi*I/3))*(int(AiryAi(t), t = -infinity .. (z*exp(+ 2*Pi*I/3))))-AiryAi(z*exp(+ 2*Pi*I/3))*(int(AiryBi(t), t = -infinity .. (z*exp(+ 2*Pi*I/3))))+ I*AiryAi(z)

ScorerGi[z] == Exp[- Pi*I/3]*ScorerHi[z*Exp[+ 2*Pi*I/3]]+ I*AiryAi[z]
Failure Successful
Failed [7 / 7]
Result: -.1900131227-.1739897867*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: .1844815903-.2874294645*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Successful [Tested: 7]
9.12.E13 Gi ⁑ ( z ) = e + Ο€ ⁒ i / 3 ⁒ Hi ⁑ ( z ⁒ e - 2 ⁒ Ο€ ⁒ i / 3 ) - i ⁒ Ai ⁑ ( z ) Scorer-Gi 𝑧 superscript 𝑒 πœ‹ 𝑖 3 Scorer-Hi 𝑧 superscript 𝑒 2 πœ‹ 𝑖 3 𝑖 Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Gi}\left(z\right)=e^{+\pi i/3}\mathrm{Hi}% \left(ze^{-2\pi i/3}\right)-i\mathrm{Ai}\left(z\right)}}
\ScorerGi@{z} = e^{+\pi i/3}\ScorerHi@{ze^{- 2\pi i/3}}- i\AiryAi@{z}		 

AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))) = exp(+ Pi*I/3)*AiryBi(z*exp(- 2*Pi*I/3))*(int(AiryAi(t), t = -infinity .. (z*exp(- 2*Pi*I/3))))-AiryAi(z*exp(- 2*Pi*I/3))*(int(AiryBi(t), t = -infinity .. (z*exp(- 2*Pi*I/3))))- I*AiryAi(z)

ScorerGi[z] == Exp[+ Pi*I/3]*ScorerHi[z*Exp[- 2*Pi*I/3]]- I*AiryAi[z]
Failure Successful
Failed [7 / 7]
Result: -.1672613648-.2485864233*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: .590395216e-1-.1022594507*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Successful [Tested: 7]
9.12.E14 Hi ⁑ ( z ) = e + 2 ⁒ Ο€ ⁒ i / 3 ⁒ Hi ⁑ ( z ⁒ e + 2 ⁒ Ο€ ⁒ i / 3 ) + 2 ⁒ e - Ο€ ⁒ i / 6 ⁒ Ai ⁑ ( z ⁒ e - 2 ⁒ Ο€ ⁒ i / 3 ) Scorer-Hi 𝑧 superscript 𝑒 2 πœ‹ 𝑖 3 Scorer-Hi 𝑧 superscript 𝑒 2 πœ‹ 𝑖 3 2 superscript 𝑒 πœ‹ 𝑖 6 Airy-Ai 𝑧 superscript 𝑒 2 πœ‹ 𝑖 3 {\displaystyle{\displaystyle\mathrm{Hi}\left(z\right)=e^{+2\pi i/3}\mathrm{Hi}% \left(ze^{+2\pi i/3}\right)+2e^{-\pi i/6}\mathrm{Ai}\left(ze^{-2\pi i/3}\right% )}}
\ScorerHi@{z} = e^{+ 2\pi i/3}\ScorerHi@{ze^{+ 2\pi i/3}}+2e^{-\pi i/6}\AiryAi@{ze^{- 2\pi i/3}}		 

AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = exp(+ 2*Pi*I/3)*AiryBi(z*exp(+ 2*Pi*I/3))*(int(AiryAi(t), t = -infinity .. (z*exp(+ 2*Pi*I/3))))-AiryAi(z*exp(+ 2*Pi*I/3))*(int(AiryBi(t), t = -infinity .. (z*exp(+ 2*Pi*I/3))))+ 2*exp(- Pi*I/6)*AiryAi(z*exp(- 2*Pi*I/3))

ScorerHi[z] == Exp[+ 2*Pi*I/3]*ScorerHi[z*Exp[+ 2*Pi*I/3]]+ 2*Exp[- Pi*I/6]*AiryAi[z*Exp[- 2*Pi*I/3]]
Failure Successful
Failed [7 / 7]
Result: -.3013591505+.3291123823*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: -.4978424366-.3195314878*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Successful [Tested: 7]
9.12.E14 Hi ⁑ ( z ) = e - 2 ⁒ Ο€ ⁒ i / 3 ⁒ Hi ⁑ ( z ⁒ e - 2 ⁒ Ο€ ⁒ i / 3 ) + 2 ⁒ e + Ο€ ⁒ i / 6 ⁒ Ai ⁑ ( z ⁒ e + 2 ⁒ Ο€ ⁒ i / 3 ) Scorer-Hi 𝑧 superscript 𝑒 2 πœ‹ 𝑖 3 Scorer-Hi 𝑧 superscript 𝑒 2 πœ‹ 𝑖 3 2 superscript 𝑒 πœ‹ 𝑖 6 Airy-Ai 𝑧 superscript 𝑒 2 πœ‹ 𝑖 3 {\displaystyle{\displaystyle\mathrm{Hi}\left(z\right)=e^{-2\pi i/3}\mathrm{Hi}% \left(ze^{-2\pi i/3}\right)+2e^{+\pi i/6}\mathrm{Ai}\left(ze^{+2\pi i/3}\right% )}}
\ScorerHi@{z} = e^{- 2\pi i/3}\ScorerHi@{ze^{- 2\pi i/3}}+2e^{+\pi i/6}\AiryAi@{ze^{+ 2\pi i/3}}		 

AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = exp(- 2*Pi*I/3)*AiryBi(z*exp(- 2*Pi*I/3))*(int(AiryAi(t), t = -infinity .. (z*exp(- 2*Pi*I/3))))-AiryAi(z*exp(- 2*Pi*I/3))*(int(AiryBi(t), t = -infinity .. (z*exp(- 2*Pi*I/3))))+ 2*exp(+ Pi*I/6)*AiryAi(z*exp(+ 2*Pi*I/3))

ScorerHi[z] == Exp[- 2*Pi*I/3]*ScorerHi[z*Exp[- 2*Pi*I/3]]+ 2*Exp[+ Pi*I/6]*AiryAi[z*Exp[+ 2*Pi*I/3]]
Failure Successful
Failed [7 / 7]
Result: .430564314-.2897051813*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: .1771185635+.1022594505*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Successful [Tested: 7]
9.12.E15 Gi ⁑ ( z ) = 3 - 2 / 3 Ο€ ⁒ βˆ‘ k = 0 ∞ cos ⁑ ( 2 ⁒ k - 1 3 ⁒ Ο€ ) ⁒ Ξ“ ⁑ ( k + 1 3 ) ⁒ ( 3 1 / 3 ⁒ z ) k k ! Scorer-Gi 𝑧 superscript 3 2 3 πœ‹ superscript subscript π‘˜ 0 2 π‘˜ 1 3 πœ‹ Euler-Gamma π‘˜ 1 3 superscript superscript 3 1 3 𝑧 π‘˜ π‘˜ {\displaystyle{\displaystyle\mathrm{Gi}\left(z\right)=\frac{3^{-2/3}}{\pi}\*% \sum_{k=0}^{\infty}\cos\left(\frac{2k-1}{3}\pi\right)\Gamma\left(\frac{k+1}{3}% \right)\frac{(3^{1/3}z)^{k}}{k!}}}
\ScorerGi@{z} = \frac{3^{-2/3}}{\pi}\*\sum_{k=0}^{\infty}\cos@{\frac{2k-1}{3}\pi}\EulerGamma@{\frac{k+1}{3}}\frac{(3^{1/3}z)^{k}}{k!}		 

AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))) = ((3)^(- 2/3))/(Pi)* sum(cos((2*k - 1)/(3)*Pi)*GAMMA((k + 1)/(3))*(((3)^(1/3)* z)^(k))/(factorial(k)), k = 0..infinity)

ScorerGi[z] == Divide[(3)^(- 2/3),Pi]* Sum[Cos[Divide[2*k - 1,3]*Pi]*Gamma[Divide[k + 1,3]]*Divide[((3)^(1/3)* z)^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
9.12.E16 Gi β€² ⁑ ( z ) = 3 - 1 / 3 Ο€ ⁒ βˆ‘ k = 0 ∞ cos ⁑ ( 2 ⁒ k + 1 3 ⁒ Ο€ ) ⁒ Ξ“ ⁑ ( k + 2 3 ) ⁒ ( 3 1 / 3 ⁒ z ) k k ! diffop Scorer-Gi 1 𝑧 superscript 3 1 3 πœ‹ superscript subscript π‘˜ 0 2 π‘˜ 1 3 πœ‹ Euler-Gamma π‘˜ 2 3 superscript superscript 3 1 3 𝑧 π‘˜ π‘˜ {\displaystyle{\displaystyle\mathrm{Gi}'\left(z\right)=\frac{3^{-1/3}}{\pi}\*% \sum_{k=0}^{\infty}\cos\left(\frac{2k+1}{3}\pi\right)\Gamma\left(\frac{k+2}{3}% \right)\frac{(3^{1/3}z)^{k}}{k!}}}
\ScorerGi'@{z} = \frac{3^{-1/3}}{\pi}\*\sum_{k=0}^{\infty}\cos@{\frac{2k+1}{3}\pi}\EulerGamma@{\frac{k+2}{3}}\frac{(3^{1/3}z)^{k}}{k!}		 

diff( AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))), z$(1) ) = ((3)^(- 1/3))/(Pi)* sum(cos((2*k + 1)/(3)*Pi)*GAMMA((k + 2)/(3))*(((3)^(1/3)* z)^(k))/(factorial(k)), k = 0..infinity)

D[ScorerGi[z], {z, 1}] == Divide[(3)^(- 1/3),Pi]* Sum[Cos[Divide[2*k + 1,3]*Pi]*Gamma[Divide[k + 2,3]]*Divide[((3)^(1/3)* z)^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
9.12.E17 Hi ⁑ ( z ) = 3 - 2 / 3 Ο€ ⁒ βˆ‘ k = 0 ∞ Ξ“ ⁑ ( k + 1 3 ) ⁒ ( 3 1 / 3 ⁒ z ) k k ! Scorer-Hi 𝑧 superscript 3 2 3 πœ‹ superscript subscript π‘˜ 0 Euler-Gamma π‘˜ 1 3 superscript superscript 3 1 3 𝑧 π‘˜ π‘˜ {\displaystyle{\displaystyle\mathrm{Hi}\left(z\right)=\frac{3^{-2/3}}{\pi}\sum% _{k=0}^{\infty}\Gamma\left(\frac{k+1}{3}\right)\frac{(3^{1/3}z)^{k}}{k!}}}
\ScorerHi@{z} = \frac{3^{-2/3}}{\pi}\sum_{k=0}^{\infty}\EulerGamma@{\frac{k+1}{3}}\frac{(3^{1/3}z)^{k}}{k!}		 

AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = ((3)^(- 2/3))/(Pi)*sum(GAMMA((k + 1)/(3))*(((3)^(1/3)* z)^(k))/(factorial(k)), k = 0..infinity)

ScorerHi[z] == Divide[(3)^(- 2/3),Pi]*Sum[Gamma[Divide[k + 1,3]]*Divide[((3)^(1/3)* z)^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
9.12.E18 Hi β€² ⁑ ( z ) = 3 - 1 / 3 Ο€ ⁒ βˆ‘ k = 0 ∞ Ξ“ ⁑ ( k + 2 3 ) ⁒ ( 3 1 / 3 ⁒ z ) k k ! diffop Scorer-Hi 1 𝑧 superscript 3 1 3 πœ‹ superscript subscript π‘˜ 0 Euler-Gamma π‘˜ 2 3 superscript superscript 3 1 3 𝑧 π‘˜ π‘˜ {\displaystyle{\displaystyle\mathrm{Hi}'\left(z\right)=\frac{3^{-1/3}}{\pi}% \sum_{k=0}^{\infty}\Gamma\left(\frac{k+2}{3}\right)\frac{(3^{1/3}z)^{k}}{k!}}}
\ScorerHi'@{z} = \frac{3^{-1/3}}{\pi}\sum_{k=0}^{\infty}\EulerGamma@{\frac{k+2}{3}}\frac{(3^{1/3}z)^{k}}{k!}		 

diff( AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))), z$(1) ) = ((3)^(- 1/3))/(Pi)*sum(GAMMA((k + 2)/(3))*(((3)^(1/3)* z)^(k))/(factorial(k)), k = 0..infinity)

D[ScorerHi[z], {z, 1}] == Divide[(3)^(- 1/3),Pi]*Sum[Gamma[Divide[k + 2,3]]*Divide[((3)^(1/3)* z)^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
9.12.E19 Gi ⁑ ( x ) = 1 Ο€ ⁒ ∫ 0 ∞ sin ⁑ ( 1 3 ⁒ t 3 + x ⁒ t ) ⁒ d t Scorer-Gi π‘₯ 1 πœ‹ superscript subscript 0 1 3 superscript 𝑑 3 π‘₯ 𝑑 𝑑 {\displaystyle{\displaystyle\mathrm{Gi}\left(x\right)=\frac{1}{\pi}\int_{0}^{% \infty}\sin\left(\tfrac{1}{3}t^{3}+xt\right)\mathrm{d}t}}
\ScorerGi@{x} = \frac{1}{\pi}\int_{0}^{\infty}\sin@{\tfrac{1}{3}t^{3}+xt}\diff{t}		 

AiryBi(x)*(int(AiryAi(t), t = (x) .. infinity))+AiryAi(x)*(int(AiryBi(t), t = 0 .. (x))) = (1)/(Pi)*int(sin((1)/(3)*(t)^(3)+ x*t), t = 0..infinity)

ScorerGi[x] == Divide[1,Pi]*Integrate[Sin[Divide[1,3]*(t)^(3)+ x*t], {t, 0, Infinity}, GenerateConditions->None]
Failure Failure Successful [Tested: 3] Successful [Tested: 1]
9.12.E20 Hi ⁑ ( z ) = 1 Ο€ ⁒ ∫ 0 ∞ exp ⁑ ( - 1 3 ⁒ t 3 + z ⁒ t ) ⁒ d t Scorer-Hi 𝑧 1 πœ‹ superscript subscript 0 1 3 superscript 𝑑 3 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle\mathrm{Hi}\left(z\right)=\frac{1}{\pi}\int_{0}^{% \infty}\exp\left(-\tfrac{1}{3}t^{3}+zt\right)\mathrm{d}t}}
\ScorerHi@{z} = \frac{1}{\pi}\int_{0}^{\infty}\exp@{-\tfrac{1}{3}t^{3}+zt}\diff{t}		 

AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = (1)/(Pi)*int(exp(-(1)/(3)*(t)^(3)+ z*t), t = 0..infinity)

ScorerHi[z] == Divide[1,Pi]*Integrate[Exp[-Divide[1,3]*(t)^(3)+ z*t], {t, 0, Infinity}, GenerateConditions->None]
Failure Successful
Failed [4 / 7]
Result: .4525872086+.6186053865*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: 1.510759173-.1408206709*I
Test Values: {z = 1.5}

... skip entries to safe data
 Successful [Tested: 7]
9.12.E21 Gi ⁑ ( z ) = - 1 Ο€ ⁒ ∫ 0 ∞ exp ⁑ ( - 1 3 ⁒ t 3 - 1 2 ⁒ z ⁒ t ) ⁒ cos ⁑ ( 1 2 ⁒ 3 ⁒ z ⁒ t + 2 3 ⁒ Ο€ ) ⁒ d t Scorer-Gi 𝑧 1 πœ‹ superscript subscript 0 1 3 superscript 𝑑 3 1 2 𝑧 𝑑 1 2 3 𝑧 𝑑 2 3 πœ‹ 𝑑 {\displaystyle{\displaystyle\mathrm{Gi}\left(z\right)=-\frac{1}{\pi}\int_{0}^{% \infty}\exp\left(-\tfrac{1}{3}t^{3}-\tfrac{1}{2}zt\right)\cos\left(\tfrac{1}{2% }\sqrt{3}zt+\tfrac{2}{3}\pi\right)\mathrm{d}t}}
\ScorerGi@{z} = -\frac{1}{\pi}\int_{0}^{\infty}\exp@{-\tfrac{1}{3}t^{3}-\tfrac{1}{2}zt}\cos@{\tfrac{1}{2}\sqrt{3}zt+\tfrac{2}{3}\pi}\diff{t}		 

AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))) = -(1)/(Pi)*int(exp(-(1)/(3)*(t)^(3)-(1)/(2)*z*t)*cos((1)/(2)*sqrt(3)*z*t +(2)/(3)*Pi), t = 0..infinity)

ScorerGi[z] == -Divide[1,Pi]*Integrate[Exp[-Divide[1,3]*(t)^(3)-Divide[1,2]*z*t]*Cos[Divide[1,2]*Sqrt[3]*z*t +Divide[2,3]*Pi], {t, 0, Infinity}, GenerateConditions->None]
Failure Failure Successful [Tested: 7] Skipped - Because timed out
9.12.E22 Hi ⁑ ( - z ) = 4 ⁒ z 2 3 3 / 2 ⁒ Ο€ 2 ⁒ ∫ 0 ∞ K 1 / 3 ⁑ ( t ) ΞΆ 2 + t 2 ⁒ d t Scorer-Hi 𝑧 4 superscript 𝑧 2 superscript 3 3 2 superscript πœ‹ 2 superscript subscript 0 modified-Bessel-second-kind 1 3 𝑑 superscript 𝜁 2 superscript 𝑑 2 𝑑 {\displaystyle{\displaystyle\mathrm{Hi}\left(-z\right)=\frac{4z^{2}}{3^{3/2}% \pi^{2}}\int_{0}^{\infty}\frac{K_{1/3}\left(t\right)}{\zeta^{2}+t^{2}}\mathrm{% d}t}}
\ScorerHi@{-z} = \frac{4z^{2}}{3^{3/2}\pi^{2}}\int_{0}^{\infty}\frac{\modBesselK{1/3}@{t}}{\zeta^{2}+t^{2}}\diff{t}		 | ph ⁑ z | < 1 3 ⁒ Ο€ phase 𝑧 1 3 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{3}\pi}} 
AiryBi(- z)*(int(AiryAi(t), t = -infinity .. (- z)))-AiryAi(- z)*(int(AiryBi(t), t = -infinity .. (- z))) = (4*(z)^(2))/((3)^(3/2)* (Pi)^(2))*int((BesselK(1/3, t))/((2)/(3)*((z)^((3)/(2)))^(2)+ (t)^(2)), t = 0..infinity)

ScorerHi[- z] == Divide[4*(z)^(2),(3)^(3/2)* (Pi)^(2)]*Integrate[Divide[BesselK[1/3, t],Divide[2,3]*((z)^(Divide[3,2]))^(2)+ (t)^(2)], {t, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [4 / 4]
Result: .660208669e-1-.1388055037e-1*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: .528823090e-1
Test Values: {z = 1.5}

... skip entries to safe data
Failed [4 / 4]
Result: Complex[0.06602086668543175, -0.01388055052265768]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: 0.05288230872547964
Test Values: {Rule[z, 1.5]}

... skip entries to safe data
9.12.E24 Hi ⁑ ( z ) = 3 - 2 / 3 2 ⁒ Ο€ 2 ⁒ i ⁒ ∫ - i ⁒ ∞ i ⁒ ∞ Ξ“ ⁑ ( 1 3 + 1 3 ⁒ t ) ⁒ Ξ“ ⁑ ( - t ) ⁒ ( 3 1 / 3 ⁒ e Ο€ ⁒ i ⁒ z ) t ⁒ d t Scorer-Hi 𝑧 superscript 3 2 3 2 superscript πœ‹ 2 𝑖 superscript subscript 𝑖 𝑖 Euler-Gamma 1 3 1 3 𝑑 Euler-Gamma 𝑑 superscript superscript 3 1 3 superscript 𝑒 πœ‹ 𝑖 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle\mathrm{Hi}\left(z\right)=\frac{3^{-2/3}}{2\pi^{2}% i}\int_{-i\infty}^{i\infty}\Gamma\left(\tfrac{1}{3}+\tfrac{1}{3}t\right)\Gamma% \left(-t\right)(3^{1/3}e^{\pi i}z)^{t}\mathrm{d}t}}
\ScorerHi@{z} = \frac{3^{-2/3}}{2\pi^{2}i}\int_{-i\infty}^{i\infty}\EulerGamma@{\tfrac{1}{3}+\tfrac{1}{3}t}\EulerGamma@{-t}(3^{1/3}e^{\pi i}z)^{t}\diff{t}		 β„œ ⁑ ( 1 3 + 1 3 ⁒ t ) > 0 , β„œ ⁑ ( - t ) > 0 formulae-sequence 1 3 1 3 𝑑 0 𝑑 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{3}+\tfrac{1}{3}t)>0,\Re(-t)>0}} 
AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = ((3)^(- 2/3))/(2*(Pi)^(2)* I)*int(GAMMA((1)/(3)+(1)/(3)*t)*GAMMA(- t)*((3)^(1/3)* exp(Pi*I)*z)^(t), t = - I*infinity..I*infinity)

ScorerHi[z] == Divide[(3)^(- 2/3),2*(Pi)^(2)* I]*Integrate[Gamma[Divide[1,3]+Divide[1,3]*t]*Gamma[- t]*((3)^(1/3)* Exp[Pi*I]*z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
Failure Aborted Manual Skip! Skipped - Because timed out
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