Incomplete Gamma and Related Functions - 9.2 Differential Equation

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
9.2.E2 w = Ai ⁡ ( z ) , Bi ⁡ ( z ) , Ai ⁡ ( z ⁢ e - 2 ⁢ π ⁢ i / 3 ) 𝑤 Airy-Ai 𝑧 Airy-Bi 𝑧 Airy-Ai 𝑧 superscript 𝑒 2 𝜋 imaginary-unit 3 {\displaystyle{\displaystyle w=\mathrm{Ai}\left(z\right),\;\mathrm{Bi}\left(z% \right),\;\mathrm{Ai}\left(ze^{-2\pi\mathrm{i}/3}\right)}}
w = \AiryAi@{z},\;\AiryBi@{z},\;\AiryAi@{ze^{- 2\pi\iunit/3}}		 

w = AiryAi(z); AiryBi(z), AiryAi(z*exp(- 2*Pi*I/3))

w == AiryAi[z]
 AiryBi[z], AiryAi[z*Exp[- 2*Pi*I/3]]
Failure Failure Error Error
9.2.E2 w = Ai ⁡ ( z ) , Bi ⁡ ( z ) , Ai ⁡ ( z ⁢ e + 2 ⁢ π ⁢ i / 3 ) 𝑤 Airy-Ai 𝑧 Airy-Bi 𝑧 Airy-Ai 𝑧 superscript 𝑒 2 𝜋 imaginary-unit 3 {\displaystyle{\displaystyle w=\mathrm{Ai}\left(z\right),\;\mathrm{Bi}\left(z% \right),\;\mathrm{Ai}\left(ze^{+2\pi\mathrm{i}/3}\right)}}
w = \AiryAi@{z},\;\AiryBi@{z},\;\AiryAi@{ze^{+ 2\pi\iunit/3}}		 

w = AiryAi(z); AiryBi(z), AiryAi(z*exp(+ 2*Pi*I/3))

w == AiryAi[z]
 AiryBi[z], AiryAi[z*Exp[+ 2*Pi*I/3]]
Failure Failure Error Error
9.2.E3 Ai ⁡ ( 0 ) = 1 3 2 / 3 ⁢ Γ ⁡ ( 2 3 ) Airy-Ai 0 1 superscript 3 2 3 Euler-Gamma 2 3 {\displaystyle{\displaystyle\mathrm{Ai}\left(0\right)=\frac{1}{3^{2/3}\Gamma% \left(\tfrac{2}{3}\right)}}}
\AiryAi@{0} = \frac{1}{3^{2/3}\EulerGamma@{\tfrac{2}{3}}}		 

AiryAi(0) = (1)/((3)^(2/3)* GAMMA((2)/(3)))

AiryAi[0] == Divide[1,(3)^(2/3)* Gamma[Divide[2,3]]]
Successful Successful - Successful [Tested: 1]
9.2.E3 1 3 2 / 3 ⁢ Γ ⁡ ( 2 3 ) = 0.35502 80538 ⁢ … 1 superscript 3 2 3 Euler-Gamma 2 3 0.35502 80538 … {\displaystyle{\displaystyle\frac{1}{3^{2/3}\Gamma\left(\tfrac{2}{3}\right)}=0% .35502\;80538\ldots}}
\frac{1}{3^{2/3}\EulerGamma@{\tfrac{2}{3}}} = 0.35502\;80538\ldots		 

(1)/((3)^(2/3)* GAMMA((2)/(3))) = 0.3550280538

Divide[1,(3)^(2/3)* Gamma[Divide[2,3]]] == 0.3550280538
Successful Failure - Successful [Tested: 1]
9.2.E4 Ai ′ ⁡ ( 0 ) = - 1 3 1 / 3 ⁢ Γ ⁡ ( 1 3 ) diffop Airy-Ai 1 0 1 superscript 3 1 3 Euler-Gamma 1 3 {\displaystyle{\displaystyle\mathrm{Ai}'\left(0\right)=-\frac{1}{3^{1/3}\Gamma% \left(\tfrac{1}{3}\right)}}}
\AiryAi'@{0} = -\frac{1}{3^{1/3}\EulerGamma@{\tfrac{1}{3}}}		 

subs( temp=0, diff( AiryAi(temp), temp$(1) ) ) = -(1)/((3)^(1/3)* GAMMA((1)/(3)))

(D[AiryAi[temp], {temp, 1}]/.temp-> 0) == -Divide[1,(3)^(1/3)* Gamma[Divide[1,3]]]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 1]
9.2.E4 - 1 3 1 / 3 ⁢ Γ ⁡ ( 1 3 ) = - 0.25881 94037 ⁢ … 1 superscript 3 1 3 Euler-Gamma 1 3 0.25881 94037 … {\displaystyle{\displaystyle-\frac{1}{3^{1/3}\Gamma\left(\tfrac{1}{3}\right)}=% -0.25881\;94037\ldots}}
-\frac{1}{3^{1/3}\EulerGamma@{\tfrac{1}{3}}} = -0.25881\;94037\ldots		 

-(1)/((3)^(1/3)* GAMMA((1)/(3))) = - 0.2588194037

-Divide[1,(3)^(1/3)* Gamma[Divide[1,3]]] == - 0.2588194037
Failure Failure Successful [Tested: 0] Successful [Tested: 1]
9.2.E5 Bi ⁡ ( 0 ) = 1 3 1 / 6 ⁢ Γ ⁡ ( 2 3 ) Airy-Bi 0 1 superscript 3 1 6 Euler-Gamma 2 3 {\displaystyle{\displaystyle\mathrm{Bi}\left(0\right)=\frac{1}{3^{1/6}\Gamma% \left(\tfrac{2}{3}\right)}}}
\AiryBi@{0} = \frac{1}{3^{1/6}\EulerGamma@{\tfrac{2}{3}}}		 

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

AiryBi[0] == Divide[1,(3)^(1/6)* Gamma[Divide[2,3]]]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 1]
9.2.E5 1 3 1 / 6 ⁢ Γ ⁡ ( 2 3 ) = 0.61492 66274 ⁢ … 1 superscript 3 1 6 Euler-Gamma 2 3 0.61492 66274 … {\displaystyle{\displaystyle\frac{1}{3^{1/6}\Gamma\left(\tfrac{2}{3}\right)}=0% .61492\;66274\ldots}}
\frac{1}{3^{1/6}\EulerGamma@{\tfrac{2}{3}}} = 0.61492\;66274\ldots		 

(1)/((3)^(1/6)* GAMMA((2)/(3))) = 0.6149266274

Divide[1,(3)^(1/6)* Gamma[Divide[2,3]]] == 0.6149266274
Failure Failure Successful [Tested: 0] Successful [Tested: 1]
9.2.E6 Bi ′ ⁡ ( 0 ) = 3 1 / 6 Γ ⁡ ( 1 3 ) diffop Airy-Bi 1 0 superscript 3 1 6 Euler-Gamma 1 3 {\displaystyle{\displaystyle\mathrm{Bi}'\left(0\right)=\frac{3^{1/6}}{\Gamma% \left(\tfrac{1}{3}\right)}}}
\AiryBi'@{0} = \frac{3^{1/6}}{\EulerGamma@{\tfrac{1}{3}}}		 

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

(D[AiryBi[temp], {temp, 1}]/.temp-> 0) == Divide[(3)^(1/6),Gamma[Divide[1,3]]]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 1]
9.2.E6 3 1 / 6 Γ ⁡ ( 1 3 ) = 0.44828 83573 ⁢ … superscript 3 1 6 Euler-Gamma 1 3 0.44828 83573 … {\displaystyle{\displaystyle\frac{3^{1/6}}{\Gamma\left(\tfrac{1}{3}\right)}=0.% 44828\;83573\ldots}}
\frac{3^{1/6}}{\EulerGamma@{\tfrac{1}{3}}} = 0.44828\;83573\ldots		 

((3)^(1/6))/(GAMMA((1)/(3))) = 0.4482883573

Divide[(3)^(1/6),Gamma[Divide[1,3]]] == 0.4482883573
Failure Failure Successful [Tested: 0] Successful [Tested: 1]
9.2.E7 𝒲 ⁡ { Ai ⁡ ( z ) , Bi ⁡ ( z ) } = 1 π Wronskian Airy-Ai 𝑧 Airy-Bi 𝑧 1 𝜋 {\displaystyle{\displaystyle\mathscr{W}\left\{\mathrm{Ai}\left(z\right),% \mathrm{Bi}\left(z\right)\right\}=\frac{1}{\pi}}}
\Wronskian@{\AiryAi@{z},\AiryBi@{z}} = \frac{1}{\pi}		 

(AiryAi(z))*diff(AiryBi(z), z)-diff(AiryAi(z), z)*(AiryBi(z)) = (1)/(Pi)

Wronskian[{AiryAi[z], AiryBi[z]}, z] == Divide[1,Pi]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
9.2.E8 𝒲 ⁡ { Ai ⁡ ( z ) , Ai ⁡ ( z ⁢ e - 2 ⁢ π ⁢ i / 3 ) } = e + π ⁢ i / 6 2 ⁢ π Wronskian Airy-Ai 𝑧 Airy-Ai 𝑧 superscript 𝑒 2 𝜋 𝑖 3 superscript 𝑒 𝜋 𝑖 6 2 𝜋 {\displaystyle{\displaystyle\mathscr{W}\left\{\mathrm{Ai}\left(z\right),% \mathrm{Ai}\left(ze^{-2\pi i/3}\right)\right\}=\frac{e^{+\pi i/6}}{2\pi}}}
\Wronskian@{\AiryAi@{z},\AiryAi@{ze^{- 2\pi i/3}}} = \frac{e^{+\pi i/6}}{2\pi}		 

(AiryAi(z))*diff(AiryAi(z*exp(- 2*Pi*I/3)), z)-diff(AiryAi(z), z)*(AiryAi(z*exp(- 2*Pi*I/3))) = (exp(+ Pi*I/6))/(2*Pi)

Wronskian[{AiryAi[z], AiryAi[z*Exp[- 2*Pi*I/3]]}, z] == Divide[Exp[+ Pi*I/6],2*Pi]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
9.2.E8 𝒲 ⁡ { Ai ⁡ ( z ) , Ai ⁡ ( z ⁢ e + 2 ⁢ π ⁢ i / 3 ) } = e - π ⁢ i / 6 2 ⁢ π Wronskian Airy-Ai 𝑧 Airy-Ai 𝑧 superscript 𝑒 2 𝜋 𝑖 3 superscript 𝑒 𝜋 𝑖 6 2 𝜋 {\displaystyle{\displaystyle\mathscr{W}\left\{\mathrm{Ai}\left(z\right),% \mathrm{Ai}\left(ze^{+2\pi i/3}\right)\right\}=\frac{e^{-\pi i/6}}{2\pi}}}
\Wronskian@{\AiryAi@{z},\AiryAi@{ze^{+ 2\pi i/3}}} = \frac{e^{-\pi i/6}}{2\pi}		 

(AiryAi(z))*diff(AiryAi(z*exp(+ 2*Pi*I/3)), z)-diff(AiryAi(z), z)*(AiryAi(z*exp(+ 2*Pi*I/3))) = (exp(- Pi*I/6))/(2*Pi)

Wronskian[{AiryAi[z], AiryAi[z*Exp[+ 2*Pi*I/3]]}, z] == Divide[Exp[- Pi*I/6],2*Pi]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
9.2.E9 𝒲 ⁡ { Ai ⁡ ( z ⁢ e - 2 ⁢ π ⁢ i / 3 ) , Ai ⁡ ( z ⁢ e 2 ⁢ π ⁢ i / 3 ) } = 1 2 ⁢ π ⁢ i Wronskian Airy-Ai 𝑧 superscript 𝑒 2 𝜋 𝑖 3 Airy-Ai 𝑧 superscript 𝑒 2 𝜋 𝑖 3 1 2 𝜋 𝑖 {\displaystyle{\displaystyle\mathscr{W}\left\{\mathrm{Ai}\left(ze^{-2\pi i/3}% \right),\mathrm{Ai}\left(ze^{2\pi i/3}\right)\right\}=\frac{1}{2\pi i}}}
\Wronskian@{\AiryAi@{ze^{-2\pi i/3}},\AiryAi@{ze^{2\pi i/3}}} = \frac{1}{2\pi i}		 

(AiryAi(z*exp(- 2*Pi*I/3)))*diff(AiryAi(z*exp(2*Pi*I/3)), z)-diff(AiryAi(z*exp(- 2*Pi*I/3)), z)*(AiryAi(z*exp(2*Pi*I/3))) = (1)/(2*Pi*I)

Wronskian[{AiryAi[z*Exp[- 2*Pi*I/3]], AiryAi[z*Exp[2*Pi*I/3]]}, z] == Divide[1,2*Pi*I]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
9.2.E10 Bi ⁡ ( z ) = e - π ⁢ i / 6 ⁢ Ai ⁡ ( z ⁢ e - 2 ⁢ π ⁢ i / 3 ) + e π ⁢ i / 6 ⁢ Ai ⁡ ( z ⁢ e 2 ⁢ π ⁢ i / 3 ) Airy-Bi 𝑧 superscript 𝑒 𝜋 𝑖 6 Airy-Ai 𝑧 superscript 𝑒 2 𝜋 𝑖 3 superscript 𝑒 𝜋 𝑖 6 Airy-Ai 𝑧 superscript 𝑒 2 𝜋 𝑖 3 {\displaystyle{\displaystyle\mathrm{Bi}\left(z\right)=e^{-\pi i/6}\mathrm{Ai}% \left(ze^{-2\pi i/3}\right)+e^{\pi i/6}\mathrm{Ai}\left(ze^{2\pi i/3}\right)}}
\AiryBi@{z} = e^{-\pi i/6}\AiryAi@{ze^{-2\pi i/3}}+e^{\pi i/6}\AiryAi@{ze^{2\pi i/3}}		 

AiryBi(z) = exp(- Pi*I/6)*AiryAi(z*exp(- 2*Pi*I/3))+ exp(Pi*I/6)*AiryAi(z*exp(2*Pi*I/3))

AiryBi[z] == Exp[- Pi*I/6]*AiryAi[z*Exp[- 2*Pi*I/3]]+ Exp[Pi*I/6]*AiryAi[z*Exp[2*Pi*I/3]]
Successful Successful - Successful [Tested: 7]
9.2.E11 Ai ⁡ ( z ⁢ e - 2 ⁢ π ⁢ i / 3 ) = 1 2 ⁢ e - π ⁢ i / 3 ⁢ ( Ai ⁡ ( z ) + i ⁢ Bi ⁡ ( z ) ) Airy-Ai 𝑧 superscript 𝑒 2 𝜋 𝑖 3 1 2 superscript 𝑒 𝜋 𝑖 3 Airy-Ai 𝑧 𝑖 Airy-Bi 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(ze^{-2\pi i/3}\right)=\tfrac{1}{2% }e^{-\pi i/3}\left(\mathrm{Ai}\left(z\right)+i\mathrm{Bi}\left(z\right)\right)}}
\AiryAi@{ze^{- 2\pi i/3}} = \tfrac{1}{2}e^{-\pi i/3}\left(\AiryAi@{z}+ i\AiryBi@{z}\right)		 

AiryAi(z*exp(- 2*Pi*I/3)) = (1)/(2)*exp(- Pi*I/3)*(AiryAi(z)+ I*AiryBi(z))

AiryAi[z*Exp[- 2*Pi*I/3]] == Divide[1,2]*Exp[- Pi*I/3]*(AiryAi[z]+ I*AiryBi[z])
Successful Successful - Successful [Tested: 7]
9.2.E11 Ai ⁡ ( z ⁢ e + 2 ⁢ π ⁢ i / 3 ) = 1 2 ⁢ e + π ⁢ i / 3 ⁢ ( Ai ⁡ ( z ) - i ⁢ Bi ⁡ ( z ) ) Airy-Ai 𝑧 superscript 𝑒 2 𝜋 𝑖 3 1 2 superscript 𝑒 𝜋 𝑖 3 Airy-Ai 𝑧 𝑖 Airy-Bi 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(ze^{+2\pi i/3}\right)=\tfrac{1}{2% }e^{+\pi i/3}\left(\mathrm{Ai}\left(z\right)-i\mathrm{Bi}\left(z\right)\right)}}
\AiryAi@{ze^{+ 2\pi i/3}} = \tfrac{1}{2}e^{+\pi i/3}\left(\AiryAi@{z}- i\AiryBi@{z}\right)		 

AiryAi(z*exp(+ 2*Pi*I/3)) = (1)/(2)*exp(+ Pi*I/3)*(AiryAi(z)- I*AiryBi(z))

AiryAi[z*Exp[+ 2*Pi*I/3]] == Divide[1,2]*Exp[+ Pi*I/3]*(AiryAi[z]- I*AiryBi[z])
Successful Successful - Successful [Tested: 7]
9.2.E12 Ai ⁡ ( z ) + e - 2 ⁢ π ⁢ i / 3 ⁢ Ai ⁡ ( z ⁢ e - 2 ⁢ π ⁢ i / 3 ) + e 2 ⁢ π ⁢ i / 3 ⁢ Ai ⁡ ( z ⁢ e 2 ⁢ π ⁢ i / 3 ) = 0 Airy-Ai 𝑧 superscript 𝑒 2 𝜋 𝑖 3 Airy-Ai 𝑧 superscript 𝑒 2 𝜋 𝑖 3 superscript 𝑒 2 𝜋 𝑖 3 Airy-Ai 𝑧 superscript 𝑒 2 𝜋 𝑖 3 0 {\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)+e^{-2\pi i/3}\mathrm{Ai}% \left(ze^{-2\pi i/3}\right)+e^{2\pi i/3}\mathrm{Ai}\left(ze^{2\pi i/3}\right)=% 0}}
\AiryAi@{z}+e^{-2\pi i/3}\AiryAi@{ze^{-2\pi i/3}}+e^{2\pi i/3}\AiryAi@{ze^{2\pi i/3}} = 0		 

AiryAi(z)+ exp(- 2*Pi*I/3)*AiryAi(z*exp(- 2*Pi*I/3))+ exp(2*Pi*I/3)*AiryAi(z*exp(2*Pi*I/3)) = 0

AiryAi[z]+ Exp[- 2*Pi*I/3]*AiryAi[z*Exp[- 2*Pi*I/3]]+ Exp[2*Pi*I/3]*AiryAi[z*Exp[2*Pi*I/3]] == 0
Successful Successful - Successful [Tested: 7]
9.2.E13 Bi ⁡ ( z ) + e - 2 ⁢ π ⁢ i / 3 ⁢ Bi ⁡ ( z ⁢ e - 2 ⁢ π ⁢ i / 3 ) + e 2 ⁢ π ⁢ i / 3 ⁢ Bi ⁡ ( z ⁢ e 2 ⁢ π ⁢ i / 3 ) = 0 Airy-Bi 𝑧 superscript 𝑒 2 𝜋 𝑖 3 Airy-Bi 𝑧 superscript 𝑒 2 𝜋 𝑖 3 superscript 𝑒 2 𝜋 𝑖 3 Airy-Bi 𝑧 superscript 𝑒 2 𝜋 𝑖 3 0 {\displaystyle{\displaystyle\mathrm{Bi}\left(z\right)+e^{-2\pi i/3}\mathrm{Bi}% \left(ze^{-2\pi i/3}\right)+e^{2\pi i/3}\mathrm{Bi}\left(ze^{2\pi i/3}\right)=% 0}}
\AiryBi@{z}+e^{-2\pi i/3}\AiryBi@{ze^{-2\pi i/3}}+e^{2\pi i/3}\AiryBi@{ze^{2\pi i/3}} = 0		 

AiryBi(z)+ exp(- 2*Pi*I/3)*AiryBi(z*exp(- 2*Pi*I/3))+ exp(2*Pi*I/3)*AiryBi(z*exp(2*Pi*I/3)) = 0

AiryBi[z]+ Exp[- 2*Pi*I/3]*AiryBi[z*Exp[- 2*Pi*I/3]]+ Exp[2*Pi*I/3]*AiryBi[z*Exp[2*Pi*I/3]] == 0
Successful Successful - Successful [Tested: 7]
9.2.E14 Ai ⁡ ( - z ) = e π ⁢ i / 3 ⁢ Ai ⁡ ( z ⁢ e π ⁢ i / 3 ) + e - π ⁢ i / 3 ⁢ Ai ⁡ ( z ⁢ e - π ⁢ i / 3 ) Airy-Ai 𝑧 superscript 𝑒 𝜋 𝑖 3 Airy-Ai 𝑧 superscript 𝑒 𝜋 𝑖 3 superscript 𝑒 𝜋 𝑖 3 Airy-Ai 𝑧 superscript 𝑒 𝜋 𝑖 3 {\displaystyle{\displaystyle\mathrm{Ai}\left(-z\right)=e^{\pi i/3}\mathrm{Ai}% \left(ze^{\pi i/3}\right)+e^{-\pi i/3}\mathrm{Ai}\left(ze^{-\pi i/3}\right)}}
\AiryAi@{-z} = e^{\pi i/3}\AiryAi@{ze^{\pi i/3}}+e^{-\pi i/3}\AiryAi@{ze^{-\pi i/3}}		 

AiryAi(- z) = exp(Pi*I/3)*AiryAi(z*exp(Pi*I/3))+ exp(- Pi*I/3)*AiryAi(z*exp(- Pi*I/3))

AiryAi[- z] == Exp[Pi*I/3]*AiryAi[z*Exp[Pi*I/3]]+ Exp[- Pi*I/3]*AiryAi[z*Exp[- Pi*I/3]]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
9.2.E15 Bi ⁡ ( - z ) = e - π ⁢ i / 6 ⁢ Ai ⁡ ( z ⁢ e π ⁢ i / 3 ) + e π ⁢ i / 6 ⁢ Ai ⁡ ( z ⁢ e - π ⁢ i / 3 ) Airy-Bi 𝑧 superscript 𝑒 𝜋 𝑖 6 Airy-Ai 𝑧 superscript 𝑒 𝜋 𝑖 3 superscript 𝑒 𝜋 𝑖 6 Airy-Ai 𝑧 superscript 𝑒 𝜋 𝑖 3 {\displaystyle{\displaystyle\mathrm{Bi}\left(-z\right)=e^{-\pi i/6}\mathrm{Ai}% \left(ze^{\pi i/3}\right)+e^{\pi i/6}\mathrm{Ai}\left(ze^{-\pi i/3}\right)}}
\AiryBi@{-z} = e^{-\pi i/6}\AiryAi@{ze^{\pi i/3}}+e^{\pi i/6}\AiryAi@{ze^{-\pi i/3}}		 

AiryBi(- z) = exp(- Pi*I/6)*AiryAi(z*exp(Pi*I/3))+ exp(Pi*I/6)*AiryAi(z*exp(- Pi*I/3))

AiryBi[- z] == Exp[- Pi*I/6]*AiryAi[z*Exp[Pi*I/3]]+ Exp[Pi*I/6]*AiryAi[z*Exp[- Pi*I/3]]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
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