# Results of Airy and Related Functions

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9.2.E2 ${\displaystyle{\displaystyle w=\mathrm{Ai}\left(z\right),\;\mathrm{Bi}\left(z% \right),\;\mathrm{Ai}\left(ze^{-2\pi\mathrm{i}/3}\right)}}$ w = AiryAi(z), AiryBi(z), AiryAi(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 ${\displaystyle{\displaystyle w=\mathrm{Ai}\left(z\right),\;\mathrm{Bi}\left(z% \right),\;\mathrm{Ai}\left(ze^{+2\pi\mathrm{i}/3}\right)}}$ w = AiryAi(z), AiryBi(z), AiryAi(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 ${\displaystyle{\displaystyle\mathrm{Ai}\left(0\right)=\frac{1}{3^{2/3}\Gamma% \left(\tfrac{2}{3}\right)}}}$ AiryAi(0)=(1)/((3)^(2/ 3)* GAMMA((2)/(3))) AiryAi[0]=Divide[1,(3)^(2/ 3)* Gamma[Divide[2,3]]] Successful Successful - -
9.2.E4 ${\displaystyle{\displaystyle\mathrm{Ai}'\left(0\right)=-\frac{1}{3^{1/3}\Gamma% \left(\tfrac{1}{3}\right)}}}$ 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 - - 9.2.E5 ${\displaystyle{\displaystyle\mathrm{Bi}\left(0\right)=\frac{1}{3^{1/6}\Gamma% \left(\tfrac{2}{3}\right)}}}$ AiryBi(0)=(1)/((3)^(1/ 6)* GAMMA((2)/(3))) AiryBi[0]=Divide[1,(3)^(1/ 6)* Gamma[Divide[2,3]]] Successful Successful - - 9.2.E6 ${\displaystyle{\displaystyle\mathrm{Bi}'\left(0\right)=\frac{3^{1/6}}{\Gamma% \left(\tfrac{1}{3}\right)}}}$ 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 - -
9.2.E7 ${\displaystyle{\displaystyle\mathscr{W}\left\{\mathrm{Ai}\left(z\right),% \mathrm{Bi}\left(z\right)\right\}=\frac{1}{\pi}}}$ (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 -
9.2.E8 ${\displaystyle{\displaystyle\mathscr{W}\left\{\mathrm{Ai}\left(z\right),% \mathrm{Ai}\left(ze^{-2\pi i/3}\right)\right\}=\frac{e^{+\pi i/6}}{2\pi}}}$ (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 -
9.2.E8 ${\displaystyle{\displaystyle\mathscr{W}\left\{\mathrm{Ai}\left(z\right),% \mathrm{Ai}\left(ze^{+2\pi i/3}\right)\right\}=\frac{e^{-\pi i/6}}{2\pi}}}$ (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 -
9.2.E9 ${\displaystyle{\displaystyle\mathscr{W}\left\{\mathrm{Ai}\left(ze^{-2\pi i/3}% \right),\mathrm{Ai}\left(ze^{2\pi i/3}\right)\right\}=\frac{1}{2\pi i}}}$ (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 -
9.2.E10 ${\displaystyle{\displaystyle\mathrm{Bi}\left(z\right)=e^{-\pi i/6}\mathrm{Ai}% \left(ze^{-2\pi i/3}\right)+e^{\pi i/6}\mathrm{Ai}\left(ze^{2\pi i/3}\right)}}$ AiryBi(z)= 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]] Failure Successful Successful -
9.2.E11 ${\displaystyle{\displaystyle\mathrm{Ai}\left(ze^{-2\pi i/3}\right)=\tfrac{1}{2% }e^{-\pi i/3}\left(\mathrm{Ai}\left(z\right)+i\mathrm{Bi}\left(z\right)\right)}}$ AiryAi(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]) Failure Successful Successful -
9.2.E11 ${\displaystyle{\displaystyle\mathrm{Ai}\left(ze^{+2\pi i/3}\right)=\tfrac{1}{2% }e^{+\pi i/3}\left(\mathrm{Ai}\left(z\right)-i\mathrm{Bi}\left(z\right)\right)}}$ AiryAi(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]) Failure Successful Successful -
9.2.E12 ${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)+e^{-2\pi i/3}\mathrm{Ai}% \left(ze^{-2\pi i/3}\right)+e^{2\pi i/3}\mathrm{Ai}\left(ze^{2\pi i/3}\right)=% 0}}$ AiryAi(z)+ 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 Failure Successful Successful -
9.2.E13 ${\displaystyle{\displaystyle\mathrm{Bi}\left(z\right)+e^{-2\pi i/3}\mathrm{Bi}% \left(ze^{-2\pi i/3}\right)+e^{2\pi i/3}\mathrm{Bi}\left(ze^{2\pi i/3}\right)=% 0}}$ AiryBi(z)+ 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 Failure Successful Successful -
9.2.E14 ${\displaystyle{\displaystyle\mathrm{Ai}\left(-z\right)=e^{\pi i/3}\mathrm{Ai}% \left(ze^{\pi i/3}\right)+e^{-\pi i/3}\mathrm{Ai}\left(ze^{-\pi i/3}\right)}}$ AiryAi(- z)= 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 -
9.2.E15 ${\displaystyle{\displaystyle\mathrm{Bi}\left(-z\right)=e^{-\pi i/6}\mathrm{Ai}% \left(ze^{\pi i/3}\right)+e^{\pi i/6}\mathrm{Ai}\left(ze^{-\pi i/3}\right)}}$ AiryBi(- z)= 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 -
9.5.E1 ${\displaystyle{\displaystyle\mathrm{Ai}\left(x\right)=\frac{1}{\pi}\int_{0}^{% \infty}\cos\left(\tfrac{1}{3}t^{3}+xt\right)\mathrm{d}t}}$ AiryAi(x)=(1)/(Pi)*int(cos((1)/(3)*(t)^(3)+ x*t), t = 0..infinity) AiryAi[x]=Divide[1,Pi]*Integrate[Cos[Divide[1,3]*(t)^(3)+ x*t], {t, 0, Infinity}] Successful Failure - Skip
9.5.E2 ${\displaystyle{\displaystyle\mathrm{Ai}\left(-x\right)=\frac{x^{\ifrac{1}{2}}}% {\pi}\int_{-1}^{\infty}\cos\left(x^{\ifrac{3}{2}}(\tfrac{1}{3}t^{3}+t^{2}-% \tfrac{2}{3})\right)\mathrm{d}t}}$ AiryAi(- x)=((x)^((1)/(2)))/(Pi)*int(cos((x)^((3)/(2))*((1)/(3)*(t)^(3)+ (t)^(2)-(2)/(3))), t = - 1..infinity) AiryAi[- x]=Divide[(x)^(Divide[1,2]),Pi]*Integrate[Cos[(x)^(Divide[3,2])*(Divide[1,3]*(t)^(3)+ (t)^(2)-Divide[2,3])], {t, - 1, Infinity}] Failure Failure Skip Error
9.5.E3 ${\displaystyle{\displaystyle\mathrm{Bi}\left(x\right)=\frac{1}{\pi}\int_{0}^{% \infty}\exp\left(-{\tfrac{1}{3}}t^{3}+xt\right)\mathrm{d}t+\frac{1}{\pi}\int_{% 0}^{\infty}\sin\left(\tfrac{1}{3}t^{3}+xt\right)\mathrm{d}t}}$ AiryBi(x)=(1)/(Pi)*int(exp(-(1)/(3)*(t)^(3)+ x*t), t = 0..infinity)+(1)/(Pi)*int(sin((1)/(3)*(t)^(3)+ x*t), t = 0..infinity) AiryBi[x]=Divide[1,Pi]*Integrate[Exp[-Divide[1,3]*(t)^(3)+ x*t], {t, 0, Infinity}]+Divide[1,Pi]*Integrate[Sin[Divide[1,3]*(t)^(3)+ x*t], {t, 0, Infinity}] Failure Failure Skip
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[Plus[AiryBi[x], Times[Rational[-1, 6], Power[Pi, -1], Plus[Times[4, Pi, AiryBi[x]], Times[3, Power[x, 2], HypergeometricPFQ[{1}, {Rational[4, 3], Rational[5, 3]}, Times[Rational[1, 9], Power[x, 3]]]]]], Times[-1, Power[Pi, -1], Plus[Times[Rational[1, 3], Pi, AiryBi[x]], Times[Rational[-1, 2], Power[x, 2], HypergeometricPFQ[{1}, {Rational[2, 3], Rational[5, 6], Rational[7, 6], Rational[4, 3]}, Times[Rational[1, 1296], Power[x, 6]]]], Times[Rational[-1, 40], Power[x, 5], HypergeometricPFQ[{1}, {Rational[7, 6], Rational[4, 3], Rational[5, 3], Rational[11, 6]}, Times[Rational[1, 1296], Power[x, 6]]]]]]], And[Element[x, Reals], Less[Re[x], 0]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[Plus[AiryBi[x], Times[Rational[-1, 6], Power[Pi, -1], Plus[Times[4, Pi, AiryBi[x]], Times[3, Power[x, 2], HypergeometricPFQ[{1}, {Rational[4, 3], Rational[5, 3]}, Times[Rational[1, 9], Power[x, 3]]]]]], Times[-1, Power[Pi, -1], Plus[Times[Rational[1, 3], Pi, AiryBi[x]], Times[Rational[-1, 2], Power[x, 2], HypergeometricPFQ[{1}, {Rational[2, 3], Rational[5, 6], Rational[7, 6], Rational[4, 3]}, Times[Rational[1, 1296], Power[x, 6]]]], Times[Rational[-1, 40], Power[x, 5], HypergeometricPFQ[{1}, {Rational[7, 6], Rational[4, 3], Rational[5, 3], Rational[11, 6]}, Times[Rational[1, 1296], Power[x, 6]]]]]]], And[Element[x, Reals], Less[Re[x], 0]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[Plus[AiryBi[x], Times[Rational[-1, 6], Power[Pi, -1], Plus[Times[4, Pi, AiryBi[x]], Times[3, Power[x, 2], HypergeometricPFQ[{1}, {Rational[4, 3], Rational[5, 3]}, Times[Rational[1, 9], Power[x, 3]]]]]], Times[-1, Power[Pi, -1], Plus[Times[Rational[1, 3], Pi, AiryBi[x]], Times[Rational[-1, 2], Power[x, 2], HypergeometricPFQ[{1}, {Rational[2, 3], Rational[5, 6], Rational[7, 6], Rational[4, 3]}, Times[Rational[1, 1296], Power[x, 6]]]], Times[Rational[-1, 40], Power[x, 5], HypergeometricPFQ[{1}, {Rational[7, 6], Rational[4, 3], Rational[5, 3], Rational[11, 6]}, Times[Rational[1, 1296], Power[x, 6]]]]]]], And[Element[x, Reals], Less[Re[x], 0]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[Plus[AiryBi[x], Times[Rational[-1, 6], Power[Pi, -1], Plus[Times[4, Pi, AiryBi[x]], Times[3, Power[x, 2], HypergeometricPFQ[{1}, {Rational[4, 3], Rational[5, 3]}, Times[Rational[1, 9], Power[x, 3]]]]]], Times[-1, Power[Pi, -1], Plus[Times[Rational[1, 3], Pi, AiryBi[x]], Times[Rational[-1, 2], Power[x, 2], HypergeometricPFQ[{1}, {Rational[2, 3], Rational[5, 6], Rational[7, 6], Rational[4, 3]}, Times[Rational[1, 1296], Power[x, 6]]]], Times[Rational[-1, 40], Power[x, 5], HypergeometricPFQ[{1}, {Rational[7, 6], Rational[4, 3], Rational[5, 3], Rational[11, 6]}, Times[Rational[1, 1296], Power[x, 6]]]]]]], And[Element[x, Reals], Less[Re[x], 0]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.5.E4 ${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\frac{1}{2\pi i}\int_{% \infty e^{-\pi i/3}}^{\infty e^{\pi i/3}}\exp\left(\tfrac{1}{3}t^{3}-zt\right)% \mathrm{d}t}}$ AiryAi(z)=(1)/(2*Pi*I)*int(exp((1)/(3)*(t)^(3)- z*t), t = infinity*exp(- Pi*I/ 3)..infinity*exp(Pi*I/ 3)) AiryAi[z]=Divide[1,2*Pi*I]*Integrate[Exp[Divide[1,3]*(t)^(3)- z*t], {t, Infinity*Exp[- Pi*I/ 3], Infinity*Exp[Pi*I/ 3]}] Failure Failure Skip Error
9.5.E5 ${\displaystyle{\displaystyle\mathrm{Bi}\left(z\right)=\frac{1}{2\pi}\int_{-% \infty}^{\infty e^{\pi i/3}}\exp\left(\tfrac{1}{3}t^{3}-zt\right)\mathrm{d}t+% \dfrac{1}{2\pi}\int_{-\infty}^{\infty e^{-\pi i/3}}\exp\left(\tfrac{1}{3}t^{3}% -zt\right)\mathrm{d}t}}$ AiryBi(z)=(1)/(2*Pi)*int(exp((1)/(3)*(t)^(3)- z*t), t = - infinity..infinity*exp(Pi*I/ 3))+(1)/(2*Pi)*int(exp((1)/(3)*(t)^(3)- z*t), t = - infinity..infinity*exp(- Pi*I/ 3)) AiryBi[z]=Divide[1,2*Pi]*Integrate[Exp[Divide[1,3]*(t)^(3)- z*t], {t, - Infinity, Infinity*Exp[Pi*I/ 3]}]+Divide[1,2*Pi]*Integrate[Exp[Divide[1,3]*(t)^(3)- z*t], {t, - Infinity, Infinity*Exp[- Pi*I/ 3]}] Failure Failure Skip Error
9.5.E6 ${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\frac{\sqrt{3}}{2\pi}% \int_{0}^{\infty}\exp\left(-\frac{t^{3}}{3}-\frac{z^{3}}{3t^{3}}\right)\mathrm% {d}t}}$ AiryAi(z)=(sqrt(3))/(2*Pi)*int(exp(-((t)^(3))/(3)-((z)^(3))/(3*(t)^(3))), t = 0..infinity) AiryAi[z]=Divide[Sqrt[3],2*Pi]*Integrate[Exp[-Divide[(t)^(3),3]-Divide[(z)^(3),3*(t)^(3)]], {t, 0, Infinity}] Successful Failure - Skip
9.5.E7 ${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\frac{e^{-\zeta}}{\pi}% \int_{0}^{\infty}\exp\left(-z^{\ifrac{1}{2}}t^{2}\right)\cos\left(\tfrac{1}{3}% t^{3}\right)\mathrm{d}t}}$ AiryAi(z)=(exp(-(2)/(3)*(z)^((3)/(2))))/(Pi)*int(exp(- (z)^((1)/(2))* (t)^(2))*cos((1)/(3)*(t)^(3)), t = 0..infinity) AiryAi[z]=Divide[Exp[-Divide[2,3]*(z)^(Divide[3,2])],Pi]*Integrate[Exp[- (z)^(Divide[1,2])* (t)^(2)]*Cos[Divide[1,3]*(t)^(3)], {t, 0, Infinity}] Failure Failure Skip Successful
9.5.E8 ${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\frac{e^{-\zeta}\zeta^{% \ifrac{-1}{6}}}{\sqrt{\pi}(48)^{\ifrac{1}{6}}\Gamma\left(\frac{5}{6}\right)}% \int_{0}^{\infty}e^{-t}t^{-\ifrac{1}{6}}\left(2+\frac{t}{\zeta}\right)^{-% \ifrac{1}{6}}\mathrm{d}t}}$ AiryAi(z)=(exp(-(2)/(3)*(z)^((3)/(2)))*(2)/(3)*((z)^((3)/(2)))^((- 1)/(6)))/(sqrt(Pi)*(48)^((1)/(6))* GAMMA((5)/(6)))*int(exp(- t)*(t)^(-(1)/(6))*(2 +(t)/((2)/(3)*(z)^((3)/(2))))^(-(1)/(6)), t = 0..infinity) AiryAi[z]=Divide[Exp[-Divide[2,3]*(z)^(Divide[3,2])]*Divide[2,3]*((z)^(Divide[3,2]))^(Divide[- 1,6]),Sqrt[Pi]*(48)^(Divide[1,6])* Gamma[Divide[5,6]]]*Integrate[Exp[- t]*(t)^(-Divide[1,6])*(2 +Divide[t,Divide[2,3]*(z)^(Divide[3,2])])^(-Divide[1,6]), {t, 0, Infinity}] Failure Failure Skip Error
9.6.E2 ${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\pi^{-1}\sqrt{z/3}K_{+1/% 3}\left(\zeta\right)}}$ AiryAi(z)= (Pi)^(- 1)*sqrt(z/ 3)*BesselK(+ 1/ 3, (2)/(3)*(z)^((3)/(2))) AiryAi[z]= (Pi)^(- 1)*Sqrt[z/ 3]*BesselK[+ 1/ 3, Divide[2,3]*(z)^(Divide[3,2])] Failure Failure
Fail
2.833765278-.3039461853*I <- {z = -2^(1/2)-I*2^(1/2)}
2.833765278+.3039461853*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[2.8337652800788264, -0.3039461861802381] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8337652800788264, 0.3039461861802381] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E2 ${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\pi^{-1}\sqrt{z/3}K_{-1/% 3}\left(\zeta\right)}}$ AiryAi(z)= (Pi)^(- 1)*sqrt(z/ 3)*BesselK(- 1/ 3, (2)/(3)*(z)^((3)/(2))) AiryAi[z]= (Pi)^(- 1)*Sqrt[z/ 3]*BesselK[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])] Failure Failure
Fail
2.833765278-.3039461853*I <- {z = -2^(1/2)-I*2^(1/2)}
2.833765278+.3039461853*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[2.8337652800788264, -0.3039461861802381] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8337652800788264, 0.3039461861802381] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E2 ${\displaystyle{\displaystyle\pi^{-1}\sqrt{z/3}K_{+1/3}\left(\zeta\right)=% \tfrac{1}{3}\sqrt{z}\left(I_{-1/3}\left(\zeta\right)-I_{1/3}\left(\zeta\right)% \right)}}$ (Pi)^(- 1)*sqrt(z/ 3)*BesselK(+ 1/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(3)*sqrt(z)*(BesselI(- 1/ 3, (2)/(3)*(z)^((3)/(2)))- BesselI(1/ 3, (2)/(3)*(z)^((3)/(2)))) (Pi)^(- 1)*Sqrt[z/ 3]*BesselK[+ 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,3]*Sqrt[z]*(BesselI[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]- BesselI[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Successful Successful - -
9.6.E2 ${\displaystyle{\displaystyle\pi^{-1}\sqrt{z/3}K_{-1/3}\left(\zeta\right)=% \tfrac{1}{3}\sqrt{z}\left(I_{-1/3}\left(\zeta\right)-I_{1/3}\left(\zeta\right)% \right)}}$ (Pi)^(- 1)*sqrt(z/ 3)*BesselK(- 1/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(3)*sqrt(z)*(BesselI(- 1/ 3, (2)/(3)*(z)^((3)/(2)))- BesselI(1/ 3, (2)/(3)*(z)^((3)/(2)))) (Pi)^(- 1)*Sqrt[z/ 3]*BesselK[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,3]*Sqrt[z]*(BesselI[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]- BesselI[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Successful Successful - -
9.6.E2 ${\displaystyle{\displaystyle\tfrac{1}{3}\sqrt{z}\left(I_{-1/3}\left(\zeta% \right)-I_{1/3}\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}e^{2\pi i/3}{H^% {(1)}_{1/3}}\left(\zeta e^{\pi i/2}\right)}}$ (1)/(3)*sqrt(z)*(BesselI(- 1/ 3, (2)/(3)*(z)^((3)/(2)))- BesselI(1/ 3, (2)/(3)*(z)^((3)/(2))))=(1)/(2)*sqrt(z/ 3)*exp(2*Pi*I/ 3)*HankelH1(1/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2)) Divide[1,3]*Sqrt[z]*(BesselI[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]- BesselI[1/ 3, Divide[2,3]*(z)^(Divide[3,2])])=Divide[1,2]*Sqrt[z/ 3]*Exp[2*Pi*I/ 3]*HankelH1[1/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]] Failure Failure Skip Skip
9.6.E2 ${\displaystyle{\displaystyle\tfrac{1}{2}\sqrt{z/3}e^{2\pi i/3}{H^{(1)}_{1/3}}% \left(\zeta e^{\pi i/2}\right)=\tfrac{1}{2}\sqrt{z/3}e^{\pi i/3}{H^{(1)}_{-1/3% }}\left(\zeta e^{\pi i/2}\right)}}$ (1)/(2)*sqrt(z/ 3)*exp(2*Pi*I/ 3)*HankelH1(1/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2))=(1)/(2)*sqrt(z/ 3)*exp(Pi*I/ 3)*HankelH1(- 1/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2)) Divide[1,2]*Sqrt[z/ 3]*Exp[2*Pi*I/ 3]*HankelH1[1/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]]=Divide[1,2]*Sqrt[z/ 3]*Exp[Pi*I/ 3]*HankelH1[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]] Successful Failure - Skip
9.6.E2 ${\displaystyle{\displaystyle\tfrac{1}{2}\sqrt{z/3}e^{\pi i/3}{H^{(1)}_{-1/3}}% \left(\zeta e^{\pi i/2}\right)=\tfrac{1}{2}\sqrt{z/3}e^{-2\pi i/3}{H^{(2)}_{1/% 3}}\left(\zeta e^{-\pi i/2}\right)}}$ (1)/(2)*sqrt(z/ 3)*exp(Pi*I/ 3)*HankelH1(- 1/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2))=(1)/(2)*sqrt(z/ 3)*exp(- 2*Pi*I/ 3)*HankelH2(1/ 3, (2)/(3)*(z)^((3)/(2))*exp(- Pi*I/ 2)) Divide[1,2]*Sqrt[z/ 3]*Exp[Pi*I/ 3]*HankelH1[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]]=Divide[1,2]*Sqrt[z/ 3]*Exp[- 2*Pi*I/ 3]*HankelH2[1/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[- Pi*I/ 2]] Failure Failure Skip Skip
9.6.E2 ${\displaystyle{\displaystyle\tfrac{1}{2}\sqrt{z/3}e^{-2\pi i/3}{H^{(2)}_{1/3}}% \left(\zeta e^{-\pi i/2}\right)=\tfrac{1}{2}\sqrt{z/3}e^{-\pi i/3}{H^{(2)}_{-1% /3}}\left(\zeta e^{-\pi i/2}\right)}}$ (1)/(2)*sqrt(z/ 3)*exp(- 2*Pi*I/ 3)*HankelH2(1/ 3, (2)/(3)*(z)^((3)/(2))*exp(- Pi*I/ 2))=(1)/(2)*sqrt(z/ 3)*exp(- Pi*I/ 3)*HankelH2(- 1/ 3, (2)/(3)*(z)^((3)/(2))*exp(- Pi*I/ 2)) Divide[1,2]*Sqrt[z/ 3]*Exp[- 2*Pi*I/ 3]*HankelH2[1/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[- Pi*I/ 2]]=Divide[1,2]*Sqrt[z/ 3]*Exp[- Pi*I/ 3]*HankelH2[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[- Pi*I/ 2]] Successful Failure - Skip
9.6.E3 ${\displaystyle{\displaystyle\mathrm{Ai}'\left(z\right)=-\pi^{-1}(z/\sqrt{3})K_% {+2/3}\left(\zeta\right)}}$ subs( temp=z, diff( AiryAi(temp), temp$(1) ) )= - (Pi)^(- 1)*(z/sqrt(3))* BesselK(+ 2/ 3, (2)/(3)*(z)^((3)/(2))) (D[AiryAi[temp], {temp, 1}]/.temp-> z)= - (Pi)^(- 1)*(z/Sqrt[3])* BesselK[+ 2/ 3, Divide[2,3]*(z)^(Divide[3,2])] Failure Failure Fail -.7883076520+3.485863958*I <- {z = -2^(1/2)-I*2^(1/2)} -.7883076520-3.485863958*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[-0.7883076520663912, 3.485863960601928] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.7883076520663912, -3.485863960601928] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E3 ${\displaystyle{\displaystyle\mathrm{Ai}'\left(z\right)=-\pi^{-1}(z/\sqrt{3})K_% {-2/3}\left(\zeta\right)}}$ subs( temp=z, diff( AiryAi(temp), temp$(1) ) )= - (Pi)^(- 1)*(z/sqrt(3))* BesselK(- 2/ 3, (2)/(3)*(z)^((3)/(2))) (D[AiryAi[temp], {temp, 1}]/.temp-> z)= - (Pi)^(- 1)*(z/Sqrt[3])* BesselK[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])] Failure Failure
Fail
-.7883076520+3.485863958*I <- {z = -2^(1/2)-I*2^(1/2)}
-.7883076520-3.485863958*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-0.7883076520663912, 3.485863960601928] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.7883076520663912, -3.485863960601928] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E3 ${\displaystyle{\displaystyle-\pi^{-1}(z/\sqrt{3})K_{+2/3}\left(\zeta\right)=(z% /3)\left(I_{2/3}\left(\zeta\right)-I_{-2/3}\left(\zeta\right)\right)}}$ - (Pi)^(- 1)*(z/sqrt(3))* BesselK(+ 2/ 3, (2)/(3)*(z)^((3)/(2)))=(z/ 3)*(BesselI(2/ 3, (2)/(3)*(z)^((3)/(2)))- BesselI(- 2/ 3, (2)/(3)*(z)^((3)/(2)))) - (Pi)^(- 1)*(z/Sqrt[3])* BesselK[+ 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]=(z/ 3)*(BesselI[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]- BesselI[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Successful Successful - -
9.6.E3 ${\displaystyle{\displaystyle-\pi^{-1}(z/\sqrt{3})K_{-2/3}\left(\zeta\right)=(z% /3)\left(I_{2/3}\left(\zeta\right)-I_{-2/3}\left(\zeta\right)\right)}}$ - (Pi)^(- 1)*(z/sqrt(3))* BesselK(- 2/ 3, (2)/(3)*(z)^((3)/(2)))=(z/ 3)*(BesselI(2/ 3, (2)/(3)*(z)^((3)/(2)))- BesselI(- 2/ 3, (2)/(3)*(z)^((3)/(2)))) - (Pi)^(- 1)*(z/Sqrt[3])* BesselK[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]=(z/ 3)*(BesselI[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]- BesselI[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Successful Successful - -
9.6.E3 ${\displaystyle{\displaystyle(z/3)\left(I_{2/3}\left(\zeta\right)-I_{-2/3}\left% (\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})e^{-\pi i/6}{H^{(1)}_{2/3}}\left(% \zeta e^{\pi i/2}\right)}}$ (z/ 3)*(BesselI(2/ 3, (2)/(3)*(z)^((3)/(2)))- BesselI(- 2/ 3, (2)/(3)*(z)^((3)/(2))))=(1)/(2)*(z/sqrt(3))* exp(- Pi*I/ 6)*HankelH1(2/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2)) (z/ 3)*(BesselI[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]- BesselI[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])])=Divide[1,2]*(z/Sqrt[3])* Exp[- Pi*I/ 6]*HankelH1[2/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]] Failure Failure Skip Skip
9.6.E3 ${\displaystyle{\displaystyle\tfrac{1}{2}(z/\sqrt{3})e^{-\pi i/6}{H^{(1)}_{2/3}% }\left(\zeta e^{\pi i/2}\right)=\tfrac{1}{2}(z/\sqrt{3})e^{-5\pi i/6}{H^{(1)}_% {-2/3}}\left(\zeta e^{\pi i/2}\right)}}$ (1)/(2)*(z/sqrt(3))* exp(- Pi*I/ 6)*HankelH1(2/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2))=(1)/(2)*(z/sqrt(3))* exp(- 5*Pi*I/ 6)*HankelH1(- 2/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2)) Divide[1,2]*(z/Sqrt[3])* Exp[- Pi*I/ 6]*HankelH1[2/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]]=Divide[1,2]*(z/Sqrt[3])* Exp[- 5*Pi*I/ 6]*HankelH1[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]] Successful Failure - Skip
9.6.E3 ${\displaystyle{\displaystyle\tfrac{1}{2}(z/\sqrt{3})e^{-5\pi i/6}{H^{(1)}_{-2/% 3}}\left(\zeta e^{\pi i/2}\right)=\tfrac{1}{2}(z/\sqrt{3})e^{\pi i/6}{H^{(2)}_% {2/3}}\left(\zeta e^{-\pi i/2}\right)}}$ (1)/(2)*(z/sqrt(3))* exp(- 5*Pi*I/ 6)*HankelH1(- 2/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2))=(1)/(2)*(z/sqrt(3))* exp(Pi*I/ 6)*HankelH2(2/ 3, (2)/(3)*(z)^((3)/(2))*exp(- Pi*I/ 2)) Divide[1,2]*(z/Sqrt[3])* Exp[- 5*Pi*I/ 6]*HankelH1[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]]=Divide[1,2]*(z/Sqrt[3])* Exp[Pi*I/ 6]*HankelH2[2/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[- Pi*I/ 2]] Failure Failure Skip Skip
9.6.E3 ${\displaystyle{\displaystyle\tfrac{1}{2}(z/\sqrt{3})e^{\pi i/6}{H^{(2)}_{2/3}}% \left(\zeta e^{-\pi i/2}\right)=\tfrac{1}{2}(z/\sqrt{3})e^{5\pi i/6}{H^{(2)}_{% -2/3}}\left(\zeta e^{-\pi i/2}\right)}}$ (1)/(2)*(z/sqrt(3))* exp(Pi*I/ 6)*HankelH2(2/ 3, (2)/(3)*(z)^((3)/(2))*exp(- Pi*I/ 2))=(1)/(2)*(z/sqrt(3))* exp(5*Pi*I/ 6)*HankelH2(- 2/ 3, (2)/(3)*(z)^((3)/(2))*exp(- Pi*I/ 2)) Divide[1,2]*(z/Sqrt[3])* Exp[Pi*I/ 6]*HankelH2[2/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[- Pi*I/ 2]]=Divide[1,2]*(z/Sqrt[3])* Exp[5*Pi*I/ 6]*HankelH2[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[- Pi*I/ 2]] Successful Failure - Skip
9.6.E4 ${\displaystyle{\displaystyle\mathrm{Bi}\left(z\right)=\sqrt{z/3}\left(I_{1/3}% \left(\zeta\right)+I_{-1/3}\left(\zeta\right)\right)}}$ AiryBi(z)=sqrt(z/ 3)*(BesselI(1/ 3, (2)/(3)*(z)^((3)/(2)))+ BesselI(- 1/ 3, (2)/(3)*(z)^((3)/(2)))) AiryBi[z]=Sqrt[z/ 3]*(BesselI[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ BesselI[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Failure Failure
Fail
.323091265e-1+.116725832*I <- {z = -2^(1/2)-I*2^(1/2)}
.323091265e-1-.116725832*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[0.032309126109843156, 0.11672583064563491] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.032309126109843156, -0.11672583064563491] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E4 ${\displaystyle{\displaystyle\sqrt{z/3}\left(I_{1/3}\left(\zeta\right)+I_{-1/3}% \left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/6}{H^{(1)}_{1/3% }}\left(\zeta e^{-\pi i/2}\right)+e^{-\pi i/6}{H^{(2)}_{1/3}}\left(\zeta e^{% \pi i/2}\right)\right)}}$ sqrt(z/ 3)*(BesselI(1/ 3, (2)/(3)*(z)^((3)/(2)))+ BesselI(- 1/ 3, (2)/(3)*(z)^((3)/(2))))=(1)/(2)*sqrt(z/ 3)*(exp(Pi*I/ 6)*HankelH1(1/ 3, (2)/(3)*(z)^((3)/(2))*exp(- Pi*I/ 2))+ exp(- Pi*I/ 6)*HankelH2(1/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2))) Sqrt[z/ 3]*(BesselI[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ BesselI[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])])=Divide[1,2]*Sqrt[z/ 3]*(Exp[Pi*I/ 6]*HankelH1[1/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[- Pi*I/ 2]]+ Exp[- Pi*I/ 6]*HankelH2[1/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]]) Failure Failure
Fail
-.1681276560-1.475245556*I <- {z = -2^(1/2)-I*2^(1/2)}
-.1681276560+1.475245556*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-0.16812765614504083, -1.4752455553622306] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.16812765614504083, 1.4752455553622306] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E4 ${\displaystyle{\displaystyle\tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/6}{H^{(1)}_{1% /3}}\left(\zeta e^{-\pi i/2}\right)+e^{-\pi i/6}{H^{(2)}_{1/3}}\left(\zeta e^{% \pi i/2}\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{-\pi i/6}{H^{(1)}_{-1/3}% }\left(\zeta e^{-\pi i/2}\right)+e^{\pi i/6}{H^{(2)}_{-1/3}}\left(\zeta e^{\pi i% /2}\right)\right)}}$ (1)/(2)*sqrt(z/ 3)*(exp(Pi*I/ 6)*HankelH1(1/ 3, (2)/(3)*(z)^((3)/(2))*exp(- Pi*I/ 2))+ exp(- Pi*I/ 6)*HankelH2(1/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2)))=(1)/(2)*sqrt(z/ 3)*(exp(- Pi*I/ 6)*HankelH1(- 1/ 3, (2)/(3)*(z)^((3)/(2))*exp(- Pi*I/ 2))+ exp(Pi*I/ 6)*HankelH2(- 1/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2))) Divide[1,2]*Sqrt[z/ 3]*(Exp[Pi*I/ 6]*HankelH1[1/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[- Pi*I/ 2]]+ Exp[- Pi*I/ 6]*HankelH2[1/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]])=Divide[1,2]*Sqrt[z/ 3]*(Exp[- Pi*I/ 6]*HankelH1[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[- Pi*I/ 2]]+ Exp[Pi*I/ 6]*HankelH2[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]]) Successful Failure - Skip
9.6.E5 ${\displaystyle{\displaystyle\mathrm{Bi}'\left(z\right)=(z/\sqrt{3})\left(I_{2/% 3}\left(\zeta\right)+I_{-2/3}\left(\zeta\right)\right)}}$ subs( temp=z, diff( AiryBi(temp), temp$(1) ) )=(z/sqrt(3))*(BesselI(2/ 3, (2)/(3)*(z)^((3)/(2)))+ BesselI(- 2/ 3, (2)/(3)*(z)^((3)/(2)))) (D[AiryBi[temp], {temp, 1}]/.temp-> z)=(z/Sqrt[3])*(BesselI[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ BesselI[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Failure Failure Fail .181539689+.267445042e-1*I <- {z = -2^(1/2)-I*2^(1/2)} .181539689-.267445042e-1*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[0.18153969005752768, 0.026744504839266825] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[0.18153969005752768, -0.026744504839266825] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E5 ${\displaystyle{\displaystyle(z/\sqrt{3})\left(I_{2/3}\left(\zeta\right)+I_{-2/% 3}\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{\pi i/3}{H^{(1)}_% {2/3}}\left(\zeta e^{-\pi i/2}\right)+e^{-\pi i/3}{H^{(2)}_{2/3}}\left(\zeta e% ^{\pi i/2}\right)\right)}}$ (z/sqrt(3))*(BesselI(2/ 3, (2)/(3)*(z)^((3)/(2)))+ BesselI(- 2/ 3, (2)/(3)*(z)^((3)/(2))))=(1)/(2)*(z/sqrt(3))*(exp(Pi*I/ 3)*HankelH1(2/ 3, (2)/(3)*(z)^((3)/(2))*exp(- Pi*I/ 2))+ exp(- Pi*I/ 3)*HankelH2(2/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2))) (z/Sqrt[3])*(BesselI[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ BesselI[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])])=Divide[1,2]*(z/Sqrt[3])*(Exp[Pi*I/ 3]*HankelH1[2/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[- Pi*I/ 2]]+ Exp[- Pi*I/ 3]*HankelH2[2/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]]) Failure Failure Fail 1.652162135+.3807815744*I <- {z = -2^(1/2)-I*2^(1/2)} 1.652162135-.3807815744*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[1.6521621352721998, 0.3807815736135619] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[1.6521621352721998, -0.3807815736135619] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E5 ${\displaystyle{\displaystyle\tfrac{1}{2}(z/\sqrt{3})\left(e^{\pi i/3}{H^{(1)}_% {2/3}}\left(\zeta e^{-\pi i/2}\right)+e^{-\pi i/3}{H^{(2)}_{2/3}}\left(\zeta e% ^{\pi i/2}\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{-\pi i/3}{H^{(1)}_{-% 2/3}}\left(\zeta e^{-\pi i/2}\right)+e^{\pi i/3}{H^{(2)}_{-2/3}}\left(\zeta e^% {\pi i/2}\right)\right)}}$ (1)/(2)*(z/sqrt(3))*(exp(Pi*I/ 3)*HankelH1(2/ 3, (2)/(3)*(z)^((3)/(2))*exp(- Pi*I/ 2))+ exp(- Pi*I/ 3)*HankelH2(2/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2)))=(1)/(2)*(z/sqrt(3))*(exp(- Pi*I/ 3)*HankelH1(- 2/ 3, (2)/(3)*(z)^((3)/(2))*exp(- Pi*I/ 2))+ exp(Pi*I/ 3)*HankelH2(- 2/ 3, (2)/(3)*(z)^((3)/(2))*exp(Pi*I/ 2))) Divide[1,2]*(z/Sqrt[3])*(Exp[Pi*I/ 3]*HankelH1[2/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[- Pi*I/ 2]]+ Exp[- Pi*I/ 3]*HankelH2[2/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]])=Divide[1,2]*(z/Sqrt[3])*(Exp[- Pi*I/ 3]*HankelH1[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[- Pi*I/ 2]]+ Exp[Pi*I/ 3]*HankelH2[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])*Exp[Pi*I/ 2]]) Successful Failure - Skip 9.6.E6 ${\displaystyle{\displaystyle\mathrm{Ai}\left(-z\right)=(\sqrt{z}/3)\left(J_{1/% 3}\left(\zeta\right)+J_{-1/3}\left(\zeta\right)\right)}}$ AiryAi(- z)=(sqrt(z)/ 3)*(BesselJ(1/ 3, (2)/(3)*(z)^((3)/(2)))+ BesselJ(- 1/ 3, (2)/(3)*(z)^((3)/(2)))) AiryAi[- z]=(Sqrt[z]/ 3)*(BesselJ[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ BesselJ[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Failure Failure Fail -.5274645816-.7652257224e-1*I <- {z = -2^(1/2)-I*2^(1/2)} -.5274645816+.7652257224e-1*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[-0.5274645818155765, -0.0765225723412053] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.5274645818155765, 0.0765225723412053] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E6 ${\displaystyle{\displaystyle(\sqrt{z}/3)\left(J_{1/3}\left(\zeta\right)+J_{-1/% 3}\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/6}{H^{(1)}_{1% /3}}\left(\zeta\right)+e^{-\pi i/6}{H^{(2)}_{1/3}}\left(\zeta\right)\right)}}$ (sqrt(z)/ 3)*(BesselJ(1/ 3, (2)/(3)*(z)^((3)/(2)))+ BesselJ(- 1/ 3, (2)/(3)*(z)^((3)/(2))))=(1)/(2)*sqrt(z/ 3)*(exp(Pi*I/ 6)*HankelH1(1/ 3, (2)/(3)*(z)^((3)/(2)))+ exp(- Pi*I/ 6)*HankelH2(1/ 3, (2)/(3)*(z)^((3)/(2)))) (Sqrt[z]/ 3)*(BesselJ[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ BesselJ[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])])=Divide[1,2]*Sqrt[z/ 3]*(Exp[Pi*I/ 6]*HankelH1[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ Exp[- Pi*I/ 6]*HankelH2[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Successful Failure - Successful 9.6.E6 ${\displaystyle{\displaystyle\tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/6}{H^{(1)}_{1% /3}}\left(\zeta\right)+e^{-\pi i/6}{H^{(2)}_{1/3}}\left(\zeta\right)\right)=% \tfrac{1}{2}\sqrt{z/3}\left(e^{-\pi i/6}{H^{(1)}_{-1/3}}\left(\zeta\right)+e^{% \pi i/6}{H^{(2)}_{-1/3}}\left(\zeta\right)\right)}}$ (1)/(2)*sqrt(z/ 3)*(exp(Pi*I/ 6)*HankelH1(1/ 3, (2)/(3)*(z)^((3)/(2)))+ exp(- Pi*I/ 6)*HankelH2(1/ 3, (2)/(3)*(z)^((3)/(2))))=(1)/(2)*sqrt(z/ 3)*(exp(- Pi*I/ 6)*HankelH1(- 1/ 3, (2)/(3)*(z)^((3)/(2)))+ exp(Pi*I/ 6)*HankelH2(- 1/ 3, (2)/(3)*(z)^((3)/(2)))) Divide[1,2]*Sqrt[z/ 3]*(Exp[Pi*I/ 6]*HankelH1[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ Exp[- Pi*I/ 6]*HankelH2[1/ 3, Divide[2,3]*(z)^(Divide[3,2])])=Divide[1,2]*Sqrt[z/ 3]*(Exp[- Pi*I/ 6]*HankelH1[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ Exp[Pi*I/ 6]*HankelH2[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Successful Successful - - 9.6.E7 ${\displaystyle{\displaystyle\mathrm{Ai}'\left(-z\right)=(z/3)\left(J_{2/3}% \left(\zeta\right)-J_{-2/3}\left(\zeta\right)\right)}}$ subs( temp=- z, diff( AiryAi(temp), temp$(1) ) )=(z/ 3)*(BesselJ(2/ 3, (2)/(3)*(z)^((3)/(2)))- BesselJ(- 2/ 3, (2)/(3)*(z)^((3)/(2)))) (D[AiryAi[temp], {temp, 1}]/.temp-> - z)=(z/ 3)*(BesselJ[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]- BesselJ[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Failure Failure
Fail
-.6239178317-.1120108402*I <- {z = -2^(1/2)-I*2^(1/2)}
-.6239178317+.1120108402*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-0.6239178317433629, -0.1120108405877985] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.6239178317433629, 0.1120108405877985] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E7 ${\displaystyle{\displaystyle(z/3)\left(J_{2/3}\left(\zeta\right)-J_{-2/3}\left% (\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{-\pi i/6}{H^{(1)}_{2/3}}% \left(\zeta\right)+e^{\pi i/6}{H^{(2)}_{2/3}}\left(\zeta\right)\right)}}$ (z/ 3)*(BesselJ(2/ 3, (2)/(3)*(z)^((3)/(2)))- BesselJ(- 2/ 3, (2)/(3)*(z)^((3)/(2))))=(1)/(2)*(z/sqrt(3))*(exp(- Pi*I/ 6)*HankelH1(2/ 3, (2)/(3)*(z)^((3)/(2)))+ exp(Pi*I/ 6)*HankelH2(2/ 3, (2)/(3)*(z)^((3)/(2)))) (z/ 3)*(BesselJ[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]- BesselJ[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])])=Divide[1,2]*(z/Sqrt[3])*(Exp[- Pi*I/ 6]*HankelH1[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ Exp[Pi*I/ 6]*HankelH2[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Successful Failure - Successful
9.6.E7 ${\displaystyle{\displaystyle\tfrac{1}{2}(z/\sqrt{3})\left(e^{-\pi i/6}{H^{(1)}% _{2/3}}\left(\zeta\right)+e^{\pi i/6}{H^{(2)}_{2/3}}\left(\zeta\right)\right)=% \tfrac{1}{2}(z/\sqrt{3})\left(e^{-5\pi i/6}{H^{(1)}_{-2/3}}\left(\zeta\right)+% e^{5\pi i/6}{H^{(2)}_{-2/3}}\left(\zeta\right)\right)}}$ (1)/(2)*(z/sqrt(3))*(exp(- Pi*I/ 6)*HankelH1(2/ 3, (2)/(3)*(z)^((3)/(2)))+ exp(Pi*I/ 6)*HankelH2(2/ 3, (2)/(3)*(z)^((3)/(2))))=(1)/(2)*(z/sqrt(3))*(exp(- 5*Pi*I/ 6)*HankelH1(- 2/ 3, (2)/(3)*(z)^((3)/(2)))+ exp(5*Pi*I/ 6)*HankelH2(- 2/ 3, (2)/(3)*(z)^((3)/(2)))) Divide[1,2]*(z/Sqrt[3])*(Exp[- Pi*I/ 6]*HankelH1[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ Exp[Pi*I/ 6]*HankelH2[2/ 3, Divide[2,3]*(z)^(Divide[3,2])])=Divide[1,2]*(z/Sqrt[3])*(Exp[- 5*Pi*I/ 6]*HankelH1[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ Exp[5*Pi*I/ 6]*HankelH2[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Successful Successful - -
9.6.E8 ${\displaystyle{\displaystyle\mathrm{Bi}\left(-z\right)=\sqrt{z/3}\left(J_{-1/3% }\left(\zeta\right)-J_{1/3}\left(\zeta\right)\right)}}$ AiryBi(- z)=sqrt(z/ 3)*(BesselJ(- 1/ 3, (2)/(3)*(z)^((3)/(2)))- BesselJ(1/ 3, (2)/(3)*(z)^((3)/(2)))) AiryBi[- z]=Sqrt[z/ 3]*(BesselJ[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]- BesselJ[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Failure Failure
Fail
.4111747943+1.355498750*I <- {z = -2^(1/2)-I*2^(1/2)}
.4111747943-1.355498750*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[0.41117479327057227, 1.355498750084894] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.41117479327057227, -1.355498750084894] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E8 ${\displaystyle{\displaystyle\sqrt{z/3}\left(J_{-1/3}\left(\zeta\right)-J_{1/3}% \left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{2\pi i/3}{H^{(1)}_{1/% 3}}\left(\zeta\right)+e^{-2\pi i/3}{H^{(2)}_{1/3}}\left(\zeta\right)\right)}}$ sqrt(z/ 3)*(BesselJ(- 1/ 3, (2)/(3)*(z)^((3)/(2)))- BesselJ(1/ 3, (2)/(3)*(z)^((3)/(2))))=(1)/(2)*sqrt(z/ 3)*(exp(2*Pi*I/ 3)*HankelH1(1/ 3, (2)/(3)*(z)^((3)/(2)))+ exp(- 2*Pi*I/ 3)*HankelH2(1/ 3, (2)/(3)*(z)^((3)/(2)))) Sqrt[z/ 3]*(BesselJ[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]- BesselJ[1/ 3, Divide[2,3]*(z)^(Divide[3,2])])=Divide[1,2]*Sqrt[z/ 3]*(Exp[2*Pi*I/ 3]*HankelH1[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ Exp[- 2*Pi*I/ 3]*HankelH2[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Successful Failure - Successful
9.6.E8 ${\displaystyle{\displaystyle\tfrac{1}{2}\sqrt{z/3}\left(e^{2\pi i/3}{H^{(1)}_{% 1/3}}\left(\zeta\right)+e^{-2\pi i/3}{H^{(2)}_{1/3}}\left(\zeta\right)\right)=% \tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/3}{H^{(1)}_{-1/3}}\left(\zeta\right)+e^{-% \pi i/3}{H^{(2)}_{-1/3}}\left(\zeta\right)\right)}}$ (1)/(2)*sqrt(z/ 3)*(exp(2*Pi*I/ 3)*HankelH1(1/ 3, (2)/(3)*(z)^((3)/(2)))+ exp(- 2*Pi*I/ 3)*HankelH2(1/ 3, (2)/(3)*(z)^((3)/(2))))=(1)/(2)*sqrt(z/ 3)*(exp(Pi*I/ 3)*HankelH1(- 1/ 3, (2)/(3)*(z)^((3)/(2)))+ exp(- Pi*I/ 3)*HankelH2(- 1/ 3, (2)/(3)*(z)^((3)/(2)))) Divide[1,2]*Sqrt[z/ 3]*(Exp[2*Pi*I/ 3]*HankelH1[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ Exp[- 2*Pi*I/ 3]*HankelH2[1/ 3, Divide[2,3]*(z)^(Divide[3,2])])=Divide[1,2]*Sqrt[z/ 3]*(Exp[Pi*I/ 3]*HankelH1[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ Exp[- Pi*I/ 3]*HankelH2[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Successful Successful - -
9.6.E9 ${\displaystyle{\displaystyle\mathrm{Bi}'\left(-z\right)=(z/\sqrt{3})\left(J_{-% 2/3}\left(\zeta\right)+J_{2/3}\left(\zeta\right)\right)}}$ subs( temp=- z, diff( AiryBi(temp), temp$(1) ) )=(z/sqrt(3))*(BesselJ(- 2/ 3, (2)/(3)*(z)^((3)/(2)))+ BesselJ(2/ 3, (2)/(3)*(z)^((3)/(2)))) (D[AiryBi[temp], {temp, 1}]/.temp-> - z)=(z/Sqrt[3])*(BesselJ[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ BesselJ[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Failure Failure Fail -1.338452844+1.884861589*I <- {z = -2^(1/2)-I*2^(1/2)} -1.338452844-1.884861589*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[-1.338452844987923, 1.884861589266007] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.338452844987923, -1.884861589266007] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E9 ${\displaystyle{\displaystyle(z/\sqrt{3})\left(J_{-2/3}\left(\zeta\right)+J_{2/% 3}\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{\pi i/3}{H^{(1)}_% {2/3}}\left(\zeta\right)+e^{-\pi i/3}{H^{(2)}_{2/3}}\left(\zeta\right)\right)}}$ (z/sqrt(3))*(BesselJ(- 2/ 3, (2)/(3)*(z)^((3)/(2)))+ BesselJ(2/ 3, (2)/(3)*(z)^((3)/(2))))=(1)/(2)*(z/sqrt(3))*(exp(Pi*I/ 3)*HankelH1(2/ 3, (2)/(3)*(z)^((3)/(2)))+ exp(- Pi*I/ 3)*HankelH2(2/ 3, (2)/(3)*(z)^((3)/(2)))) (z/Sqrt[3])*(BesselJ[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ BesselJ[2/ 3, Divide[2,3]*(z)^(Divide[3,2])])=Divide[1,2]*(z/Sqrt[3])*(Exp[Pi*I/ 3]*HankelH1[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ Exp[- Pi*I/ 3]*HankelH2[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Successful Failure - Successful 9.6.E9 ${\displaystyle{\displaystyle\tfrac{1}{2}(z/\sqrt{3})\left(e^{\pi i/3}{H^{(1)}_% {2/3}}\left(\zeta\right)+e^{-\pi i/3}{H^{(2)}_{2/3}}\left(\zeta\right)\right)=% \tfrac{1}{2}(z/\sqrt{3})\left(e^{-\pi i/3}{H^{(1)}_{-2/3}}\left(\zeta\right)+e% ^{\pi i/3}{H^{(2)}_{-2/3}}\left(\zeta\right)\right)}}$ (1)/(2)*(z/sqrt(3))*(exp(Pi*I/ 3)*HankelH1(2/ 3, (2)/(3)*(z)^((3)/(2)))+ exp(- Pi*I/ 3)*HankelH2(2/ 3, (2)/(3)*(z)^((3)/(2))))=(1)/(2)*(z/sqrt(3))*(exp(- Pi*I/ 3)*HankelH1(- 2/ 3, (2)/(3)*(z)^((3)/(2)))+ exp(Pi*I/ 3)*HankelH2(- 2/ 3, (2)/(3)*(z)^((3)/(2)))) Divide[1,2]*(z/Sqrt[3])*(Exp[Pi*I/ 3]*HankelH1[2/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ Exp[- Pi*I/ 3]*HankelH2[2/ 3, Divide[2,3]*(z)^(Divide[3,2])])=Divide[1,2]*(z/Sqrt[3])*(Exp[- Pi*I/ 3]*HankelH1[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]+ Exp[Pi*I/ 3]*HankelH2[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]) Successful Successful - - 9.6.E11 ${\displaystyle{\displaystyle J_{+1/3}\left(\zeta\right)=\tfrac{1}{2}\sqrt{3/z}% \left(\sqrt{3}\mathrm{Ai}\left(-z\right)-\mathrm{Bi}\left(-z\right)\right)}}$ BesselJ(+ 1/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(2)*sqrt(3/ z)*(sqrt(3)*AiryAi(- z)- AiryBi(- z)) BesselJ[+ 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*Sqrt[3/ z]*(Sqrt[3]*AiryAi[- z]- AiryBi[- z]) Failure Failure Fail -.5314179186+1.098214195*I <- {z = -2^(1/2)-I*2^(1/2)} -.5314179186-1.098214195*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[-0.531417918807772, 1.09821419470116] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.531417918807772, -1.09821419470116] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E11 ${\displaystyle{\displaystyle J_{-1/3}\left(\zeta\right)=\tfrac{1}{2}\sqrt{3/z}% \left(\sqrt{3}\mathrm{Ai}\left(-z\right)+\mathrm{Bi}\left(-z\right)\right)}}$ BesselJ(- 1/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(2)*sqrt(3/ z)*(sqrt(3)*AiryAi(- z)+ AiryBi(- z)) BesselJ[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*Sqrt[3/ z]*(Sqrt[3]*AiryAi[- z]+ AiryBi[- z]) Failure Failure Fail .8096382420-.23450870e-2*I <- {z = -2^(1/2)-I*2^(1/2)} .8096382420+.23450870e-2*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[0.8096382422763857, -0.0023450862884800416] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[0.8096382422763857, 0.0023450862884800416] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E12 ${\displaystyle{\displaystyle J_{+2/3}\left(\zeta\right)=\tfrac{1}{2}(\sqrt{3}/% z)\left(+\sqrt{3}\mathrm{Ai}'\left(-z\right)+\mathrm{Bi}'\left(-z\right)\right% )}}$ BesselJ(+ 2/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(2)*(sqrt(3)/ z)*(+sqrt(3)*subs( temp=- z, diff( AiryAi(temp), temp$(1) ) )+ subs( temp=- z, diff( AiryBi(temp), temp$(1) ) )) BesselJ[+ 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*(Sqrt[3]/ z)*(+Sqrt[3]*(D[AiryAi[temp], {temp, 1}]/.temp-> - z)+ (D[AiryBi[temp], {temp, 1}]/.temp-> - z)) Failure Failure Fail -.2229822889+1.258414134*I <- {z = -2^(1/2)-I*2^(1/2)} -.2229822889-1.258414134*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[-0.22298228919670726, 1.2584141341459216] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.22298228919670726, -1.2584141341459216] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E12 ${\displaystyle{\displaystyle J_{-2/3}\left(\zeta\right)=\tfrac{1}{2}(\sqrt{3}/% z)\left(-\sqrt{3}\mathrm{Ai}'\left(-z\right)+\mathrm{Bi}'\left(-z\right)\right% )}}$ BesselJ(- 2/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(2)*(sqrt(3)/ z)*(-sqrt(3)*subs( temp=- z, diff( AiryAi(temp), temp$(1) ) )+ subs( temp=- z, diff( AiryBi(temp), temp$(1) ) )) BesselJ[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*(Sqrt[3]/ z)*(-Sqrt[3]*(D[AiryAi[temp], {temp, 1}]/.temp-> - z)+ (D[AiryBi[temp], {temp, 1}]/.temp-> - z)) Failure Failure Fail .5575879430+.7154547765*I <- {z = -2^(1/2)-I*2^(1/2)} .5575879430-.7154547765*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[0.5575879428157583, 0.7154547769715692] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[0.5575879428157583, -0.7154547769715692] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E13 ${\displaystyle{\displaystyle I_{+1/3}\left(\zeta\right)=\tfrac{1}{2}\sqrt{3/z}% \left(-\sqrt{3}\mathrm{Ai}\left(z\right)+\mathrm{Bi}\left(z\right)\right)}}$ BesselI(+ 1/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(2)*sqrt(3/ z)*(-sqrt(3)*AiryAi(z)+ AiryBi(z)) BesselI[+ 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*Sqrt[3/ z]*(-Sqrt[3]*AiryAi[z]+ AiryBi[z]) Failure Failure Fail 1.506527799+2.607865458*I <- {z = -2^(1/2)-I*2^(1/2)} 1.506527799-2.607865458*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[1.5065278006053275, 2.6078654586738588] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[1.5065278006053275, -2.6078654586738588] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E13 ${\displaystyle{\displaystyle I_{-1/3}\left(\zeta\right)=\tfrac{1}{2}\sqrt{3/z}% \left(+\sqrt{3}\mathrm{Ai}\left(z\right)+\mathrm{Bi}\left(z\right)\right)}}$ BesselI(- 1/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(2)*sqrt(3/ z)*(+sqrt(3)*AiryAi(z)+ AiryBi(z)) BesselI[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*Sqrt[3/ z]*(+Sqrt[3]*AiryAi[z]+ AiryBi[z]) Failure Failure Fail -1.389593520-2.699131954*I <- {z = -2^(1/2)-I*2^(1/2)} -1.389593520+2.699131954*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[-1.3895935221249858, -2.6991319545588484] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.3895935221249858, 2.6991319545588484] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E14 ${\displaystyle{\displaystyle I_{+2/3}\left(\zeta\right)=\tfrac{1}{2}(\sqrt{3}/% z)\left(+\sqrt{3}\mathrm{Ai}'\left(z\right)+\mathrm{Bi}'\left(z\right)\right)}}$ BesselI(+ 2/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(2)*(sqrt(3)/ z)*(+sqrt(3)*subs( temp=z, diff( AiryAi(temp), temp$(1) ) )+ subs( temp=z, diff( AiryBi(temp), temp$(1) ) )) BesselI[+ 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*(Sqrt[3]/ z)*(+Sqrt[3]*(D[AiryAi[temp], {temp, 1}]/.temp-> z)+ (D[AiryBi[temp], {temp, 1}]/.temp-> z)) Failure Failure Fail 1.494369018+2.219325646*I <- {z = -2^(1/2)-I*2^(1/2)} 1.494369018-2.219325646*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[1.4943690186714658, 2.219325646151564] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[1.4943690186714658, -2.219325646151564] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E14 ${\displaystyle{\displaystyle I_{-2/3}\left(\zeta\right)=\tfrac{1}{2}(\sqrt{3}/% z)\left(-\sqrt{3}\mathrm{Ai}'\left(z\right)+\mathrm{Bi}'\left(z\right)\right)}}$ BesselI(- 2/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(2)*(sqrt(3)/ z)*(-sqrt(3)*subs( temp=z, diff( AiryAi(temp), temp$(1) ) )+ subs( temp=z, diff( AiryBi(temp), temp$(1) ) )) BesselI[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*(Sqrt[3]/ z)*(-Sqrt[3]*(D[AiryAi[temp], {temp, 1}]/.temp-> z)+ (D[AiryBi[temp], {temp, 1}]/.temp-> z)) Failure Failure Fail -1.366821518-2.314117950*I <- {z = -2^(1/2)-I*2^(1/2)} -1.366821518+2.314117950*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[-1.366821518925578, -2.3141179507576517] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.366821518925578, 2.3141179507576517] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E15 ${\displaystyle{\displaystyle K_{+1/3}\left(\zeta\right)=\pi\sqrt{3/z}\mathrm{% Ai}\left(z\right)}}$ BesselK(+ 1/ 3, (2)/(3)*(z)^((3)/(2)))= Pi*sqrt(3/ z)*AiryAi(z) BesselK[+ 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]= Pi*Sqrt[3/ z]*AiryAi[z] Failure Failure Fail -5.252983010-9.625828533*I <- {z = -2^(1/2)-I*2^(1/2)} -5.252983010+9.625828533*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[-5.252983013913404, -9.625828534114124] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-5.252983013913404, 9.625828534114124] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E15 ${\displaystyle{\displaystyle K_{-1/3}\left(\zeta\right)=\pi\sqrt{3/z}\mathrm{% Ai}\left(z\right)}}$ BesselK(- 1/ 3, (2)/(3)*(z)^((3)/(2)))= Pi*sqrt(3/ z)*AiryAi(z) BesselK[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]= Pi*Sqrt[3/ z]*AiryAi[z] Failure Failure Fail -5.252983010-9.625828533*I <- {z = -2^(1/2)-I*2^(1/2)} -5.252983010+9.625828533*I <- {z = -2^(1/2)+I*2^(1/2)} Fail Complex[-5.252983013913404, -9.625828534114124] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-5.252983013913404, 9.625828534114124] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 9.6.E16 ${\displaystyle{\displaystyle K_{+2/3}\left(\zeta\right)=-\pi(\sqrt{3}/z)% \mathrm{Ai}'\left(z\right)}}$ BesselK(+ 2/ 3, (2)/(3)*(z)^((3)/(2)))= - Pi*(sqrt(3)/ z)* subs( temp=z, diff( AiryAi(temp), temp$(1) ) ) BesselK[+ 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]= - Pi*(Sqrt[3]/ z)* (D[AiryAi[temp], {temp, 1}]/.temp-> z) Failure Failure
Fail
-5.189625577-8.222757114*I <- {z = -2^(1/2)-I*2^(1/2)}
-5.189625577+8.222757114*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-5.189625578046477, -8.222757113865619] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-5.189625578046477, 8.222757113865619] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E16 ${\displaystyle{\displaystyle K_{-2/3}\left(\zeta\right)=-\pi(\sqrt{3}/z)% \mathrm{Ai}'\left(z\right)}}$ BesselK(- 2/ 3, (2)/(3)*(z)^((3)/(2)))= - Pi*(sqrt(3)/ z)* subs( temp=z, diff( AiryAi(temp), temp\$(1) ) ) BesselK[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]= - Pi*(Sqrt[3]/ z)* (D[AiryAi[temp], {temp, 1}]/.temp-> z) Failure Failure
Fail
-5.189625577-8.222757114*I <- {z = -2^(1/2)-I*2^(1/2)}
-5.189625577+8.222757114*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-5.189625578046477, -8.222757113865619] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-5.189625578046477, 8.222757113865619] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E17 ${\displaystyle{\displaystyle{H^{(1)}_{1/3}}\left(\zeta\right)=e^{-\pi i/3}{H^{% (1)}_{-1/3}}\left(\zeta\right)}}$ HankelH1(1/ 3, (2)/(3)*(z)^((3)/(2)))= exp(- Pi*I/ 3)*HankelH1(- 1/ 3, (2)/(3)*(z)^((3)/(2))) HankelH1[1/ 3, Divide[2,3]*(z)^(Divide[3,2])]= Exp[- Pi*I/ 3]*HankelH1[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])] Successful Successful - -
9.6.E17