Results of Airy and Related Functions

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DLMF Formula Maple Mathematica Symbolic
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Symbolic
Mathematica
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Maple
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Mathematica
9.2.E2 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 = 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 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 subs( temp=0, diff( AiryAi(temp), temp$(1) ) )= -(1)/((3)^(1/ 3)* GAMMA((1)/(3))) (D[AiryAi[temp], {temp, 1}]/.temp-> 0)= -Divide[1,(3)^(1/ 3)* Gamma[Divide[1,3]]] Successful Successful - -
9.2.E5 AiryBi(0)=(1)/((3)^(1/ 6)* GAMMA((2)/(3))) AiryBi[0]=Divide[1,(3)^(1/ 6)* Gamma[Divide[2,3]]] Successful Successful - -
9.2.E6 subs( temp=0, diff( AiryBi(temp), temp$(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 (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 (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 (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 (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 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 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 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 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 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 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 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 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 - Successful
9.5.E2 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 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 Successful
9.5.E4 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 Skip
9.5.E5 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 Skip
9.5.E6 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 - Successful
9.5.E7 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 Skip
9.5.E8 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
Fail
Complex[-0.014252654553766713, -0.04024893384084034] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.014252654553766713, 0.04024893384084034] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
9.6.E2 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 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 (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 (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 (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
Fail
Complex[-2.8337652800788247, 0.30394618618023783] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
9.6.E2 (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 - Successful
9.6.E2 (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
Fail
Complex[2.8337652800788256, -0.3039461861802379] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.8337652800788247, -0.3039461861802372] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E2 (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 - Successful
9.6.E3 subs( temp=z, diff( AiryAi(temp), temp$(1) ) )= - (Pi)^(- 1)*(z/sqrt(3))* BesselK(+ 2/ 3, (2)/(3)*(z)^((3)/(2))) (D[AiryAi[temp], {temp, 1}]/.temp-> z)= - (Pi)^(- 1)*(z/Sqrt[3])* BesselK[+ 2/ 3, Divide[2,3]*(z)^(Divide[3,2])] Failure Failure
Fail
-.7883076520+3.485863958*I <- {z = -2^(1/2)-I*2^(1/2)}
-.7883076520-3.485863958*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-0.7883076520663912, 3.485863960601928] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.7883076520663912, -3.485863960601928] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E3 subs( temp=z, diff( AiryAi(temp), temp$(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 - (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 - (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 (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
Fail
Complex[0.7883076520663918, -3.485863960601928] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
9.6.E3 (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 - Successful
9.6.E3 (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
Fail
Complex[-0.7883076520663909, 3.485863960601928] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.7883076520663926, 3.485863960601928] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E3 (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 - Successful
9.6.E4 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 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 (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 - Successful
9.6.E5 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 (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 (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 - Successful
9.6.E6 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 (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 (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 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 (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 (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 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 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 (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 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 (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 (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 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 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 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$(1) ) )) BesselJ[+ 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*(Sqrt[3]/ z)*(+Sqrt[3]*(D[AiryAi[temp], {temp, 1}]/.temp-> - z)+ (D[AiryBi[temp], {temp, 1}]/.temp-> - z)) Failure Failure
Fail
-.2229822889+1.258414134*I <- {z = -2^(1/2)-I*2^(1/2)}
-.2229822889-1.258414134*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-0.22298228919670726, 1.2584141341459216] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.22298228919670726, -1.2584141341459216] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E12 BesselJ(- 2/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(2)*(sqrt(3)/ z)*(-sqrt(3)*subs( temp=- z, diff( AiryAi(temp), temp$(1) ) )+ subs( temp=- z, diff( AiryBi(temp), temp$(1) ) )) BesselJ[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*(Sqrt[3]/ z)*(-Sqrt[3]*(D[AiryAi[temp], {temp, 1}]/.temp-> - z)+ (D[AiryBi[temp], {temp, 1}]/.temp-> - z)) Failure Failure
Fail
.5575879430+.7154547765*I <- {z = -2^(1/2)-I*2^(1/2)}
.5575879430-.7154547765*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[0.5575879428157583, 0.7154547769715692] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.5575879428157583, -0.7154547769715692] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E13 BesselI(+ 1/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(2)*sqrt(3/ z)*(-sqrt(3)*AiryAi(z)+ AiryBi(z)) BesselI[+ 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*Sqrt[3/ z]*(-Sqrt[3]*AiryAi[z]+ AiryBi[z]) Failure Failure
Fail
1.506527799+2.607865458*I <- {z = -2^(1/2)-I*2^(1/2)}
1.506527799-2.607865458*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[1.5065278006053275, 2.6078654586738588] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.5065278006053275, -2.6078654586738588] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E13 BesselI(- 1/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(2)*sqrt(3/ z)*(+sqrt(3)*AiryAi(z)+ AiryBi(z)) BesselI[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*Sqrt[3/ z]*(+Sqrt[3]*AiryAi[z]+ AiryBi[z]) Failure Failure
Fail
-1.389593520-2.699131954*I <- {z = -2^(1/2)-I*2^(1/2)}
-1.389593520+2.699131954*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-1.3895935221249858, -2.6991319545588484] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.3895935221249858, 2.6991319545588484] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E14 BesselI(+ 2/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(2)*(sqrt(3)/ z)*(+sqrt(3)*subs( temp=z, diff( AiryAi(temp), temp$(1) ) )+ subs( temp=z, diff( AiryBi(temp), temp$(1) ) )) BesselI[+ 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*(Sqrt[3]/ z)*(+Sqrt[3]*(D[AiryAi[temp], {temp, 1}]/.temp-> z)+ (D[AiryBi[temp], {temp, 1}]/.temp-> z)) Failure Failure
Fail
1.494369018+2.219325646*I <- {z = -2^(1/2)-I*2^(1/2)}
1.494369018-2.219325646*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[1.4943690186714658, 2.219325646151564] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.4943690186714658, -2.219325646151564] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E14 BesselI(- 2/ 3, (2)/(3)*(z)^((3)/(2)))=(1)/(2)*(sqrt(3)/ z)*(-sqrt(3)*subs( temp=z, diff( AiryAi(temp), temp$(1) ) )+ subs( temp=z, diff( AiryBi(temp), temp$(1) ) )) BesselI[- 2/ 3, Divide[2,3]*(z)^(Divide[3,2])]=Divide[1,2]*(Sqrt[3]/ z)*(-Sqrt[3]*(D[AiryAi[temp], {temp, 1}]/.temp-> z)+ (D[AiryBi[temp], {temp, 1}]/.temp-> z)) Failure Failure
Fail
-1.366821518-2.314117950*I <- {z = -2^(1/2)-I*2^(1/2)}
-1.366821518+2.314117950*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-1.366821518925578, -2.3141179507576517] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.366821518925578, 2.3141179507576517] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E15 BesselK(+ 1/ 3, (2)/(3)*(z)^((3)/(2)))= Pi*sqrt(3/ z)*AiryAi(z) BesselK[+ 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]= Pi*Sqrt[3/ z]*AiryAi[z] Failure Failure
Fail
-5.252983010-9.625828533*I <- {z = -2^(1/2)-I*2^(1/2)}
-5.252983010+9.625828533*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-5.252983013913404, -9.625828534114124] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-5.252983013913404, 9.625828534114124] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E15 BesselK(- 1/ 3, (2)/(3)*(z)^((3)/(2)))= Pi*sqrt(3/ z)*AiryAi(z) BesselK[- 1/ 3, Divide[2,3]*(z)^(Divide[3,2])]= Pi*Sqrt[3/ z]*AiryAi[z] Failure Failure
Fail
-5.252983010-9.625828533*I <- {z = -2^(1/2)-I*2^(1/2)}
-5.252983010+9.625828533*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-5.252983013913404, -9.625828534114124] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-5.252983013913404, 9.625828534114124] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
9.6.E16 BesselK(+ 2/ 3, (2)/(3)*(z)^((3)/(2)))= - Pi*(sqrt(3)/ z)* subs( temp=z, diff( AiryAi(temp), temp$(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 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 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