Parabolic Cylinder Functions - 12.8 Recurrence Relations and Derivatives

From testwiki
Revision as of 11:31, 28 June 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
12.8.E1 z ⁢ U ⁡ ( a , z ) - U ⁡ ( a - 1 , z ) + ( a + 1 2 ) ⁢ U ⁡ ( a + 1 , z ) = 0 𝑧 parabolic-U 𝑎 𝑧 parabolic-U 𝑎 1 𝑧 𝑎 1 2 parabolic-U 𝑎 1 𝑧 0 {\displaystyle{\displaystyle zU\left(a,z\right)-U\left(a-1,z\right)+(a+\tfrac{% 1}{2})U\left(a+1,z\right)=0}}
z\paraU@{a}{z}-\paraU@{a-1}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0		 

z*CylinderU(a, z)- CylinderU(a - 1, z)+(a +(1)/(2))*CylinderU(a + 1, z) = 0

z*ParabolicCylinderD[- 1/2 -(a), z]- ParabolicCylinderD[- 1/2 -(a - 1), z]+(a +Divide[1,2])*ParabolicCylinderD[- 1/2 -(a + 1), z] == 0
Successful Successful - Successful [Tested: 42]
12.8.E2 U ′ ⁡ ( a , z ) + 1 2 ⁢ z ⁢ U ⁡ ( a , z ) + ( a + 1 2 ) ⁢ U ⁡ ( a + 1 , z ) = 0 diffop parabolic-U 1 𝑎 𝑧 1 2 𝑧 parabolic-U 𝑎 𝑧 𝑎 1 2 parabolic-U 𝑎 1 𝑧 0 {\displaystyle{\displaystyle U'\left(a,z\right)+\tfrac{1}{2}zU\left(a,z\right)% +(a+\tfrac{1}{2})U\left(a+1,z\right)=0}}
\paraU'@{a}{z}+\tfrac{1}{2}z\paraU@{a}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0		 

diff( CylinderU(a, z), z$(1) )+(1)/(2)*z*CylinderU(a, z)+(a +(1)/(2))*CylinderU(a + 1, z) = 0

D[ParabolicCylinderD[- 1/2 -(a), z], {z, 1}]+Divide[1,2]*z*ParabolicCylinderD[- 1/2 -(a), z]+(a +Divide[1,2])*ParabolicCylinderD[- 1/2 -(a + 1), z] == 0
Successful Successful - Successful [Tested: 42]
12.8.E3 U ′ ⁡ ( a , z ) - 1 2 ⁢ z ⁢ U ⁡ ( a , z ) + U ⁡ ( a - 1 , z ) = 0 diffop parabolic-U 1 𝑎 𝑧 1 2 𝑧 parabolic-U 𝑎 𝑧 parabolic-U 𝑎 1 𝑧 0 {\displaystyle{\displaystyle U'\left(a,z\right)-\tfrac{1}{2}zU\left(a,z\right)% +U\left(a-1,z\right)=0}}
\paraU'@{a}{z}-\tfrac{1}{2}z\paraU@{a}{z}+\paraU@{a-1}{z} = 0		 

diff( CylinderU(a, z), z$(1) )-(1)/(2)*z*CylinderU(a, z)+ CylinderU(a - 1, z) = 0

D[ParabolicCylinderD[- 1/2 -(a), z], {z, 1}]-Divide[1,2]*z*ParabolicCylinderD[- 1/2 -(a), z]+ ParabolicCylinderD[- 1/2 -(a - 1), z] == 0
Successful Successful - Successful [Tested: 42]
12.8.E4 2 ⁢ U ′ ⁡ ( a , z ) + U ⁡ ( a - 1 , z ) + ( a + 1 2 ) ⁢ U ⁡ ( a + 1 , z ) = 0 2 diffop parabolic-U 1 𝑎 𝑧 parabolic-U 𝑎 1 𝑧 𝑎 1 2 parabolic-U 𝑎 1 𝑧 0 {\displaystyle{\displaystyle 2U'\left(a,z\right)+U\left(a-1,z\right)+(a+\tfrac% {1}{2})U\left(a+1,z\right)=0}}
2\paraU'@{a}{z}+\paraU@{a-1}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0		 

2*diff( CylinderU(a, z), z$(1) )+ CylinderU(a - 1, z)+(a +(1)/(2))*CylinderU(a + 1, z) = 0

2*D[ParabolicCylinderD[- 1/2 -(a), z], {z, 1}]+ ParabolicCylinderD[- 1/2 -(a - 1), z]+(a +Divide[1,2])*ParabolicCylinderD[- 1/2 -(a + 1), z] == 0
Successful Successful - Successful [Tested: 42]
12.8.E5 z ⁢ V ⁡ ( a , z ) - V ⁡ ( a + 1 , z ) + ( a - 1 2 ) ⁢ V ⁡ ( a - 1 , z ) = 0 𝑧 parabolic-V 𝑎 𝑧 parabolic-V 𝑎 1 𝑧 𝑎 1 2 parabolic-V 𝑎 1 𝑧 0 {\displaystyle{\displaystyle zV\left(a,z\right)-V\left(a+1,z\right)+(a-\tfrac{% 1}{2})V\left(a-1,z\right)=0}}
z\paraV@{a}{z}-\paraV@{a+1}{z}+(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0		 

z*CylinderV(a, z)- CylinderV(a + 1, z)+(a -(1)/(2))*CylinderV(a - 1, z) = 0

z*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])- Divide[GAMMA[1/2 + a + 1], Pi]*(Sin[Pi*(a + 1)] * ParabolicCylinderD[-(a + 1) - 1/2, z] + ParabolicCylinderD[-(a + 1) - 1/2, -(z)])+(a -Divide[1,2])*Divide[GAMMA[1/2 + a - 1], Pi]*(Sin[Pi*(a - 1)] * ParabolicCylinderD[-(a - 1) - 1/2, z] + ParabolicCylinderD[-(a - 1) - 1/2, -(z)]) == 0
Successful Failure -
Failed [42 / 42]
Result: Plus[Times[Complex[7.067899292141149*^-17, 0.0], GAMMA[-2.0]], Times[Complex[3.060490169192143*^-17, 1.7669748230352868*^-17], GAMMA[-1.0]], Times[Complex[0.0, -8.834874115176436*^-18], GAMMA[0.0]]]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Times[Complex[1.4135798584282297*^-16, 0.0], GAMMA[-2.0]], Times[Complex[-8.361414638298002*^-17, 7.414495684541142*^-17], GAMMA[-1.0]], Times[Complex[-7.067899292141149*^-17, -8.834874115176436*^-18], GAMMA[0.0]]]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
12.8.E6 V ′ ⁡ ( a , z ) - 1 2 ⁢ z ⁢ V ⁡ ( a , z ) - ( a - 1 2 ) ⁢ V ⁡ ( a - 1 , z ) = 0 diffop parabolic-V 1 𝑎 𝑧 1 2 𝑧 parabolic-V 𝑎 𝑧 𝑎 1 2 parabolic-V 𝑎 1 𝑧 0 {\displaystyle{\displaystyle V'\left(a,z\right)-\tfrac{1}{2}zV\left(a,z\right)% -(a-\tfrac{1}{2})V\left(a-1,z\right)=0}}
\paraV'@{a}{z}-\tfrac{1}{2}z\paraV@{a}{z}-(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0		 

diff( CylinderV(a, z), z$(1) )-(1)/(2)*z*CylinderV(a, z)-(a -(1)/(2))*CylinderV(a - 1, z) = 0

D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, 1}]-Divide[1,2]*z*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])-(a -Divide[1,2])*Divide[GAMMA[1/2 + a - 1], Pi]*(Sin[Pi*(a - 1)] * ParabolicCylinderD[-(a - 1) - 1/2, z] + ParabolicCylinderD[-(a - 1) - 1/2, -(z)]) == 0
Successful Failure -
Failed [39 / 42]
Result: Plus[Complex[0.0, 0.0], Times[Complex[-7.067899292141149*^-17, 0.0], GAMMA[-2.0]], Times[Complex[-1.5302450845960716*^-17, -8.834874115176434*^-18], GAMMA[-1.0]]]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Times[Complex[-1.4135798584282297*^-16, 0.0], GAMMA[-2.0]], Times[Complex[-9.955091265133296*^-17, -1.7329819619999673*^-18], GAMMA[-1.0]]]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
12.8.E7 V ′ ⁡ ( a , z ) + 1 2 ⁢ z ⁢ V ⁡ ( a , z ) - V ⁡ ( a + 1 , z ) = 0 diffop parabolic-V 1 𝑎 𝑧 1 2 𝑧 parabolic-V 𝑎 𝑧 parabolic-V 𝑎 1 𝑧 0 {\displaystyle{\displaystyle V'\left(a,z\right)+\tfrac{1}{2}zV\left(a,z\right)% -V\left(a+1,z\right)=0}}
\paraV'@{a}{z}+\tfrac{1}{2}z\paraV@{a}{z}-\paraV@{a+1}{z} = 0		 

diff( CylinderV(a, z), z$(1) )+(1)/(2)*z*CylinderV(a, z)- CylinderV(a + 1, z) = 0

D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, 1}]+Divide[1,2]*z*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])- Divide[GAMMA[1/2 + a + 1], Pi]*(Sin[Pi*(a + 1)] * ParabolicCylinderD[-(a + 1) - 1/2, z] + ParabolicCylinderD[-(a + 1) - 1/2, -(z)]) == 0
Successful Failure -
Failed [42 / 42]
Result: Plus[Complex[0.0, 0.0], Times[Complex[1.5302450845960716*^-17, 8.834874115176434*^-18], GAMMA[-1.0]], Times[Complex[0.0, -8.834874115176436*^-18], GAMMA[0.0]]]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Times[Complex[-1.83165059034313*^-16, 7.241197488341145*^-17], GAMMA[-1.0]], Times[Complex[-7.067899292141149*^-17, -8.834874115176436*^-18], GAMMA[0.0]]]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
12.8.E8 2 ⁢ V ′ ⁡ ( a , z ) - V ⁡ ( a + 1 , z ) - ( a - 1 2 ) ⁢ V ⁡ ( a - 1 , z ) = 0 2 diffop parabolic-V 1 𝑎 𝑧 parabolic-V 𝑎 1 𝑧 𝑎 1 2 parabolic-V 𝑎 1 𝑧 0 {\displaystyle{\displaystyle 2V'\left(a,z\right)-V\left(a+1,z\right)-(a-\tfrac% {1}{2})V\left(a-1,z\right)=0}}
2\paraV'@{a}{z}-\paraV@{a+1}{z}-(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0		 

2*diff( CylinderV(a, z), z$(1) )- CylinderV(a + 1, z)-(a -(1)/(2))*CylinderV(a - 1, z) = 0

2*D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, 1}]- Divide[GAMMA[1/2 + a + 1], Pi]*(Sin[Pi*(a + 1)] * ParabolicCylinderD[-(a + 1) - 1/2, z] + ParabolicCylinderD[-(a + 1) - 1/2, -(z)])-(a -Divide[1,2])*Divide[GAMMA[1/2 + a - 1], Pi]*(Sin[Pi*(a - 1)] * ParabolicCylinderD[-(a - 1) - 1/2, z] + ParabolicCylinderD[-(a - 1) - 1/2, -(z)]) == 0
Successful Failure -
Failed [42 / 42]
Result: Plus[Complex[0.0, 0.0], Times[Complex[-7.067899292141149*^-17, 0.0], GAMMA[-2.0]], Times[Complex[0.0, -8.834874115176436*^-18], GAMMA[0.0]]]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Times[Complex[-1.4135798584282297*^-16, 0.0], GAMMA[-2.0]], Times[Complex[-2.8271597168564594*^-16, 7.067899292141149*^-17], GAMMA[-1.0]], Times[Complex[-7.067899292141149*^-17, -8.834874115176436*^-18], GAMMA[0.0]]]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
12.8.E9 d m d z m ⁡ ( e 1 4 ⁢ z 2 ⁢ U ⁡ ( a , z ) ) = ( - 1 ) m ⁢ ( 1 2 + a ) m ⁢ e 1 4 ⁢ z 2 ⁢ U ⁡ ( a + m , z ) derivative 𝑧 𝑚 superscript 𝑒 1 4 superscript 𝑧 2 parabolic-U 𝑎 𝑧 superscript 1 𝑚 Pochhammer 1 2 𝑎 𝑚 superscript 𝑒 1 4 superscript 𝑧 2 parabolic-U 𝑎 𝑚 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^% {\frac{1}{4}z^{2}}U\left(a,z\right)\right)=(-1)^{m}{\left(\tfrac{1}{2}+a\right% )_{m}}e^{\frac{1}{4}z^{2}}U\left(a+m,z\right)}}
\deriv[m]{}{z}\left(e^{\frac{1}{4}z^{2}}\paraU@{a}{z}\right) = (-1)^{m}\Pochhammersym{\tfrac{1}{2}+a}{m}e^{\frac{1}{4}z^{2}}\paraU@{a+m}{z}		 

diff(exp((1)/(4)*(z)^(2))*CylinderU(a, z), [z$(m)]) = (- 1)^(m)* pochhammer((1)/(2)+ a, m)*exp((1)/(4)*(z)^(2))*CylinderU(a + m, z)

D[Exp[Divide[1,4]*(z)^(2)]*ParabolicCylinderD[- 1/2 -(a), z], {z, m}] == (- 1)^(m)* Pochhammer[Divide[1,2]+ a, m]*Exp[Divide[1,4]*(z)^(2)]*ParabolicCylinderD[- 1/2 -(a + m), z]
Failure Failure Error
Failed [96 / 126]
Result: Plus[Complex[-1.0, 0.0], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][1.0]], {Rule[a, -1.5], Rule[m, 1], Rule[z, Power[E, Times<syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]], {Rule[a, -1.5], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
12.8.E10 d m d z m ⁡ ( e - 1 4 ⁢ z 2 ⁢ U ⁡ ( a , z ) ) = ( - 1 ) m ⁢ e - 1 4 ⁢ z 2 ⁢ U ⁡ ( a - m , z ) derivative 𝑧 𝑚 superscript 𝑒 1 4 superscript 𝑧 2 parabolic-U 𝑎 𝑧 superscript 1 𝑚 superscript 𝑒 1 4 superscript 𝑧 2 parabolic-U 𝑎 𝑚 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^% {-\frac{1}{4}z^{2}}U\left(a,z\right)\right)=(-1)^{m}e^{-\frac{1}{4}z^{2}}U% \left(a-m,z\right)}}
\deriv[m]{}{z}\left(e^{-\frac{1}{4}z^{2}}\paraU@{a}{z}\right) = (-1)^{m}e^{-\frac{1}{4}z^{2}}\paraU@{a-m}{z}		 

diff(exp(-(1)/(4)*(z)^(2))*CylinderU(a, z), [z$(m)]) = (- 1)^(m)* exp(-(1)/(4)*(z)^(2))*CylinderU(a - m, z)

D[Exp[-Divide[1,4]*(z)^(2)]*ParabolicCylinderD[- 1/2 -(a), z], {z, m}] == (- 1)^(m)* Exp[-Divide[1,4]*(z)^(2)]*ParabolicCylinderD[- 1/2 -(a - m), z]
Failure Failure Error
Failed [96 / 126]
Result: Plus[Complex[-0.07045205979755337, 0.7756076114781977], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[-1, Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]}]][1.0]], {Rule[a, -1.5], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[2.000032302229117, -0.49556574541480647], Times[2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[-1, Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]}]][2.0]]], {Rule[a, -1.5], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
12.8.E11 d m d z m ⁡ ( e 1 4 ⁢ z 2 ⁢ V ⁡ ( a , z ) ) = e 1 4 ⁢ z 2 ⁢ V ⁡ ( a + m , z ) derivative 𝑧 𝑚 superscript 𝑒 1 4 superscript 𝑧 2 parabolic-V 𝑎 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 parabolic-V 𝑎 𝑚 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^% {\frac{1}{4}z^{2}}V\left(a,z\right)\right)=e^{\frac{1}{4}z^{2}}V\left(a+m,z% \right)}}
\deriv[m]{}{z}\left(e^{\frac{1}{4}z^{2}}\paraV@{a}{z}\right) = e^{\frac{1}{4}z^{2}}\paraV@{a+m}{z}		 

diff(exp((1)/(4)*(z)^(2))*CylinderV(a, z), [z$(m)]) = exp((1)/(4)*(z)^(2))*CylinderV(a + m, z)

D[Exp[Divide[1,4]*(z)^(2)]*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, m}] == Exp[Divide[1,4]*(z)^(2)]*Divide[GAMMA[1/2 + a + m], Pi]*(Sin[Pi*(a + m)] * ParabolicCylinderD[-(a + m) - 1/2, z] + ParabolicCylinderD[-(a + m) - 1/2, -(z)])
Failure Failure Error
Failed [126 / 126]
Result: Plus[Times[Complex[2.150599663294456*^-18, -9.777500999643939*^-18], GAMMA[0.0]], Times[0.3183098861837907, GAMMA[-1.0], Plus[DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], <syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[-0.9299481905237211, -0.4298894311242862], GAMMA[1.0]], Times[0.6366197723675814, GAMMA[-1.0], Plus[DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0], Times[1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]]]], {Rule[a, -1.5], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
12.8.E12 d m d z m ⁡ ( e - 1 4 ⁢ z 2 ⁢ V ⁡ ( a , z ) ) = ( - 1 ) m ⁢ ( 1 2 - a ) m ⁢ e - 1 4 ⁢ z 2 ⁢ V ⁡ ( a - m , z ) derivative 𝑧 𝑚 superscript 𝑒 1 4 superscript 𝑧 2 parabolic-V 𝑎 𝑧 superscript 1 𝑚 Pochhammer 1 2 𝑎 𝑚 superscript 𝑒 1 4 superscript 𝑧 2 parabolic-V 𝑎 𝑚 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^% {-\frac{1}{4}z^{2}}V\left(a,z\right)\right)=(-1)^{m}{\left(\tfrac{1}{2}-a% \right)_{m}}e^{-\frac{1}{4}z^{2}}V\left(a-m,z\right)}}
\deriv[m]{}{z}\left(e^{-\frac{1}{4}z^{2}}\paraV@{a}{z}\right) = (-1)^{m}\Pochhammersym{\tfrac{1}{2}-a}{m}e^{-\frac{1}{4}z^{2}}\paraV@{a-m}{z}		 

diff(exp(-(1)/(4)*(z)^(2))*CylinderV(a, z), [z$(m)]) = (- 1)^(m)* pochhammer((1)/(2)- a, m)*exp(-(1)/(4)*(z)^(2))*CylinderV(a - m, z)

D[Exp[-Divide[1,4]*(z)^(2)]*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, m}] == (- 1)^(m)* Pochhammer[Divide[1,2]- a, m]*Exp[-Divide[1,4]*(z)^(2)]*Divide[GAMMA[1/2 + a - m], Pi]*(Sin[Pi*(a - m)] * ParabolicCylinderD[-(a - m) - 1/2, z] + ParabolicCylinderD[-(a - m) - 1/2, -(z)])
Failure Failure Error
Failed [126 / 126]
Result: Plus[Times[Complex[-6.091780348003315*^-17, 1.3399109614774574*^-17], GAMMA[-2.0]], Times[0.3183098861837907, GAMMA[-1.0], Plus[DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]}]][1.0], Times[1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Ti<syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[1.6052302675286988*^-15, 3.2948039393826443*^-16], GAMMA[-3.0]], Times[0.6366197723675814, GAMMA[-1.0], Plus[DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]}]][2.0], Times[1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[-1, Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]}]][2.0]]]]], {Rule[a, -1.5], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
Retrieved from "https://lct.wmflabs.org/w/index.php?title=12.8&oldid=58244"