Bessel Functions - 10.18 Modulus and Phase Functions

From testwiki
Revision as of 12:23, 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
10.18#Ex7 M ν ( x ) = ( J ν 2 ( x ) + Y ν 2 ( x ) ) 1 2 modulus-Bessel-M 𝜈 𝑥 superscript Bessel-J 𝜈 2 𝑥 Bessel-Y-Weber 𝜈 2 𝑥 1 2 {\displaystyle{\displaystyle M_{\nu}\left(x\right)=\left({J_{\nu}^{2}}\left(x% \right)+{Y_{\nu}^{2}}\left(x\right)\right)^{\frac{1}{2}}}}
\HankelmodM{\nu}@{x} = \left(\BesselJ{\nu}^{2}@{x}+\BesselY{\nu}^{2}@{x}\right)^{\frac{1}{2}}
( ν + k + 1 ) > 0 , ( ( - ν ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}
Error
Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2] == ((BesselJ[\[Nu], x])^(2)+ (BesselY[\[Nu], x])^(2))^(Divide[1,2])
Missing Macro Error Failure -
Failed [30 / 30]
Result: Complex[0.19554332981034928, -0.3390785475644471]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.7197518351343698, 1.0182547128018542]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.18#Ex8 N ν ( x ) = ( J ν 2 ( x ) + Y ν 2 ( x ) ) 1 2 modulus-Bessel-N 𝜈 𝑥 superscript diffop Bessel-J 𝜈 1 2 𝑥 diffop Bessel-Y-Weber 𝜈 1 2 𝑥 1 2 {\displaystyle{\displaystyle N_{\nu}\left(x\right)=\left({J_{\nu}'^{2}}\left(x% \right)+{Y_{\nu}'^{2}}\left(x\right)\right)^{\frac{1}{2}}}}
\HankelmodderivN{\nu}@{x} = \left(\BesselJ{\nu}'^{2}@{x}+\BesselY{\nu}'^{2}@{x}\right)^{\frac{1}{2}}

Error
Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2] == ((D[BesselJ[\[Nu], x], {x, 1}])^(2)+ (D[BesselY[\[Nu], x], {x, 1}])^(2))^(Divide[1,2])
Missing Macro Error Failure -
Failed [30 / 30]
Result: Complex[-0.3065654786420606, 0.09106250304027241]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-0.41179972752410343, -0.08651542233456301]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.18.E10 ( x 2 - ν 2 ) M ν ( x ) M ν ( x ) + x 2 N ν ( x ) N ν ( x ) + x N ν 2 ( x ) = 0 superscript 𝑥 2 superscript 𝜈 2 modulus-Bessel-M 𝜈 𝑥 diffop modulus-Bessel-M 𝜈 1 𝑥 superscript 𝑥 2 modulus-Bessel-N 𝜈 𝑥 diffop modulus-Bessel-N 𝜈 1 𝑥 𝑥 modulus-Bessel-N 𝜈 2 𝑥 0 {\displaystyle{\displaystyle(x^{2}-\nu^{2})M_{\nu}\left(x\right)M_{\nu}'\left(% x\right)+x^{2}N_{\nu}\left(x\right)N_{\nu}'\left(x\right)+x{N_{\nu}^{2}}\left(% x\right)=0}}
(x^{2}-\nu^{2})\HankelmodM{\nu}@{x}\HankelmodM{\nu}'@{x}+x^{2}\HankelmodderivN{\nu}@{x}\HankelmodderivN{\nu}'@{x}+x\HankelmodderivN{\nu}^{2}@{x} = 0

Error
((x)^(2)- \[Nu]^(2))*Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2]*D[Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2], {x, 1}]+ (x)^(2)* Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2]*D[Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2], {x, 1}]+ x*(Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2])^(2) == 0
Missing Macro Error Aborted -
Failed [30 / 30]
Result: Complex[0.7620133104065328, -0.7345190431210711]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-3.2607567755462643, -4.475082123070706]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.18.E13 x 2 M ν ′′ ( x ) + x M ν ( x ) + ( x 2 - ν 2 ) M ν ( x ) = 4 π 2 M ν 3 ( x ) superscript 𝑥 2 diffop modulus-Bessel-M 𝜈 2 𝑥 𝑥 diffop modulus-Bessel-M 𝜈 1 𝑥 superscript 𝑥 2 superscript 𝜈 2 modulus-Bessel-M 𝜈 𝑥 4 superscript 𝜋 2 modulus-Bessel-M 𝜈 3 𝑥 {\displaystyle{\displaystyle x^{2}M_{\nu}''\left(x\right)+xM_{\nu}'\left(x% \right)+(x^{2}-\nu^{2})M_{\nu}\left(x\right)=\frac{4}{\pi^{2}{{M_{\nu}^{3}}(x)% }}}}
x^{2}\HankelmodM{\nu}''@{x}+x\HankelmodM{\nu}'@{x}+(x^{2}-\nu^{2})\HankelmodM{\nu}@{x} = \frac{4}{\pi^{2}{\HankelmodM{\nu}^{3}(x)}}

Error
(x)^(2)* D[Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2], {x, 2}]+ x*D[Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2], {x, 1}]+((x)^(2)- \[Nu]^(2))*Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2] == Divide[4,(Pi)^(2)*(Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2])^(3)]
Missing Macro Error Translation Error - -