Bessel Functions - 10.57 Uniform Asymptotic Expansions for Large Order
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 10.57.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sphBesselJ{n}'@{(n+\tfrac{1}{2})z} = \frac{\pi^{\frac{1}{2}}}{((2n+1)z)^{\frac{1}{2}}}\BesselJ{n+\frac{1}{2}}'@{(n+\tfrac{1}{2})z}-\frac{\pi^{\frac{1}{2}}}{((2n+1)z)^{\frac{3}{2}}}\BesselJ{n+\frac{1}{2}}@{(n+\tfrac{1}{2})z}}
\sphBesselJ{n}'@{(n+\tfrac{1}{2})z} = \frac{\pi^{\frac{1}{2}}}{((2n+1)z)^{\frac{1}{2}}}\BesselJ{n+\frac{1}{2}}'@{(n+\tfrac{1}{2})z}-\frac{\pi^{\frac{1}{2}}}{((2n+1)z)^{\frac{3}{2}}}\BesselJ{n+\frac{1}{2}}@{(n+\tfrac{1}{2})z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0} | Error
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D[SphericalBesselJ[n, (n +Divide[1,2])*z], {(n +Divide[1,2])*z, 1}] == Divide[(Pi)^(Divide[1,2]),((2*n + 1)*z)^(Divide[1,2])]*D[BesselJ[n +Divide[1,2], (n +Divide[1,2])*z], {(n +Divide[1,2])*z, 1}]-Divide[(Pi)^(Divide[1,2]),((2*n + 1)*z)^(Divide[3,2])]*BesselJ[n +Divide[1,2], (n +Divide[1,2])*z]
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Missing Macro Error | Failure | - | Failed [21 / 21]
Result: Plus[Complex[0.14653389603833195, -0.029869009956249915], Times[Complex[-0.988457695936884, 0.2648564413786163], D[Complex[0.36567703182522004, 0.24184221354059504]
Test Values: {Complex[1.299038105676658, 0.7499999999999999], 1.0}]], D[Complex[0.425509744388485, 0.14219887983348967], {Complex[1.299038105676658, 0.7499999999999999], 1.0}]], {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Plus[Complex[0.06710374092328811, 0.007963502819859997], Times[Complex[-0.7656560389588212, 0.20515691731902835], D[Complex[0.2637838125883578, 0.3348231997381719]
Test Values: {Complex[2.165063509461097, 1.2499999999999998], 1.0}]], D[Complex[0.27065896459303473, 0.20224233103375913], {Complex[2.165063509461097, 1.2499999999999998], 1.0}]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |