Bessel Functions - 10.24 Functions of Imaginary Order
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 10.24.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(x^{2}+\nu^{2})w = 0}
x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(x^{2}+\nu^{2})w = 0 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)+((x)^(2)+ (nu)^(2))*w = 0
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(x)^(2)* D[w, {x, 2}]+ x*D[w, x]+((x)^(2)+ \[Nu]^(2))*w == 0
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Failure | Failure | Failed [300 / 300] Result: 1.948557159+2.125000000*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2}
Result: .2165063513+1.125000001*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 1/2}
... skip entries to safe data |
Failed [300 / 300]
Result: Complex[1.9485571585149875, 2.125]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[1.948557158514987, 0.12499999999999989]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 10.24#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BesselJimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselJ{i\nu}@{x}}}
\BesselJimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselJ{i\nu}@{x}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{((\iunit \nu)+k+1)} > 0} | sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselJ(I*nu, x))
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Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselJ[I*\[Nu], x]]
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Successful | Successful | - | Successful [Tested: 30] |
| 10.24#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BesselYimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselY{i\nu}@{x}}}
\BesselYimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselY{i\nu}@{x}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{((\iunit \nu)+k+1)} > 0, \realpart@@{((-(\iunit \nu))+k+1)} > 0} | sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselY(I*nu, x))
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Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselY[I*\[Nu], x]]
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Successful | Successful | - | Successful [Tested: 30] |
| 10.24.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{1+i\nu} = \left(\frac{\pi\nu}{\sinh@{\pi\nu}}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}}
\EulerGamma@{1+i\nu} = \left(\frac{\pi\nu}{\sinh@{\pi\nu}}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(1+\iunit \nu)} > 0} | GAMMA(1 + I*nu) = ((Pi*nu)/(sinh(Pi*nu)))^((1)/(2))* exp(I*gamma[nu])
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Gamma[1 + I*\[Nu]] == (Divide[Pi*\[Nu],Sinh[Pi*\[Nu]]])^(Divide[1,2])* Exp[I*Subscript[\[Gamma], \[Nu]]]
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Failure | Failure | Failed [300 / 300] Result: .131682196e-1-.6479738907*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = 1/2*3^(1/2)+1/2*I}
Result: .2393622021-.2867640040*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Failed [300 / 300]
Result: Complex[0.013168219691258531, -0.6479738909120968]
Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[γ, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[0.23936220222535412, -0.28676400411697583]
Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[γ, ν], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 10.24#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BesselJimag{-\nu}@{x} = \BesselJimag{\nu}@{x}}
\BesselJimag{-\nu}@{x} = \BesselJimag{\nu}@{x} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | sech((1/2)*Pi*(- nu))*Re(BesselJ(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x))
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Sech[1/2 Pi - \[Nu]] Re[BesselJ[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]]
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Failure | Failure | Failed [12 / 30] Result: .1765981285-.1547836875*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}
Result: -1.059084556+.9282601935*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}
... skip entries to safe data |
Failed [30 / 30]
Result: Complex[-0.6353785354467336, 0.04153700144653363]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[0.2910880978413849, 0.681683596996288]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 10.24#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BesselYimag{-\nu}@{x} = \BesselYimag{\nu}@{x}}
\BesselYimag{-\nu}@{x} = \BesselYimag{\nu}@{x} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{((\iunit (-\nu))+k+1)} > 0, \realpart@@{((\iunit \nu)+k+1)} > 0, \realpart@@{((-(\iunit (-\nu)))+k+1)} > 0, \realpart@@{((-(\iunit \nu))+k+1)} > 0} | sech((1/2)*Pi*(- nu))*Re(BesselY(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))
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Sech[1/2 Pi - \[Nu]] Re[BesselY[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]
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Failure | Failure | Failed [12 / 30] Result: -.6730010946+.5898680353*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}
Result: -.1980888923+.1736197856*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}
... skip entries to safe data |
Failed [30 / 30]
Result: Complex[0.16541121369118172, 0.7534126929509344]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[-0.3242468905843751, -0.9796849117084342]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 10.24.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Wronskian@{\BesselJimag{\nu}@{x},\BesselYimag{\nu}@{x}} = 2/(\pi x)}
\Wronskian@{\BesselJimag{\nu}@{x},\BesselYimag{\nu}@{x}} = 2/(\pi x) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{((\iunit \nu)+k+1)} > 0, \realpart@@{((-(\iunit \nu))+k+1)} > 0} | (sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)))*diff(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)), x)-diff(sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)), x)*(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))) = 2/(Pi*x)
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Wronskian[{Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]], Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]}, x] == 2/(Pi*x)
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Failure | Failure | Failed [12 / 30] Result: -.3214564733-.7786157192*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}
Result: -.6431025084-4.765445687*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}
... skip entries to safe data |
Failed [30 / 30]
Result: Plus[-0.4244131815783876, Times[Complex[0.017184424665049866, -0.12995814793225188], Plus[Times[Complex[5.94457417937745, -0.08806734388290616], Derivative[1][Re][Complex[0.5424102683642863, 1.3820413572565333]]], Times[Complex[0.04670634387761448, 2.0064149502593187], Derivative[1][Re][Complex[1.5013396639532606, -0.5145465005058608]]]]]]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Plus[-0.4244131815783876, Times[Complex[-0.5062208144169521, 0.3689208146583662], Plus[Times[Complex[1.2690034139339206, -1.428145592425075], Derivative[1][Re][Complex[-0.5230512553281585, -0.7250724679588263]]], Times[Complex[0.9907135967899046, 0.5862869255257461], Derivative[1][Re][Complex[0.9118063408652576, -0.381897212811936]]]]]]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 10.24.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BesselYimag{0}@{x} = \BesselY{0}@{x}}
\BesselYimag{0}@{x} = \BesselY{0}@{x} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(0+k+1)} > 0, \realpart@@{((-0)+k+1)} > 0, \realpart@@{((\iunit 0)+k+1)} > 0, \realpart@@{((-(\iunit 0))+k+1)} > 0} | sech((1/2)*Pi*(0))*Re(BesselY(I*(0), x)) = BesselY(0, x)
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Sech[1/2 Pi 0] Re[BesselY[I 0, x]] == BesselY[0, x]
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Failure | Failure | Successful [Tested: 3] | Successful [Tested: 3] |