Bessel Functions - 10.8 Power Series
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 10.8.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BesselY{n}@{z} = -\frac{(\tfrac{1}{2}z)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\left(\tfrac{1}{4}z^{2}\right)^{k}+\frac{2}{\pi}\ln@{\tfrac{1}{2}z}\BesselJ{n}@{z}-\frac{(\tfrac{1}{2}z)^{n}}{\pi}\sum_{k=0}^{\infty}(\digamma@{k+1}+\digamma@{n+k+1})\frac{(-\tfrac{1}{4}z^{2})^{k}}{k!(n+k)!}}
\BesselY{n}@{z} = -\frac{(\tfrac{1}{2}z)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\left(\tfrac{1}{4}z^{2}\right)^{k}+\frac{2}{\pi}\ln@{\tfrac{1}{2}z}\BesselJ{n}@{z}-\frac{(\tfrac{1}{2}z)^{n}}{\pi}\sum_{k=0}^{\infty}(\digamma@{k+1}+\digamma@{n+k+1})\frac{(-\tfrac{1}{4}z^{2})^{k}}{k!(n+k)!} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(n+k+1)} > 0, \realpart@@{((-n)+k+1)} > 0} | BesselY(n, z) = -(((1)/(2)*z)^(- n))/(Pi)*sum((factorial(n - k - 1))/(factorial(k))*((1)/(4)*(z)^(2))^(k), k = 0..n - 1)+(2)/(Pi)*ln((1)/(2)*z)*BesselJ(n, z)-(((1)/(2)*z)^(n))/(Pi)*sum((Psi(k + 1)+ Psi(n + k + 1))*((-(1)/(4)*(z)^(2))^(k))/(factorial(k)*factorial(n + k)), k = 0..infinity)
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BesselY[n, z] == -Divide[(Divide[1,2]*z)^(- n),Pi]*Sum[Divide[(n - k - 1)!,(k)!]*(Divide[1,4]*(z)^(2))^(k), {k, 0, n - 1}, GenerateConditions->None]+Divide[2,Pi]*Log[Divide[1,2]*z]*BesselJ[n, z]-Divide[(Divide[1,2]*z)^(n),Pi]*Sum[(PolyGamma[k + 1]+ PolyGamma[n + k + 1])*Divide[(-Divide[1,4]*(z)^(2))^(k),(k)!*(n + k)!], {k, 0, Infinity}, GenerateConditions->None]
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Failure | Failure | Skipped - Because timed out | Failed [6 / 21]
Result: Plus[-0.4244131815783875, Times[0.4244131815783876, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-4, []], Times[Plus[12, Times[8, ]], [Plus[1, ]]], Times[Plus[-16, Times[-16, ], Times[-4, Power[, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[Plus[2, ], Power[1.5, 2], [Plus[3, ]]]], 0], Equal[[1], 1], Equal[[2], Plus[1, Times[4, Power[1.5, -2]]]], Equal[[3], Plus[Rational[1, 2], Times[16, Power[1.5, -4], Plus[2, Times[Rational[1, 4], Power[1.5, 2]]]]]], Equal[[4], Times[Rational[32, 3], Power[1.5, -6], Plus[3, Times[Rational[1, 4], Power[1.5, 2]]], Plus[12, Times[Rational[1, 16], Power[1.5, 4]]]]]}]][1.0]]], {Rule[n, 1], Rule[z, 1.5]}
Result: Plus[-0.8841941282883073, Times[0.3183098861837907, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-4, []], Times[Plus[12, Times[8, ]], [Plus[1, ]]], Times[Plus[-16, Times[-16, ], Times[-4, Power[, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[Plus[2, ], Power[1.5, 2], [Plus[3, ]]]], 0], Equal[[1], 1], Equal[[2], Plus[1, Times[4, Power[1.5, -2]]]], Equal[[3], Plus[Rational[1, 2], Times[16, Power[1.5, -4], Plus[2, Times[Rational[1, 4], Power[1.5, 2]]]]]], Equal[[4], Times[Rational[32, 3], Power[1.5, -6], Plus[3, Times[Rational[1, 4], Power[1.5, 2]]], Plus[12, Times[Rational[1, 16], Power[1.5, 4]]]]]}]][2.0]]], {Rule[n, 2], Rule[z, 1.5]}
... skip entries to safe data |
| 10.8.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BesselY{0}@{z} = \frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}+\EulerConstant\right)\BesselJ{0}@{z}+\frac{2}{\pi}\left(\frac{\tfrac{1}{4}z^{2}}{(1!)^{2}}-(1+\tfrac{1}{2})\frac{(\tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{1}{3})\frac{(\tfrac{1}{4}z^{2})^{3}}{(3!)^{2}}-\dotsi\right)}
\BesselY{0}@{z} = \frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}+\EulerConstant\right)\BesselJ{0}@{z}+\frac{2}{\pi}\left(\frac{\tfrac{1}{4}z^{2}}{(1!)^{2}}-(1+\tfrac{1}{2})\frac{(\tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{1}{3})\frac{(\tfrac{1}{4}z^{2})^{3}}{(3!)^{2}}-\dotsi\right) |
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} | BesselY(0, z) = (2)/(Pi)*(ln((1)/(2)*z)+ gamma)*BesselJ(0, z)+(2)/(Pi)*(((1)/(4)*(z)^(2))/((factorial(1))^(2))-(1 +(1)/(2))*(((1)/(4)*(z)^(2))^(2))/((factorial(2))^(2))+(1 +(1)/(2)+(1)/(3))*(((1)/(4)*(z)^(2))^(3))/((factorial(3))^(2))- ..)
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BesselY[0, z] == Divide[2,Pi]*(Log[Divide[1,2]*z]+ EulerGamma)*BesselJ[0, z]+Divide[2,Pi]*(Divide[Divide[1,4]*(z)^(2),((1)!)^(2)]-(1 +Divide[1,2])*Divide[(Divide[1,4]*(z)^(2))^(2),((2)!)^(2)]+(1 +Divide[1,2]+Divide[1,3])*Divide[(Divide[1,4]*(z)^(2))^(3),((3)!)^(2)]- \[Ellipsis])
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Error | Failure | - | Failed [7 / 7]
Result: Plus[Complex[0.08653583575184755, 0.12491815695491987], Times[-0.6366197723675814, Plus[Complex[0.13592303240740744, 0.19620888054491187], Times[-1.0, …]]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Plus[Complex[-0.07160606681826986, -0.15074612001799426], Times[-0.6366197723675814, Plus[Complex[-0.11248553240740736, -0.23680382134730746], Times[-1.0, …]]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 10.8.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BesselJ{\nu}@{z}\BesselJ{\mu}@{z} = (\tfrac{1}{2}z)^{\nu+\mu}\sum_{k=0}^{\infty}\frac{(\nu+\mu+k+1)_{k}(-\tfrac{1}{4}z^{2})^{k}}{k!\EulerGamma@{\nu+k+1}\EulerGamma@{\mu+k+1}}}
\BesselJ{\nu}@{z}\BesselJ{\mu}@{z} = (\tfrac{1}{2}z)^{\nu+\mu}\sum_{k=0}^{\infty}\frac{(\nu+\mu+k+1)_{k}(-\tfrac{1}{4}z^{2})^{k}}{k!\EulerGamma@{\nu+k+1}\EulerGamma@{\mu+k+1}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\nu+k+1)} > 0, \realpart@@{((\mu)+k+1)} > 0, \realpart@@{(\mu+k+1)} > 0} | BesselJ(nu, z)*BesselJ(mu, z) = ((1)/(2)*z)^(nu + mu)* sum((nu + mu + k + 1[k]*(-(1)/(4)*(z)^(2))^(k))/(factorial(k)*GAMMA(nu + k + 1)*GAMMA(mu + k + 1)), k = 0..infinity)
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BesselJ[\[Nu], z]*BesselJ[\[Mu], z] == (Divide[1,2]*z)^(\[Nu]+ \[Mu])* Sum[Divide[Subscript[\[Nu]+ \[Mu]+ k + 1, k]*(-Divide[1,4]*(z)^(2))^(k),(k)!*Gamma[\[Nu]+ k + 1]*Gamma[\[Mu]+ k + 1]], {k, 0, Infinity}, GenerateConditions->None]
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Failure | Failure | Skipped - Because timed out | Failed [300 / 300]
Result: Plus[Complex[0.18482793500467376, -0.06270111308873656], Times[Complex[-0.17426361621858172, -0.037827155645948574], NSum[Times[Power[Times[Rational[-1, 4], Power[E, Times[Complex[0, Rational[1, 3]], Pi]]], k], Power[Factorial[k], -1], Power[Gamma[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k]], -2], Subscript[Plus[1, Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], k], k]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Plus[Complex[0.47215054540190965, -0.036453907426047115], Times[Complex[-0.27630938504679325, 0.26010894184513544], NSum[Times[Power[Times[Rational[-1, 4], Power[E, Times[Complex[0, Rational[1, 3]], Pi]]], k], Power[Factorial[k], -1], Power[Gamma[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k]], -1], Power[Gamma[Plus[1, Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k]], -1], Subscript[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], k]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |