Bessel Functions - 10.66 Expansions in Series of Bessel Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.66.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Kelvinber{\nu}@@{x}+i\Kelvinbei{\nu}@@{x} = \sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}\BesselJ{\nu+k}@{x}}{2^{k/2}k!}}
\Kelvinber{\nu}@@{x}+i\Kelvinbei{\nu}@@{x} = \sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}\BesselJ{\nu+k}@{x}}{2^{k/2}k!}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{((\nu+k)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0} 
KelvinBer(nu, x)+ I*KelvinBei(nu, x) = sum((exp((3*nu + k)*Pi*I/4)*(x)^(k)* BesselJ(nu + k, x))/((2)^(k/2)* factorial(k)), k = 0..infinity)

KelvinBer[\[Nu], x]+ I*KelvinBei[\[Nu], x] == Sum[Divide[Exp[(3*\[Nu]+ k)*Pi*I/4]*(x)^(k)* BesselJ[\[Nu]+ k, x],(2)^(k/2)* (k)!], {k, 0, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out
Failed [30 / 30]
Result: Plus[Complex[-0.12257968900025018, 0.2735107661041647], Times[-1.0, NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], k], Pi]], BesselJ[Plus[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], 1.5], Power[Factorial[k], -1]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[0.3467793075651209, -0.08562995402477025], Times[-1.0, NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], k], Pi]], BesselJ[Plus[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], 1.5], Power[Factorial[k], -1]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.66.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}\BesselJ{\nu+k}@{x}}{2^{k/2}k!} = \sum_{k=0}^{\infty}\frac{e^{(3\nu+3k)\pi i/4}x^{k}\modBesselI{\nu+k}@{x}}{2^{k/2}k!}}
\sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}\BesselJ{\nu+k}@{x}}{2^{k/2}k!} = \sum_{k=0}^{\infty}\frac{e^{(3\nu+3k)\pi i/4}x^{k}\modBesselI{\nu+k}@{x}}{2^{k/2}k!}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{((\nu+k)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0} 
sum((exp((3*nu + k)*Pi*I/4)*(x)^(k)* BesselJ(nu + k, x))/((2)^(k/2)* factorial(k)), k = 0..infinity) = sum((exp((3*nu + 3*k)*Pi*I/4)*(x)^(k)* BesselI(nu + k, x))/((2)^(k/2)* factorial(k)), k = 0..infinity)

Sum[Divide[Exp[(3*\[Nu]+ k)*Pi*I/4]*(x)^(k)* BesselJ[\[Nu]+ k, x],(2)^(k/2)* (k)!], {k, 0, Infinity}, GenerateConditions->None] == Sum[Divide[Exp[(3*\[Nu]+ 3*k)*Pi*I/4]*(x)^(k)* BesselI[\[Nu]+ k, x],(2)^(k/2)* (k)!], {k, 0, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out
Failed [30 / 30]
Result: Plus[Times[-1.0, NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[3, k]], Pi]], BesselI[Plus[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], 1.5], Power[Factorial[k], -1]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], k], Pi]], BesselJ[Plus[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], 1.5], Power[Factorial[k], -1]], {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Times[-1.0, NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Times[3, k]], Pi]], BesselI[Plus[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], 1.5], Power[Factorial[k], -1]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], k], Pi]], BesselJ[Plus[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], 1.5], Power[Factorial[k], -1]], {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.66#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Kelvinber{n}@{x\sqrt{2}} = \sum_{k=-\infty}^{\infty}(-1)^{n+k}\BesselJ{n+2k}@{x}\modBesselI{2k}@{x}}
\Kelvinber{n}@{x\sqrt{2}} = \sum_{k=-\infty}^{\infty}(-1)^{n+k}\BesselJ{n+2k}@{x}\modBesselI{2k}@{x}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{((n+2k)+k+1)} > 0, \realpart@@{(n+k+1)} > 0, \realpart@@{((2k)+k+1)} > 0} 
KelvinBer(n, x*sqrt(2)) = sum((- 1)^(n + k)* BesselJ(n + 2*k, x)*BesselI(2*k, x), k = - infinity..infinity)

KelvinBer[n, x*Sqrt[2]] == Sum[(- 1)^(n + k)* BesselJ[n + 2*k, x]*BesselI[2*k, x], {k, - Infinity, Infinity}, GenerateConditions->None]
Failure Aborted Successful [Tested: 9] Skipped - Because timed out
10.66#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Kelvinbei{n}@{x\sqrt{2}} = \sum_{k=-\infty}^{\infty}(-1)^{n+k}\BesselJ{n+2k+1}@{x}\modBesselI{2k+1}@{x}}
\Kelvinbei{n}@{x\sqrt{2}} = \sum_{k=-\infty}^{\infty}(-1)^{n+k}\BesselJ{n+2k+1}@{x}\modBesselI{2k+1}@{x}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{((n+2k+1)+k+1)} > 0, \realpart@@{((2k+1)+k+1)} > 0} 
KelvinBei(n, x*sqrt(2)) = sum((- 1)^(n + k)* BesselJ(n + 2*k + 1, x)*BesselI(2*k + 1, x), k = - infinity..infinity)

KelvinBei[n, x*Sqrt[2]] == Sum[(- 1)^(n + k)* BesselJ[n + 2*k + 1, x]*BesselI[2*k + 1, x], {k, - Infinity, Infinity}, GenerateConditions->None]
Failure Aborted Successful [Tested: 9] Skipped - Because timed out
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