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 ber ν ⁡ x + i ⁢ bei ν ⁡ x = ∑ k = 0 ∞ e ( 3 ⁢ ν + k ) ⁢ π ⁢ i / 4 ⁢ x k ⁢ J ν + k ⁡ ( x ) 2 k / 2 ⁢ k ! Kelvin-ber 𝜈 𝑥 𝑖 Kelvin-bei 𝜈 𝑥 superscript subscript 𝑘 0 superscript 𝑒 3 𝜈 𝑘 𝜋 𝑖 4 superscript 𝑥 𝑘 Bessel-J 𝜈 𝑘 𝑥 superscript 2 𝑘 2 𝑘 {\displaystyle{\displaystyle\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu% }x=\sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}J_{\nu+k}\left(x\right)}{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!}		 ℜ ⁡ ( ( ν + k ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((\nu+k)+k+1)>0,\Re(\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 ∑ k = 0 ∞ e ( 3 ⁢ ν + k ) ⁢ π ⁢ i / 4 ⁢ x k ⁢ J ν + k ⁡ ( x ) 2 k / 2 ⁢ k ! = ∑ k = 0 ∞ e ( 3 ⁢ ν + 3 ⁢ k ) ⁢ π ⁢ i / 4 ⁢ x k ⁢ I ν + k ⁡ ( x ) 2 k / 2 ⁢ k ! superscript subscript 𝑘 0 superscript 𝑒 3 𝜈 𝑘 𝜋 𝑖 4 superscript 𝑥 𝑘 Bessel-J 𝜈 𝑘 𝑥 superscript 2 𝑘 2 𝑘 superscript subscript 𝑘 0 superscript 𝑒 3 𝜈 3 𝑘 𝜋 𝑖 4 superscript 𝑥 𝑘 modified-Bessel-first-kind 𝜈 𝑘 𝑥 superscript 2 𝑘 2 𝑘 {\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}J% _{\nu+k}\left(x\right)}{2^{k/2}k!}=\sum_{k=0}^{\infty}\frac{e^{(3\nu+3k)\pi i/% 4}x^{k}I_{\nu+k}\left(x\right)}{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!}		 ℜ ⁡ ( ( ν + k ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((\nu+k)+k+1)>0,\Re(\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 ber n ⁡ ( x ⁢ 2 ) = ∑ k = - ∞ ∞ ( - 1 ) n + k ⁢ J n + 2 ⁢ k ⁡ ( x ) ⁢ I 2 ⁢ k ⁡ ( x ) Kelvin-ber 𝑛 𝑥 2 superscript subscript 𝑘 superscript 1 𝑛 𝑘 Bessel-J 𝑛 2 𝑘 𝑥 modified-Bessel-first-kind 2 𝑘 𝑥 {\displaystyle{\displaystyle\operatorname{ber}_{n}\left(x\sqrt{2}\right)=\sum_% {k=-\infty}^{\infty}(-1)^{n+k}J_{n+2k}\left(x\right)I_{2k}\left(x\right)}}
\Kelvinber{n}@{x\sqrt{2}} = \sum_{k=-\infty}^{\infty}(-1)^{n+k}\BesselJ{n+2k}@{x}\modBesselI{2k}@{x}		 ℜ ⁡ ( ( n + 2 ⁢ k ) + k + 1 ) > 0 , ℜ ⁡ ( n + k + 1 ) > 0 , ℜ ⁡ ( ( 2 ⁢ k ) + k + 1 ) > 0 formulae-sequence 𝑛 2 𝑘 𝑘 1 0 formulae-sequence 𝑛 𝑘 1 0 2 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+2k)+k+1)>0,\Re(n+k+1)>0,\Re((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 bei n ⁡ ( x ⁢ 2 ) = ∑ k = - ∞ ∞ ( - 1 ) n + k ⁢ J n + 2 ⁢ k + 1 ⁡ ( x ) ⁢ I 2 ⁢ k + 1 ⁡ ( x ) Kelvin-bei 𝑛 𝑥 2 superscript subscript 𝑘 superscript 1 𝑛 𝑘 Bessel-J 𝑛 2 𝑘 1 𝑥 modified-Bessel-first-kind 2 𝑘 1 𝑥 {\displaystyle{\displaystyle\operatorname{bei}_{n}\left(x\sqrt{2}\right)=\sum_% {k=-\infty}^{\infty}(-1)^{n+k}J_{n+2k+1}\left(x\right)I_{2k+1}\left(x\right)}}
\Kelvinbei{n}@{x\sqrt{2}} = \sum_{k=-\infty}^{\infty}(-1)^{n+k}\BesselJ{n+2k+1}@{x}\modBesselI{2k+1}@{x}		 ℜ ⁡ ( ( n + 2 ⁢ k + 1 ) + k + 1 ) > 0 , ℜ ⁡ ( ( 2 ⁢ k + 1 ) + k + 1 ) > 0 formulae-sequence 𝑛 2 𝑘 1 𝑘 1 0 2 𝑘 1 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+2k+1)+k+1)>0,\Re((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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