Bessel Functions - 10.44 Sums

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.44#Ex1 I ν ⁡ ( z ) = ∑ k = 0 ∞ z k k ! ⁢ J ν + k ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 superscript subscript 𝑘 0 superscript 𝑧 𝑘 𝑘 Bessel-J 𝜈 𝑘 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\sum_{k=0}^{\infty}\frac{z^% {k}}{k!}J_{\nu+k}\left(z\right)}}
\modBesselI{\nu}@{z} = \sum_{k=0}^{\infty}\frac{z^{k}}{k!}\BesselJ{\nu+k}@{z}		 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}} 
BesselI(nu, z) = sum(((z)^(k))/(factorial(k))*BesselJ(nu + k, z), k = 0..infinity)

BesselI[\[Nu], z] == Sum[Divide[(z)^(k),(k)!]*BesselJ[\[Nu]+ k, z], {k, 0, Infinity}, GenerateConditions->None]
Failure Successful Skipped - Because timed out Successful [Tested: 70]
10.44#Ex2 J ν ⁡ ( z ) = ∑ k = 0 ∞ ( - 1 ) k ⁢ z k k ! ⁢ I ν + k ⁡ ( z ) Bessel-J 𝜈 𝑧 superscript subscript 𝑘 0 superscript 1 𝑘 superscript 𝑧 𝑘 𝑘 modified-Bessel-first-kind 𝜈 𝑘 𝑧 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\sum_{k=0}^{\infty}(-1)^{k}% \frac{z^{k}}{k!}I_{\nu+k}\left(z\right)}}
\BesselJ{\nu}@{z} = \sum_{k=0}^{\infty}(-1)^{k}\frac{z^{k}}{k!}\modBesselI{\nu+k}@{z}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν + k ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+k)+k+1)>0}} 
BesselJ(nu, z) = sum((- 1)^(k)*((z)^(k))/(factorial(k))*BesselI(nu + k, z), k = 0..infinity)

BesselJ[\[Nu], z] == Sum[(- 1)^(k)*Divide[(z)^(k),(k)!]*BesselI[\[Nu]+ k, z], {k, 0, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out
Failed [70 / 70]
Result: Plus[Complex[0.4358908643715884, -0.07192294931339177], Times[-1.0, NSum[Times[Power[-1, k], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], BesselI[Plus[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Factorial[k], -1]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[1.0679098760861825, 0.09257666026367889], Times[-1.0, NSum[Times[Power[-1, k], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], BesselI[Plus[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Factorial[k], -1]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, 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
10.44.E4 ( 1 2 ⁢ z ) ν = ∑ k = 0 ∞ ( - 1 ) k ⁢ ( ν + 2 ⁢ k ) ⁢ Γ ⁡ ( ν + k ) k ! ⁢ I ν + 2 ⁢ k ⁡ ( z ) superscript 1 2 𝑧 𝜈 superscript subscript 𝑘 0 superscript 1 𝑘 𝜈 2 𝑘 Euler-Gamma 𝜈 𝑘 𝑘 modified-Bessel-first-kind 𝜈 2 𝑘 𝑧 {\displaystyle{\displaystyle\left(\tfrac{1}{2}z\right)^{\nu}=\sum_{k=0}^{% \infty}(-1)^{k}\frac{(\nu+2k)\Gamma\left(\nu+k\right)}{k!}I_{\nu+2k}\left(z% \right)}}
\left(\tfrac{1}{2}z\right)^{\nu} = \sum_{k=0}^{\infty}(-1)^{k}\frac{(\nu+2k)\EulerGamma@{\nu+k}}{k!}\modBesselI{\nu+2k}@{z}		 ℜ ⁡ ( ν + k ) > 0 , ℜ ⁡ ( ( ν + 2 ⁢ k ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 0 𝜈 2 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k)>0,\Re((\nu+2k)+k+1)>0}} 
((1)/(2)*z)^(nu) = sum((- 1)^(k)*((nu + 2*k)*GAMMA(nu + k))/(factorial(k))*BesselI(nu + 2*k, z), k = 0..infinity)

(Divide[1,2]*z)^\[Nu] == Sum[(- 1)^(k)*Divide[(\[Nu]+ 2*k)*Gamma[\[Nu]+ k],(k)!]*BesselI[\[Nu]+ 2*k, z], {k, 0, Infinity}, GenerateConditions->None]
Failure Failure Manual Skip!
Failed [7 / 7]
Result: Plus[Complex[0.43301270189221935, 0.24999999999999997], Times[-1.0, NSum[Times[Power[-1, k], Plus[1, Times[2, k]], BesselI[Plus[1, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Factorial[k], -1], Gamma[Plus[1, k]]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 1]}


Result: Plus[Complex[-0.2499999999999999, 0.43301270189221935], Times[-1.0, NSum[Times[Power[-1, k], Plus[1, Times[2, k]], BesselI[Plus[1, Times[2, k]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Power[Factorial[k], -1], Gamma[Plus[1, k]]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, 1]}

... skip entries to safe data
10.44.E5 K 0 ⁡ ( z ) = - ( ln ⁡ ( 1 2 ⁢ z ) + γ ) ⁢ I 0 ⁡ ( z ) + 2 ⁢ ∑ k = 1 ∞ I 2 ⁢ k ⁡ ( z ) k modified-Bessel-second-kind 0 𝑧 1 2 𝑧 modified-Bessel-first-kind 0 𝑧 2 superscript subscript 𝑘 1 modified-Bessel-first-kind 2 𝑘 𝑧 𝑘 {\displaystyle{\displaystyle K_{0}\left(z\right)=-\left(\ln\left(\tfrac{1}{2}z% \right)+\gamma\right)I_{0}\left(z\right)+2\sum_{k=1}^{\infty}\frac{I_{2k}\left% (z\right)}{k}}}
\modBesselK{0}@{z} = -\left(\ln@{\tfrac{1}{2}z}+\EulerConstant\right)\modBesselI{0}@{z}+2\sum_{k=1}^{\infty}\frac{\modBesselI{2k}@{z}}{k}		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( ( 2 ⁢ k ) + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 2 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((2k)+k+1)>0}} 
BesselK(0, z) = -(ln((1)/(2)*z)+ gamma)*BesselI(0, z)+ 2*sum((BesselI(2*k, z))/(k), k = 1..infinity)

BesselK[0, z] == -(Log[Divide[1,2]*z]+ EulerGamma)*BesselI[0, z]+ 2*Sum[Divide[BesselI[2*k, z],k], {k, 1, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
10.44.E6 K n ⁡ ( z ) = n ! ⁢ ( 1 2 ⁢ z ) - n 2 ⁢ ∑ k = 0 n - 1 ( - 1 ) k ⁢ ( 1 2 ⁢ z ) k ⁢ I k ⁡ ( z ) k ! ⁢ ( n - k ) + ( - 1 ) n - 1 ⁢ ( ln ⁡ ( 1 2 ⁢ z ) - ψ ⁡ ( n + 1 ) ) ⁢ I n ⁡ ( z ) + ( - 1 ) n ⁢ ∑ k = 1 ∞ ( n + 2 ⁢ k ) ⁢ I n + 2 ⁢ k ⁡ ( z ) k ⁢ ( n + k ) modified-Bessel-second-kind 𝑛 𝑧 𝑛 superscript 1 2 𝑧 𝑛 2 superscript subscript 𝑘 0 𝑛 1 superscript 1 𝑘 superscript 1 2 𝑧 𝑘 modified-Bessel-first-kind 𝑘 𝑧 𝑘 𝑛 𝑘 superscript 1 𝑛 1 1 2 𝑧 digamma 𝑛 1 modified-Bessel-first-kind 𝑛 𝑧 superscript 1 𝑛 superscript subscript 𝑘 1 𝑛 2 𝑘 modified-Bessel-first-kind 𝑛 2 𝑘 𝑧 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle K_{n}\left(z\right)=\frac{n!(\tfrac{1}{2}z)^{-n}}% {2}\sum_{k=0}^{n-1}(-1)^{k}\frac{(\tfrac{1}{2}z)^{k}I_{k}\left(z\right)}{k!(n-% k)}+(-1)^{n-1}\left(\ln\left(\tfrac{1}{2}z\right)-\psi\left(n+1\right)\right)I% _{n}\left(z\right)+(-1)^{n}\sum_{k=1}^{\infty}\frac{(n+2k)I_{n+2k}\left(z% \right)}{k(n+k)}}}
\modBesselK{n}@{z} = \frac{n!(\tfrac{1}{2}z)^{-n}}{2}\sum_{k=0}^{n-1}(-1)^{k}\frac{(\tfrac{1}{2}z)^{k}\modBesselI{k}@{z}}{k!(n-k)}+(-1)^{n-1}\left(\ln@{\tfrac{1}{2}z}-\digamma@{n+1}\right)\modBesselI{n}@{z}+(-1)^{n}\sum_{k=1}^{\infty}\frac{(n+2k)\modBesselI{n+2k}@{z}}{k(n+k)}		 ℜ ⁡ ( n + k + 1 ) > 0 , ℜ ⁡ ( k + k + 1 ) > 0 , ℜ ⁡ ( ( n + 2 ⁢ k ) + k + 1 ) > 0 formulae-sequence 𝑛 𝑘 1 0 formulae-sequence 𝑘 𝑘 1 0 𝑛 2 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0,\Re(k+k+1)>0,\Re((n+2k)+k+1)>0}} 
BesselK(n, z) = (factorial(n)*((1)/(2)*z)^(- n))/(2)*sum((- 1)^(k)*(((1)/(2)*z)^(k)* BesselI(k, z))/(factorial(k)*(n - k)), k = 0..n - 1)+(- 1)^(n - 1)*(ln((1)/(2)*z)- Psi(n + 1))*BesselI(n, z)+(- 1)^(n)* sum(((n + 2*k)*BesselI(n + 2*k, z))/(k*(n + k)), k = 1..infinity)

BesselK[n, z] == Divide[(n)!*(Divide[1,2]*z)^(- n),2]*Sum[(- 1)^(k)*Divide[(Divide[1,2]*z)^(k)* BesselI[k, z],(k)!*(n - k)], {k, 0, n - 1}, GenerateConditions->None]+(- 1)^(n - 1)*(Log[Divide[1,2]*z]- PolyGamma[n + 1])*BesselI[n, z]+(- 1)^(n)* Sum[Divide[(n + 2*k)*BesselI[n + 2*k, z],k*(n + k)], {k, 1, Infinity}, GenerateConditions->None]
Failure Aborted Manual Skip!
Failed [21 / 21]
Result: Plus[Complex[1.084080291505059, -0.3914662527648858], NSum[Times[Power[k, -1], Power[Plus[1, k], -1], Plus[1, Times[2, k]], BesselI[Plus[1, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]], Times[Complex[-0.8660254037844387, 0.49999999999999994], DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Plus[Times[-1, ], 1], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], []], Times[Plus[4, Times[12, ], Times[12, Power[, 2]], Times[4, Power[, 3]], Times[-4, 1], Times[-8, , 1], Times[-4, Power[, 2], 1], Times[-1, , Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[1, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], [Plus[1, ]]], Times[4, Plus[1, ], Plus[-5, Times[-6, ], Times[-2, Power[, 2]], Times[3, 1], Times[2, , 1]], [Plus[2, ]]], Times[-4, Plus[1, ], Plus[2, ], Plus[-2, Times[-1, ], 1], [Plus[3, ]<syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.001928095904955185, 0.0030033056761246957], Times[-1.0, NSum[Times[Power[k, -1], Power[Plus[2, k], -1], Plus[2, Times[2, k]], BesselI[Plus[2, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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