Bessel Functions - 10.71 Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.71.E1 ∫ x 1 + ν ⁢ f ν ⁢ d x = - x 1 + ν 2 ⁢ ( f ν + 1 - g ν + 1 ) superscript 𝑥 1 𝜈 subscript 𝑓 𝜈 𝑥 superscript 𝑥 1 𝜈 2 subscript 𝑓 𝜈 1 subscript 𝑔 𝜈 1 {\displaystyle{\displaystyle\int x^{1+\nu}f_{\nu}\mathrm{d}x=-\frac{x^{1+\nu}}% {\sqrt{2}}(f_{\nu+1}-g_{\nu+1})}}
\int x^{1+\nu}f_{\nu}\diff{x} = -\frac{x^{1+\nu}}{\sqrt{2}}(f_{\nu+1}-g_{\nu+1})		 

int((x)^(1 + nu)* f[nu], x) = -((x)^(1 + nu))/(sqrt(2))*(f[nu + 1]- g[nu + 1])

Integrate[(x)^(1 + \[Nu])* Subscript[f, \[Nu]], x, GenerateConditions->None] == -Divide[(x)^(1 + \[Nu]),Sqrt[2]]*(Subscript[f, \[Nu]+ 1]- Subscript[g, \[Nu]+ 1])
Failure Failure
Failed [300 / 300]
Result: .9346151411+.5776724966*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, f[nu] = 1/2*3^(1/2)+1/2*I, f[1+nu] = 1/2*3^(1/2)+1/2*I, g[1+nu] = 1/2*3^(1/2)+1/2*I}


Result: 3.061934630+.4518721345*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, f[nu] = 1/2*3^(1/2)+1/2*I, f[1+nu] = 1/2*3^(1/2)+1/2*I, g[1+nu] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.9346151408625077, 0.5776724967688012]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[3.061934629891139, 0.45187213490403344]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[1, ν]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.71.E2 ∫ x 1 - ν ⁢ f ν ⁢ d x = x 1 - ν 2 ⁢ ( f ν - 1 - g ν - 1 ) superscript 𝑥 1 𝜈 subscript 𝑓 𝜈 𝑥 superscript 𝑥 1 𝜈 2 subscript 𝑓 𝜈 1 subscript 𝑔 𝜈 1 {\displaystyle{\displaystyle\int x^{1-\nu}f_{\nu}\mathrm{d}x=\frac{x^{1-\nu}}{% \sqrt{2}}(f_{\nu-1}-g_{\nu-1})}}
\int x^{1-\nu}f_{\nu}\diff{x} = \frac{x^{1-\nu}}{\sqrt{2}}(f_{\nu-1}-g_{\nu-1})		 

int((x)^(1 - nu)* f[nu], x) = ((x)^(1 - nu))/(sqrt(2))*(f[nu - 1]- g[nu - 1])

Integrate[(x)^(1 - \[Nu])* Subscript[f, \[Nu]], x, GenerateConditions->None] == Divide[(x)^(1 - \[Nu]),Sqrt[2]]*(Subscript[f, \[Nu]- 1]- Subscript[g, \[Nu]- 1])
Failure Failure
Failed [300 / 300]
Result: .9470105611+.8580421171*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, f[nu] = 1/2*3^(1/2)+1/2*I, f[nu-1] = 1/2*3^(1/2)+1/2*I, g[nu-1] = 1/2*3^(1/2)+1/2*I}


Result: .30703090e-2+1.331056152*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, f[nu] = 1/2*3^(1/2)+1/2*I, f[nu-1] = 1/2*3^(1/2)+1/2*I, g[nu-1] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.9470105613079453, 0.8580421172974921]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[-1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[-1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.0030703089818392426, 1.3310561520338196]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[-1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[-1, ν]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.71.E6 ∫ x ⁢ f ν ⁢ g ν ⁢ d x = 1 4 ⁢ x 2 ⁢ ( 2 ⁢ f ν ⁢ g ν - f ν - 1 ⁢ g ν + 1 - f ν + 1 ⁢ g ν - 1 ) 𝑥 subscript 𝑓 𝜈 subscript 𝑔 𝜈 𝑥 1 4 superscript 𝑥 2 2 subscript 𝑓 𝜈 subscript 𝑔 𝜈 subscript 𝑓 𝜈 1 subscript 𝑔 𝜈 1 subscript 𝑓 𝜈 1 subscript 𝑔 𝜈 1 {\displaystyle{\displaystyle\int xf_{\nu}g_{\nu}\mathrm{d}x=\tfrac{1}{4}x^{2}% \left(2f_{\nu}g_{\nu}-f_{\nu-1}g_{\nu+1}-f_{\nu+1}g_{\nu-1}\right)}}
\int xf_{\nu}g_{\nu}\diff{x} = \tfrac{1}{4}x^{2}\left(2f_{\nu}g_{\nu}-f_{\nu-1}g_{\nu+1}-f_{\nu+1}g_{\nu-1}\right)		 

int(x*f[nu]*g[nu], x) = (1)/(4)*(x)^(2)*(2*f[nu]*g[nu]- f[nu - 1]*g[nu + 1]- f[nu + 1]*g[nu - 1])

Integrate[x*Subscript[f, \[Nu]]*Subscript[g, \[Nu]], x, GenerateConditions->None] == Divide[1,4]*(x)^(2)*(2*Subscript[f, \[Nu]]*Subscript[g, \[Nu]]- Subscript[f, \[Nu]- 1]*Subscript[g, \[Nu]+ 1]- Subscript[f, \[Nu]+ 1]*Subscript[g, \[Nu]- 1])
Failure Failure
Failed [270 / 300]
Result: .5625000004+.9742785795*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, f[nu] = 1/2*3^(1/2)+1/2*I, f[1+nu] = 1/2*3^(1/2)+1/2*I, f[nu-1] = 1/2*3^(1/2)+1/2*I, g[nu] = 1/2*3^(1/2)+1/2*I, g[1+nu] = 1/2*3^(1/2)+1/2*I, g[nu-1] = 1/2*3^(1/2)+1/2*I}


Result: -.2058892896+.7683892900*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, f[nu] = 1/2*3^(1/2)+1/2*I, f[1+nu] = 1/2*3^(1/2)+1/2*I, f[nu-1] = 1/2*3^(1/2)+1/2*I, g[nu] = 1/2*3^(1/2)+1/2*I, g[1+nu] = 1/2*3^(1/2)+1/2*I, g[nu-1] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Skipped - Because timed out
10.71.E7 ∫ x ⁢ ( f ν 2 - g ν 2 ) ⁢ d x = 1 2 ⁢ x 2 ⁢ ( f ν 2 - f ν - 1 ⁢ f ν + 1 - g ν 2 + g ν - 1 ⁢ g ν + 1 ) 𝑥 superscript subscript 𝑓 𝜈 2 superscript subscript 𝑔 𝜈 2 𝑥 1 2 superscript 𝑥 2 superscript subscript 𝑓 𝜈 2 subscript 𝑓 𝜈 1 subscript 𝑓 𝜈 1 superscript subscript 𝑔 𝜈 2 subscript 𝑔 𝜈 1 subscript 𝑔 𝜈 1 {\displaystyle{\displaystyle\int x(f_{\nu}^{2}-g_{\nu}^{2})\mathrm{d}x=\tfrac{% 1}{2}x^{2}\left(f_{\nu}^{2}-f_{\nu-1}f_{\nu+1}-g_{\nu}^{2}+g_{\nu-1}g_{\nu+1}% \right)}}
\int x(f_{\nu}^{2}-g_{\nu}^{2})\diff{x} = \tfrac{1}{2}x^{2}\left(f_{\nu}^{2}-f_{\nu-1}f_{\nu+1}-g_{\nu}^{2}+g_{\nu-1}g_{\nu+1}\right)		 

int(x*((f[nu])^(2)- (g[nu])^(2)), x) = (1)/(2)*(x)^(2)*((f[nu])^(2)- f[nu - 1]*f[nu + 1]- (g[nu])^(2)+ g[nu - 1]*g[nu + 1])

Integrate[x*((Subscript[f, \[Nu]])^(2)- (Subscript[g, \[Nu]])^(2)), x, GenerateConditions->None] == Divide[1,2]*(x)^(2)*((Subscript[f, \[Nu]])^(2)- Subscript[f, \[Nu]- 1]*Subscript[f, \[Nu]+ 1]- (Subscript[g, \[Nu]])^(2)+ Subscript[g, \[Nu]- 1]*Subscript[g, \[Nu]+ 1])
Failure Failure Error Error
10.71#Ex1 ∫ x ⁢ M ν 2 ⁡ ( x ) ⁢ d x = x ⁢ ( ber ν ⁡ x ⁢ bei ν ′ ⁡ x - ber ν ′ ⁡ x ⁢ bei ν ⁡ x ) 𝑥 modulus-Bessel-M 𝜈 2 𝑥 𝑥 𝑥 Kelvin-ber 𝜈 𝑥 diffop Kelvin-bei 𝜈 1 𝑥 diffop Kelvin-ber 𝜈 1 𝑥 Kelvin-bei 𝜈 𝑥 {\displaystyle{\displaystyle\int x{M_{\nu}^{2}}\left(x\right)\mathrm{d}x=x(% \operatorname{ber}_{\nu}x\operatorname{bei}_{\nu}'x-\operatorname{ber}_{\nu}'x% \operatorname{bei}_{\nu}x)}}
\int x\HankelmodM{\nu}^{2}@{x}\diff{x} = x(\Kelvinber{\nu}@@{x}\Kelvinbei{\nu}'@@{x}-\Kelvinber{\nu}'@@{x}\Kelvinbei{\nu}@@{x})		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
Error

Integrate[x*(Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2])^(2), x, GenerateConditions->None] == x*(KelvinBer[\[Nu], x]*D[KelvinBei[\[Nu], x], {x, 1}]- D[KelvinBer[\[Nu], x], {x, 1}]*KelvinBei[\[Nu], x])
Missing Macro Error Successful - Successful [Tested: 30]
10.71#Ex2 ∫ x ⁢ N ν 2 ⁡ ( x ) ⁢ d x = x ⁢ ( ker ν ⁡ x ⁢ kei ν ′ ⁡ x - ker ν ′ ⁡ x ⁢ kei ν ⁡ x ) 𝑥 modulus-Bessel-N 𝜈 2 𝑥 𝑥 𝑥 Kelvin-ker 𝜈 𝑥 diffop Kelvin-kei 𝜈 1 𝑥 diffop Kelvin-ker 𝜈 1 𝑥 Kelvin-kei 𝜈 𝑥 {\displaystyle{\displaystyle\int x{N_{\nu}^{2}}\left(x\right)\mathrm{d}x=x(% \operatorname{ker}_{\nu}x\operatorname{kei}_{\nu}'x-\operatorname{ker}_{\nu}'x% \operatorname{kei}_{\nu}x)}}
\int x\HankelmodderivN{\nu}^{2}@{x}\diff{x} = x(\Kelvinker{\nu}@@{x}\Kelvinkei{\nu}'@@{x}-\Kelvinker{\nu}'@@{x}\Kelvinkei{\nu}@@{x})		 

Error

Integrate[x*(Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2])^(2), x, GenerateConditions->None] == x*(KelvinKer[\[Nu], x]*D[KelvinKei[\[Nu], x], {x, 1}]- D[KelvinKer[\[Nu], x], {x, 1}]*KelvinKei[\[Nu], x])
Missing Macro Error Successful - Successful [Tested: 30]
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