Bessel Functions - 10.61 Definitions and Basic Properties

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
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10.61.E1 ber ν ⁡ x + i ⁢ bei ν ⁡ x = J ν ⁡ ( x ⁢ e 3 ⁢ π ⁢ i / 4 ) Kelvin-ber 𝜈 𝑥 𝑖 Kelvin-bei 𝜈 𝑥 Bessel-J 𝜈 𝑥 superscript 𝑒 3 𝜋 𝑖 4 {\displaystyle{\displaystyle\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu% }x=J_{\nu}\left(xe^{3\pi i/4}\right)}}
\Kelvinber{\nu}@@{x}+i\Kelvinbei{\nu}@@{x} = \BesselJ{\nu}@{xe^{3\pi i/4}}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
KelvinBer(nu, x)+ I*KelvinBei(nu, x) = BesselJ(nu, x*exp(3*Pi*I/4))

KelvinBer[\[Nu], x]+ I*KelvinBei[\[Nu], x] == BesselJ[\[Nu], x*Exp[3*Pi*I/4]]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 30]
10.61.E1 J ν ⁡ ( x ⁢ e 3 ⁢ π ⁢ i / 4 ) = e ν ⁢ π ⁢ i ⁢ J ν ⁡ ( x ⁢ e - π ⁢ i / 4 ) Bessel-J 𝜈 𝑥 superscript 𝑒 3 𝜋 𝑖 4 superscript 𝑒 𝜈 𝜋 𝑖 Bessel-J 𝜈 𝑥 superscript 𝑒 𝜋 𝑖 4 {\displaystyle{\displaystyle J_{\nu}\left(xe^{3\pi i/4}\right)=e^{\nu\pi i}J_{% \nu}\left(xe^{-\pi i/4}\right)}}
\BesselJ{\nu}@{xe^{3\pi i/4}} = e^{\nu\pi i}\BesselJ{\nu}@{xe^{-\pi i/4}}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselJ(nu, x*exp(3*Pi*I/4)) = exp(nu*Pi*I)*BesselJ(nu, x*exp(- Pi*I/4))

BesselJ[\[Nu], x*Exp[3*Pi*I/4]] == Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], x*Exp[- Pi*I/4]]
Failure Failure Successful [Tested: 30] Successful [Tested: 30]
10.61.E1 e ν ⁢ π ⁢ i ⁢ J ν ⁡ ( x ⁢ e - π ⁢ i / 4 ) = e ν ⁢ π ⁢ i / 2 ⁢ I ν ⁡ ( x ⁢ e π ⁢ i / 4 ) superscript 𝑒 𝜈 𝜋 𝑖 Bessel-J 𝜈 𝑥 superscript 𝑒 𝜋 𝑖 4 superscript 𝑒 𝜈 𝜋 𝑖 2 modified-Bessel-first-kind 𝜈 𝑥 superscript 𝑒 𝜋 𝑖 4 {\displaystyle{\displaystyle e^{\nu\pi i}J_{\nu}\left(xe^{-\pi i/4}\right)=e^{% \nu\pi i/2}I_{\nu}\left(xe^{\pi i/4}\right)}}
e^{\nu\pi i}\BesselJ{\nu}@{xe^{-\pi i/4}} = e^{\nu\pi i/2}\modBesselI{\nu}@{xe^{\pi i/4}}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
exp(nu*Pi*I)*BesselJ(nu, x*exp(- Pi*I/4)) = exp(nu*Pi*I/2)*BesselI(nu, x*exp(Pi*I/4))

Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], x*Exp[- Pi*I/4]] == Exp[\[Nu]*Pi*I/2]*BesselI[\[Nu], x*Exp[Pi*I/4]]
Failure Failure Successful [Tested: 30] Successful [Tested: 30]
10.61.E1 e ν ⁢ π ⁢ i / 2 ⁢ I ν ⁡ ( x ⁢ e π ⁢ i / 4 ) = e 3 ⁢ ν ⁢ π ⁢ i / 2 ⁢ I ν ⁡ ( x ⁢ e - 3 ⁢ π ⁢ i / 4 ) superscript 𝑒 𝜈 𝜋 𝑖 2 modified-Bessel-first-kind 𝜈 𝑥 superscript 𝑒 𝜋 𝑖 4 superscript 𝑒 3 𝜈 𝜋 𝑖 2 modified-Bessel-first-kind 𝜈 𝑥 superscript 𝑒 3 𝜋 𝑖 4 {\displaystyle{\displaystyle e^{\nu\pi i/2}I_{\nu}\left(xe^{\pi i/4}\right)=e^% {3\nu\pi i/2}I_{\nu}\left(xe^{-3\pi i/4}\right)}}
e^{\nu\pi i/2}\modBesselI{\nu}@{xe^{\pi i/4}} = e^{3\nu\pi i/2}\modBesselI{\nu}@{xe^{-3\pi i/4}}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
exp(nu*Pi*I/2)*BesselI(nu, x*exp(Pi*I/4)) = exp(3*nu*Pi*I/2)*BesselI(nu, x*exp(- 3*Pi*I/4))

Exp[\[Nu]*Pi*I/2]*BesselI[\[Nu], x*Exp[Pi*I/4]] == Exp[3*\[Nu]*Pi*I/2]*BesselI[\[Nu], x*Exp[- 3*Pi*I/4]]
Failure Failure Successful [Tested: 30] Successful [Tested: 30]
10.61.E2 ker ν ⁡ x + i ⁢ kei ν ⁡ x = e - ν ⁢ π ⁢ i / 2 ⁢ K ν ⁡ ( x ⁢ e π ⁢ i / 4 ) Kelvin-ker 𝜈 𝑥 𝑖 Kelvin-kei 𝜈 𝑥 superscript 𝑒 𝜈 𝜋 𝑖 2 modified-Bessel-second-kind 𝜈 𝑥 superscript 𝑒 𝜋 𝑖 4 {\displaystyle{\displaystyle\operatorname{ker}_{\nu}x+i\operatorname{kei}_{\nu% }x=e^{-\nu\pi i/2}K_{\nu}\left(xe^{\pi i/4}\right)}}
\Kelvinker{\nu}@@{x}+i\Kelvinkei{\nu}@@{x} = e^{-\nu\pi i/2}\modBesselK{\nu}@{xe^{\pi i/4}}		 

KelvinKer(nu, x)+ I*KelvinKei(nu, x) = exp(- nu*Pi*I/2)*BesselK(nu, x*exp(Pi*I/4))

KelvinKer[\[Nu], x]+ I*KelvinKei[\[Nu], x] == Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], x*Exp[Pi*I/4]]
Failure Failure Successful [Tested: 30] Successful [Tested: 30]
10.61.E2 e - ν ⁢ π ⁢ i / 2 ⁢ K ν ⁡ ( x ⁢ e π ⁢ i / 4 ) = 1 2 ⁢ π ⁢ i ⁢ H ν ( 1 ) ⁡ ( x ⁢ e 3 ⁢ π ⁢ i / 4 ) superscript 𝑒 𝜈 𝜋 𝑖 2 modified-Bessel-second-kind 𝜈 𝑥 superscript 𝑒 𝜋 𝑖 4 1 2 𝜋 𝑖 Hankel-H-1-Bessel-third-kind 𝜈 𝑥 superscript 𝑒 3 𝜋 𝑖 4 {\displaystyle{\displaystyle e^{-\nu\pi i/2}K_{\nu}\left(xe^{\pi i/4}\right)=% \tfrac{1}{2}\pi i{H^{(1)}_{\nu}}\left(xe^{3\pi i/4}\right)}}
e^{-\nu\pi i/2}\modBesselK{\nu}@{xe^{\pi i/4}} = \tfrac{1}{2}\pi i\HankelH{1}{\nu}@{xe^{3\pi i/4}}		 

exp(- nu*Pi*I/2)*BesselK(nu, x*exp(Pi*I/4)) = (1)/(2)*Pi*I*HankelH1(nu, x*exp(3*Pi*I/4))

Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], x*Exp[Pi*I/4]] == Divide[1,2]*Pi*I*HankelH1[\[Nu], x*Exp[3*Pi*I/4]]
Failure Failure Successful [Tested: 30] Successful [Tested: 30]
10.61.E2 1 2 ⁢ π ⁢ i ⁢ H ν ( 1 ) ⁡ ( x ⁢ e 3 ⁢ π ⁢ i / 4 ) = - 1 2 ⁢ π ⁢ i ⁢ e - ν ⁢ π ⁢ i ⁢ H ν ( 2 ) ⁡ ( x ⁢ e - π ⁢ i / 4 ) 1 2 𝜋 𝑖 Hankel-H-1-Bessel-third-kind 𝜈 𝑥 superscript 𝑒 3 𝜋 𝑖 4 1 2 𝜋 𝑖 superscript 𝑒 𝜈 𝜋 𝑖 Hankel-H-2-Bessel-third-kind 𝜈 𝑥 superscript 𝑒 𝜋 𝑖 4 {\displaystyle{\displaystyle\tfrac{1}{2}\pi i{H^{(1)}_{\nu}}\left(xe^{3\pi i/4% }\right)=-\tfrac{1}{2}\pi ie^{-\nu\pi i}{H^{(2)}_{\nu}}\left(xe^{-\pi i/4}% \right)}}
\tfrac{1}{2}\pi i\HankelH{1}{\nu}@{xe^{3\pi i/4}} = -\tfrac{1}{2}\pi ie^{-\nu\pi i}\HankelH{2}{\nu}@{xe^{-\pi i/4}}		 

(1)/(2)*Pi*I*HankelH1(nu, x*exp(3*Pi*I/4)) = -(1)/(2)*Pi*I*exp(- nu*Pi*I)*HankelH2(nu, x*exp(- Pi*I/4))

Divide[1,2]*Pi*I*HankelH1[\[Nu], x*Exp[3*Pi*I/4]] == -Divide[1,2]*Pi*I*Exp[- \[Nu]*Pi*I]*HankelH2[\[Nu], x*Exp[- Pi*I/4]]
Failure Failure Successful [Tested: 30] Successful [Tested: 30]
10.61.E3 x 2 ⁢ d 2 w d x 2 + x ⁢ d w d x - ( i ⁢ x 2 + ν 2 ) ⁢ w = 0 superscript 𝑥 2 derivative 𝑤 𝑥 2 𝑥 derivative 𝑤 𝑥 𝑖 superscript 𝑥 2 superscript 𝜈 2 𝑤 0 {\displaystyle{\displaystyle x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+% x\frac{\mathrm{d}w}{\mathrm{d}x}-(ix^{2}+\nu^{2})w=0}}
x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}-(ix^{2}+\nu^{2})w = 0		 

(x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)-(I*(x)^(2)+ (nu)^(2))*w = 0

(x)^(2)* D[w, {x, 2}]+ x*D[w, x]-(I*(x)^(2)+ \[Nu]^(2))*w == 0
Failure Failure
Failed [300 / 300]
Result: 1.125000000-2.948557160*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: .1249999997-1.216506352*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.1249999999999996, -2.948557158514987]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.1249999999999996, -0.9485571585149869]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.61.E4 x 4 ⁢ d 4 w d x 4 + 2 ⁢ x 3 ⁢ d 3 w d x 3 - ( 1 + 2 ⁢ ν 2 ) ⁢ ( x 2 ⁢ d 2 w d x 2 - x ⁢ d w d x ) + ( ν 4 - 4 ⁢ ν 2 + x 4 ) ⁢ w = 0 superscript 𝑥 4 derivative 𝑤 𝑥 4 2 superscript 𝑥 3 derivative 𝑤 𝑥 3 1 2 superscript 𝜈 2 superscript 𝑥 2 derivative 𝑤 𝑥 2 𝑥 derivative 𝑤 𝑥 superscript 𝜈 4 4 superscript 𝜈 2 superscript 𝑥 4 𝑤 0 {\displaystyle{\displaystyle x^{4}\frac{{\mathrm{d}}^{4}w}{{\mathrm{d}x}^{4}}+% 2x^{3}\frac{{\mathrm{d}}^{3}w}{{\mathrm{d}x}^{3}}-(1+2\nu^{2})\left(x^{2}\frac% {{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-x\frac{\mathrm{d}w}{\mathrm{d}x}\right)% +(\nu^{4}-4\nu^{2}+x^{4})w=0}}
x^{4}\deriv[4]{w}{x}+2x^{3}\deriv[3]{w}{x}-(1+2\nu^{2})\left(x^{2}\deriv[2]{w}{x}-x\deriv{w}{x}\right)+(\nu^{4}-4\nu^{2}+x^{4})w = 0		 w = ber + ν ⁡ x , w = ber - ν ⁡ x formulae-sequence 𝑤 Kelvin-ber 𝜈 𝑥 𝑤 Kelvin-ber 𝜈 𝑥 {\displaystyle{\displaystyle w=\operatorname{ber}_{+\nu}x,w=\operatorname{ber}% _{-\nu}x}} 
(x)^(4)* diff(w, [x$(4)])+ 2*(x)^(3)* diff(w, [x$(3)])-(1 + 2*(nu)^(2))*((x)^(2)* diff(w, [x$(2)])- x*diff(w, x))+((nu)^(4)- 4*(nu)^(2)+ (x)^(4))*w = 0

(x)^(4)* D[w, {x, 4}]+ 2*(x)^(3)* D[w, {x, 3}]-(1 + 2*\[Nu]^(2))*((x)^(2)* D[w, {x, 2}]- x*D[w, x])+(\[Nu]^(4)- 4*\[Nu]^(2)+ (x)^(4))*w == 0
Error Failure - Skip - No test values generated
10.61#Ex1 ber n ⁡ ( - x ) = ( - 1 ) n ⁢ ber n ⁡ x Kelvin-ber 𝑛 𝑥 superscript 1 𝑛 Kelvin-ber 𝑛 𝑥 {\displaystyle{\displaystyle\operatorname{ber}_{n}\left(-x\right)=(-1)^{n}% \operatorname{ber}_{n}x}}
\Kelvinber{n}@{-x} = (-1)^{n}\Kelvinber{n}@@{x}		 ℜ ⁡ ( n + k + 1 ) > 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0}} 
KelvinBer(n, - x) = (- 1)^(n)* KelvinBer(n, x)

KelvinBer[n, - x] == (- 1)^(n)* KelvinBer[n, x]
Successful Failure - Successful [Tested: 9]
10.61#Ex2 bei n ⁡ ( - x ) = ( - 1 ) n ⁢ bei n ⁡ x Kelvin-bei 𝑛 𝑥 superscript 1 𝑛 Kelvin-bei 𝑛 𝑥 {\displaystyle{\displaystyle\operatorname{bei}_{n}\left(-x\right)=(-1)^{n}% \operatorname{bei}_{n}x}}
\Kelvinbei{n}@{-x} = (-1)^{n}\Kelvinbei{n}@@{x}		 

KelvinBei(n, - x) = (- 1)^(n)* KelvinBei(n, x)

KelvinBei[n, - x] == (- 1)^(n)* KelvinBei[n, x]
Successful Failure - Successful [Tested: 9]
10.61#Ex3 ber - ν ⁡ x = cos ⁡ ( ν ⁢ π ) ⁢ ber ν ⁡ x + sin ⁡ ( ν ⁢ π ) ⁢ bei ν ⁡ x + ( 2 / π ) ⁢ sin ⁡ ( ν ⁢ π ) ⁢ ker ν ⁡ x Kelvin-ber 𝜈 𝑥 𝜈 𝜋 Kelvin-ber 𝜈 𝑥 𝜈 𝜋 Kelvin-bei 𝜈 𝑥 2 𝜋 𝜈 𝜋 Kelvin-ker 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{ber}_{-\nu}x=\cos\left(\nu\pi\right)% \operatorname{ber}_{\nu}x+\sin\left(\nu\pi\right)\operatorname{bei}_{\nu}x+(2/% \pi)\sin\left(\nu\pi\right)\operatorname{ker}_{\nu}x}}
\Kelvinber{-\nu}@@{x} = \cos@{\nu\pi}\Kelvinber{\nu}@@{x}+\sin@{\nu\pi}\Kelvinbei{\nu}@@{x}+(2/\pi)\sin@{\nu\pi}\Kelvinker{\nu}@@{x}		 ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0}} 
KelvinBer(- nu, x) = cos(nu*Pi)*KelvinBer(nu, x)+ sin(nu*Pi)*KelvinBei(nu, x)+(2/Pi)*sin(nu*Pi)*KelvinKer(nu, x)

KelvinBer[- \[Nu], x] == Cos[\[Nu]*Pi]*KelvinBer[\[Nu], x]+ Sin[\[Nu]*Pi]*KelvinBei[\[Nu], x]+(2/Pi)*Sin[\[Nu]*Pi]*KelvinKer[\[Nu], x]
Failure Failure Successful [Tested: 30] Successful [Tested: 30]
10.61#Ex4 bei - ν ⁡ x = - sin ⁡ ( ν ⁢ π ) ⁢ ber ν ⁡ x + cos ⁡ ( ν ⁢ π ) ⁢ bei ν ⁡ x + ( 2 / π ) ⁢ sin ⁡ ( ν ⁢ π ) ⁢ kei ν ⁡ x Kelvin-bei 𝜈 𝑥 𝜈 𝜋 Kelvin-ber 𝜈 𝑥 𝜈 𝜋 Kelvin-bei 𝜈 𝑥 2 𝜋 𝜈 𝜋 Kelvin-kei 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{bei}_{-\nu}x=-\sin\left(\nu\pi\right% )\operatorname{ber}_{\nu}x+\cos\left(\nu\pi\right)\operatorname{bei}_{\nu}x+(2% /\pi)\sin\left(\nu\pi\right)\operatorname{kei}_{\nu}x}}
\Kelvinbei{-\nu}@@{x} = -\sin@{\nu\pi}\Kelvinber{\nu}@@{x}+\cos@{\nu\pi}\Kelvinbei{\nu}@@{x}+(2/\pi)\sin@{\nu\pi}\Kelvinkei{\nu}@@{x}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
KelvinBei(- nu, x) = - sin(nu*Pi)*KelvinBer(nu, x)+ cos(nu*Pi)*KelvinBei(nu, x)+(2/Pi)*sin(nu*Pi)*KelvinKei(nu, x)

KelvinBei[- \[Nu], x] == - Sin[\[Nu]*Pi]*KelvinBer[\[Nu], x]+ Cos[\[Nu]*Pi]*KelvinBei[\[Nu], x]+(2/Pi)*Sin[\[Nu]*Pi]*KelvinKei[\[Nu], x]
Failure Failure Successful [Tested: 30] Successful [Tested: 30]
10.61#Ex5 ker - ν ⁡ x = cos ⁡ ( ν ⁢ π ) ⁢ ker ν ⁡ x - sin ⁡ ( ν ⁢ π ) ⁢ kei ν ⁡ x Kelvin-ker 𝜈 𝑥 𝜈 𝜋 Kelvin-ker 𝜈 𝑥 𝜈 𝜋 Kelvin-kei 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{ker}_{-\nu}x=\cos\left(\nu\pi\right)% \operatorname{ker}_{\nu}x-\sin\left(\nu\pi\right)\operatorname{kei}_{\nu}x}}
\Kelvinker{-\nu}@@{x} = \cos@{\nu\pi}\Kelvinker{\nu}@@{x}-\sin@{\nu\pi}\Kelvinkei{\nu}@@{x}		 

KelvinKer(- nu, x) = cos(nu*Pi)*KelvinKer(nu, x)- sin(nu*Pi)*KelvinKei(nu, x)

KelvinKer[- \[Nu], x] == Cos[\[Nu]*Pi]*KelvinKer[\[Nu], x]- Sin[\[Nu]*Pi]*KelvinKei[\[Nu], x]
Successful Failure - Successful [Tested: 30]
10.61#Ex6 kei - ν ⁡ x = sin ⁡ ( ν ⁢ π ) ⁢ ker ν ⁡ x + cos ⁡ ( ν ⁢ π ) ⁢ kei ν ⁡ x Kelvin-kei 𝜈 𝑥 𝜈 𝜋 Kelvin-ker 𝜈 𝑥 𝜈 𝜋 Kelvin-kei 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{kei}_{-\nu}x=\sin\left(\nu\pi\right)% \operatorname{ker}_{\nu}x+\cos\left(\nu\pi\right)\operatorname{kei}_{\nu}x}}
\Kelvinkei{-\nu}@@{x} = \sin@{\nu\pi}\Kelvinker{\nu}@@{x}+\cos@{\nu\pi}\Kelvinkei{\nu}@@{x}		 

KelvinKei(- nu, x) = sin(nu*Pi)*KelvinKer(nu, x)+ cos(nu*Pi)*KelvinKei(nu, x)

KelvinKei[- \[Nu], x] == Sin[\[Nu]*Pi]*KelvinKer[\[Nu], x]+ Cos[\[Nu]*Pi]*KelvinKei[\[Nu], x]
Successful Failure - Successful [Tested: 30]
10.61#Ex7 ber - n ⁡ x = ( - 1 ) n ⁢ ber n ⁡ x , bei - n ⁡ x Kelvin-ber 𝑛 𝑥 superscript 1 𝑛 Kelvin-ber 𝑛 𝑥 Kelvin-bei 𝑛 𝑥 {\displaystyle{\displaystyle\operatorname{ber}_{-n}x=(-1)^{n}\operatorname{ber% }_{n}x,~{}{}\operatorname{bei}_{-n}x}}
\Kelvinber{-n}@@{x} = (-1)^{n}\Kelvinber{n}@@{x},~{}\Kelvinbei{-n}@@{x}		 ℜ ⁡ ( ( - n ) + k + 1 ) > 0 , ℜ ⁡ ( n + k + 1 ) > 0 formulae-sequence 𝑛 𝑘 1 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re((-n)+k+1)>0,\Re(n+k+1)>0}} 
KelvinBer(- n, x) = (- 1)^(n)* KelvinBer(n, x); *KelvinBei(- n, x)

KelvinBer[- n, x] == (- 1)^(n)* KelvinBer[n, x]
 *KelvinBei[- n, x]
Error Failure - Error
10.61#Ex7 ( - 1 ) n ⁢ ber n ⁡ x , bei - n ⁡ x = ( - 1 ) n ⁢ bei n ⁡ x superscript 1 𝑛 Kelvin-ber 𝑛 𝑥 Kelvin-bei 𝑛 𝑥 superscript 1 𝑛 Kelvin-bei 𝑛 𝑥 {\displaystyle{\displaystyle(-1)^{n}\operatorname{ber}_{n}x,~{}{}\operatorname% {bei}_{-n}x=(-1)^{n}\operatorname{bei}_{n}x}}
(-1)^{n}\Kelvinber{n}@@{x},~{}\Kelvinbei{-n}@@{x} = (-1)^{n}\Kelvinbei{n}@@{x}		 ℜ ⁡ ( ( - n ) + k + 1 ) > 0 , ℜ ⁡ ( n + k + 1 ) > 0 formulae-sequence 𝑛 𝑘 1 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re((-n)+k+1)>0,\Re(n+k+1)>0}} 
(- 1)^(n)* KelvinBer(n, x),*KelvinBei(- n, x) = (- 1)^(n)* KelvinBei(n, x)

(- 1)^(n)* KelvinBer[n, x],*KelvinBei[- n, x] == (- 1)^(n)* KelvinBei[n, x]
Error Failure - Error
10.61#Ex8 ker - n ⁡ x = ( - 1 ) n ⁢ ker n ⁡ x , kei - n ⁡ x Kelvin-ker 𝑛 𝑥 superscript 1 𝑛 Kelvin-ker 𝑛 𝑥 Kelvin-kei 𝑛 𝑥 {\displaystyle{\displaystyle\operatorname{ker}_{-n}x=(-1)^{n}\operatorname{ker% }_{n}x,~{}{}\operatorname{kei}_{-n}x}}
\Kelvinker{-n}@@{x} = (-1)^{n}\Kelvinker{n}@@{x},~{}\Kelvinkei{-n}@@{x}		 

KelvinKer(- n, x) = (- 1)^(n)* KelvinKer(n, x); *KelvinKei(- n, x)

KelvinKer[- n, x] == (- 1)^(n)* KelvinKer[n, x]
 *KelvinKei[- n, x]
Error Failure - Error
10.61#Ex8 ( - 1 ) n ⁢ ker n ⁡ x , kei - n ⁡ x = ( - 1 ) n ⁢ kei n ⁡ x superscript 1 𝑛 Kelvin-ker 𝑛 𝑥 Kelvin-kei 𝑛 𝑥 superscript 1 𝑛 Kelvin-kei 𝑛 𝑥 {\displaystyle{\displaystyle(-1)^{n}\operatorname{ker}_{n}x,~{}{}\operatorname% {kei}_{-n}x=(-1)^{n}\operatorname{kei}_{n}x}}
(-1)^{n}\Kelvinker{n}@@{x},~{}\Kelvinkei{-n}@@{x} = (-1)^{n}\Kelvinkei{n}@@{x}		 

(- 1)^(n)* KelvinKer(n, x),*KelvinKei(- n, x) = (- 1)^(n)* KelvinKei(n, x)

(- 1)^(n)* KelvinKer[n, x],*KelvinKei[- n, x] == (- 1)^(n)* KelvinKei[n, x]
Error Failure - Error
10.61#Ex9 ber 1 2 ⁡ ( x ⁢ 2 ) = 2 - 3 4 π ⁢ x ⁢ ( e x ⁢ cos ⁡ ( x + π 8 ) - e - x ⁢ cos ⁡ ( x - π 8 ) ) Kelvin-ber 1 2 𝑥 2 superscript 2 3 4 𝜋 𝑥 superscript 𝑒 𝑥 𝑥 𝜋 8 superscript 𝑒 𝑥 𝑥 𝜋 8 {\displaystyle{\displaystyle\operatorname{ber}_{\frac{1}{2}}\left(x\sqrt{2}% \right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos\left(x+\frac{\pi}% {8}\right)-e^{-x}\cos\left(x-\frac{\pi}{8}\right)\right)}}
\Kelvinber{\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos@{x+\frac{\pi}{8}}-e^{-x}\cos@{x-\frac{\pi}{8}}\right)		 ℜ ⁡ ( ( 1 2 ) + k + 1 ) > 0 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((\frac{1}{2})+k+1)>0}} 
KelvinBer((1)/(2), x*sqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*cos(x +(Pi)/(8))- exp(- x)*cos(x -(Pi)/(8)))

KelvinBer[Divide[1,2], x*Sqrt[2]] == Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Cos[x +Divide[Pi,8]]- Exp[- x]*Cos[x -Divide[Pi,8]])
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
10.61#Ex10 bei 1 2 ⁡ ( x ⁢ 2 ) = 2 - 3 4 π ⁢ x ⁢ ( e x ⁢ sin ⁡ ( x + π 8 ) + e - x ⁢ sin ⁡ ( x - π 8 ) ) Kelvin-bei 1 2 𝑥 2 superscript 2 3 4 𝜋 𝑥 superscript 𝑒 𝑥 𝑥 𝜋 8 superscript 𝑒 𝑥 𝑥 𝜋 8 {\displaystyle{\displaystyle\operatorname{bei}_{\frac{1}{2}}\left(x\sqrt{2}% \right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin\left(x+\frac{\pi}% {8}\right)+\,e^{-x}\sin\left(x-\frac{\pi}{8}\right)\right)}}
\Kelvinbei{\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin@{x+\frac{\pi}{8}}+\,e^{-x}\sin@{x-\frac{\pi}{8}}\right)		 

KelvinBei((1)/(2), x*sqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*sin(x +(Pi)/(8))+ exp(- x)*sin(x -(Pi)/(8)))

KelvinBei[Divide[1,2], x*Sqrt[2]] == Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Sin[x +Divide[Pi,8]]+ Exp[- x]*Sin[x -Divide[Pi,8]])
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
10.61#Ex11 ber - 1 2 ⁡ ( x ⁢ 2 ) = 2 - 3 4 π ⁢ x ⁢ ( e x ⁢ sin ⁡ ( x + π 8 ) - e - x ⁢ sin ⁡ ( x - π 8 ) ) Kelvin-ber 1 2 𝑥 2 superscript 2 3 4 𝜋 𝑥 superscript 𝑒 𝑥 𝑥 𝜋 8 superscript 𝑒 𝑥 𝑥 𝜋 8 {\displaystyle{\displaystyle\operatorname{ber}_{-\frac{1}{2}}\left(x\sqrt{2}% \right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin\left(x+\frac{\pi}% {8}\right)-e^{-x}\sin\left(x-\frac{\pi}{8}\right)\right)}}
\Kelvinber{-\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin@{x+\frac{\pi}{8}}-e^{-x}\sin@{x-\frac{\pi}{8}}\right)		 ℜ ⁡ ( ( - 1 2 ) + k + 1 ) > 0 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((-\frac{1}{2})+k+1)>0}} 
KelvinBer(-(1)/(2), x*sqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*sin(x +(Pi)/(8))- exp(- x)*sin(x -(Pi)/(8)))

KelvinBer[-Divide[1,2], x*Sqrt[2]] == Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Sin[x +Divide[Pi,8]]- Exp[- x]*Sin[x -Divide[Pi,8]])
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
10.61#Ex12 bei - 1 2 ⁡ ( x ⁢ 2 ) = - 2 - 3 4 π ⁢ x ⁢ ( e x ⁢ cos ⁡ ( x + π 8 ) + e - x ⁢ cos ⁡ ( x - π 8 ) ) Kelvin-bei 1 2 𝑥 2 superscript 2 3 4 𝜋 𝑥 superscript 𝑒 𝑥 𝑥 𝜋 8 superscript 𝑒 𝑥 𝑥 𝜋 8 {\displaystyle{\displaystyle\operatorname{bei}_{-\frac{1}{2}}\left(x\sqrt{2}% \right)=-\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos\left(x+\frac{\pi% }{8}\right)+e^{-x}\cos\left(x-\frac{\pi}{8}\right)\right)}}
\Kelvinbei{-\frac{1}{2}}@{x\sqrt{2}} = -\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos@{x+\frac{\pi}{8}}+e^{-x}\cos@{x-\frac{\pi}{8}}\right)		 

KelvinBei(-(1)/(2), x*sqrt(2)) = -((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*cos(x +(Pi)/(8))+ exp(- x)*cos(x -(Pi)/(8)))

KelvinBei[-Divide[1,2], x*Sqrt[2]] == -Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Cos[x +Divide[Pi,8]]+ Exp[- x]*Cos[x -Divide[Pi,8]])
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
10.61.E11 ker 1 2 ⁡ ( x ⁢ 2 ) = kei - 1 2 ⁡ ( x ⁢ 2 ) Kelvin-ker 1 2 𝑥 2 Kelvin-kei 1 2 𝑥 2 {\displaystyle{\displaystyle\operatorname{ker}_{\frac{1}{2}}\left(x\sqrt{2}% \right)=\operatorname{kei}_{-\frac{1}{2}}\left(x\sqrt{2}\right)}}
\Kelvinker{\frac{1}{2}}@{x\sqrt{2}} = \Kelvinkei{-\frac{1}{2}}@{x\sqrt{2}}		 

KelvinKer((1)/(2), x*sqrt(2)) = KelvinKei(-(1)/(2), x*sqrt(2))

KelvinKer[Divide[1,2], x*Sqrt[2]] == KelvinKei[-Divide[1,2], x*Sqrt[2]]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 3]
10.61.E11 kei - 1 2 ⁡ ( x ⁢ 2 ) = - 2 - 3 4 ⁢ π x ⁢ e - x ⁢ sin ⁡ ( x - π 8 ) Kelvin-kei 1 2 𝑥 2 superscript 2 3 4 𝜋 𝑥 superscript 𝑒 𝑥 𝑥 𝜋 8 {\displaystyle{\displaystyle\operatorname{kei}_{-\frac{1}{2}}\left(x\sqrt{2}% \right)=-2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\sin\left(x-\frac{\pi}{8}% \right)}}
\Kelvinkei{-\frac{1}{2}}@{x\sqrt{2}} = -2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\sin@{x-\frac{\pi}{8}}		 

KelvinKei(-(1)/(2), x*sqrt(2)) = - (2)^(-(3)/(4))*sqrt((Pi)/(x))*exp(- x)*sin(x -(Pi)/(8))

KelvinKei[-Divide[1,2], x*Sqrt[2]] == - (2)^(-Divide[3,4])*Sqrt[Divide[Pi,x]]*Exp[- x]*Sin[x -Divide[Pi,8]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
10.61.E12 kei 1 2 ⁡ ( x ⁢ 2 ) = - ker - 1 2 ⁡ ( x ⁢ 2 ) Kelvin-kei 1 2 𝑥 2 Kelvin-ker 1 2 𝑥 2 {\displaystyle{\displaystyle\operatorname{kei}_{\frac{1}{2}}\left(x\sqrt{2}% \right)=-\operatorname{ker}_{-\frac{1}{2}}\left(x\sqrt{2}\right)}}
\Kelvinkei{\frac{1}{2}}@{x\sqrt{2}} = -\Kelvinker{-\frac{1}{2}}@{x\sqrt{2}}		 

KelvinKei((1)/(2), x*sqrt(2)) = - KelvinKer(-(1)/(2), x*sqrt(2))

KelvinKei[Divide[1,2], x*Sqrt[2]] == - KelvinKer[-Divide[1,2], x*Sqrt[2]]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 3]
10.61.E12 - ker - 1 2 ⁡ ( x ⁢ 2 ) = - 2 - 3 4 ⁢ π x ⁢ e - x ⁢ cos ⁡ ( x - π 8 ) Kelvin-ker 1 2 𝑥 2 superscript 2 3 4 𝜋 𝑥 superscript 𝑒 𝑥 𝑥 𝜋 8 {\displaystyle{\displaystyle-\operatorname{ker}_{-\frac{1}{2}}\left(x\sqrt{2}% \right)=-2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\cos\left(x-\frac{\pi}{8}% \right)}}
-\Kelvinker{-\frac{1}{2}}@{x\sqrt{2}} = -2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\cos@{x-\frac{\pi}{8}}		 

- KelvinKer(-(1)/(2), x*sqrt(2)) = - (2)^(-(3)/(4))*sqrt((Pi)/(x))*exp(- x)*cos(x -(Pi)/(8))

- KelvinKer[-Divide[1,2], x*Sqrt[2]] == - (2)^(-Divide[3,4])*Sqrt[Divide[Pi,x]]*Exp[- x]*Cos[x -Divide[Pi,8]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
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