Bessel Functions - 10.4 Connection Formulas

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DLMF Formula Constraints Maple Mathematica Symbolic
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Maple 		Numeric
Mathematica
10.4#Ex1 J - n ⁡ ( z ) = ( - 1 ) n ⁢ J n ⁡ ( z ) Bessel-J 𝑛 𝑧 superscript 1 𝑛 Bessel-J 𝑛 𝑧 {\displaystyle{\displaystyle J_{-n}\left(z\right)=(-1)^{n}J_{n}\left(z\right)}}
\BesselJ{-n}@{z} = (-1)^{n}\BesselJ{n}@{z}		 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}} 
BesselJ(- n, z) = (- 1)^(n)* BesselJ(n, z)

BesselJ[- n, z] == (- 1)^(n)* BesselJ[n, z]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
10.4#Ex2 Y - n ⁡ ( z ) = ( - 1 ) n ⁢ Y n ⁡ ( z ) Bessel-Y-Weber 𝑛 𝑧 superscript 1 𝑛 Bessel-Y-Weber 𝑛 𝑧 {\displaystyle{\displaystyle Y_{-n}\left(z\right)=(-1)^{n}Y_{n}\left(z\right)}}
\BesselY{-n}@{z} = (-1)^{n}\BesselY{n}@{z}		 ℜ ⁡ ( ( - n ) + k + 1 ) > 0 , ℜ ⁡ ( n + k + 1 ) > 0 , ℜ ⁡ ( ( - ( - n ) ) + k + 1 ) > 0 formulae-sequence 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,\Re((-(-n))+k+1)>0}} 
BesselY(- n, z) = (- 1)^(n)* BesselY(n, z)

BesselY[- n, z] == (- 1)^(n)* BesselY[n, z]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
10.4#Ex3 H - n ( 1 ) ⁡ ( z ) = ( - 1 ) n ⁢ H n ( 1 ) ⁡ ( z ) Hankel-H-1-Bessel-third-kind 𝑛 𝑧 superscript 1 𝑛 Hankel-H-1-Bessel-third-kind 𝑛 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{-n}}\left(z\right)=(-1)^{n}{H^{(1)}_{n}}% \left(z\right)}}
\HankelH{1}{-n}@{z} = (-1)^{n}\HankelH{1}{n}@{z}		 

HankelH1(- n, z) = (- 1)^(n)* HankelH1(n, z)

HankelH1[- n, z] == (- 1)^(n)* HankelH1[n, z]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
10.4#Ex4 H - n ( 2 ) ⁡ ( z ) = ( - 1 ) n ⁢ H n ( 2 ) ⁡ ( z ) Hankel-H-2-Bessel-third-kind 𝑛 𝑧 superscript 1 𝑛 Hankel-H-2-Bessel-third-kind 𝑛 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{-n}}\left(z\right)=(-1)^{n}{H^{(2)}_{n}}% \left(z\right)}}
\HankelH{2}{-n}@{z} = (-1)^{n}\HankelH{2}{n}@{z}		 

HankelH2(- n, z) = (- 1)^(n)* HankelH2(n, z)

HankelH2[- n, z] == (- 1)^(n)* HankelH2[n, z]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
10.4#Ex5 H ν ( 1 ) ⁡ ( z ) = J ν ⁡ ( z ) + i ⁢ Y ν ⁡ ( z ) Hankel-H-1-Bessel-third-kind 𝜈 𝑧 Bessel-J 𝜈 𝑧 𝑖 Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=J_{\nu}\left(z\right% )+iY_{\nu}\left(z\right)}}
\HankelH{1}{\nu}@{z} = \BesselJ{\nu}@{z}+i\BesselY{\nu}@{z}		 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}} 
HankelH1(nu, z) = BesselJ(nu, z)+ I*BesselY(nu, z)

HankelH1[\[Nu], z] == BesselJ[\[Nu], z]+ I*BesselY[\[Nu], z]
Successful Successful - Successful [Tested: 70]
10.4#Ex6 H ν ( 2 ) ⁡ ( z ) = J ν ⁡ ( z ) - i ⁢ Y ν ⁡ ( z ) Hankel-H-2-Bessel-third-kind 𝜈 𝑧 Bessel-J 𝜈 𝑧 𝑖 Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=J_{\nu}\left(z\right% )-iY_{\nu}\left(z\right)}}
\HankelH{2}{\nu}@{z} = \BesselJ{\nu}@{z}-i\BesselY{\nu}@{z}		 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}} 
HankelH2(nu, z) = BesselJ(nu, z)- I*BesselY(nu, z)

HankelH2[\[Nu], z] == BesselJ[\[Nu], z]- I*BesselY[\[Nu], z]
Successful Successful - Successful [Tested: 70]
10.4#Ex7 J ν ⁡ ( z ) = 1 2 ⁢ ( H ν ( 1 ) ⁡ ( z ) + H ν ( 2 ) ⁡ ( z ) ) Bessel-J 𝜈 𝑧 1 2 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{1}{2}\left({H^{(1)}_{% \nu}}\left(z\right)+{H^{(2)}_{\nu}}\left(z\right)\right)}}
\BesselJ{\nu}@{z} = \frac{1}{2}\left(\HankelH{1}{\nu}@{z}+\HankelH{2}{\nu}@{z}\right)		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselJ(nu, z) = (1)/(2)*(HankelH1(nu, z)+ HankelH2(nu, z))

BesselJ[\[Nu], z] == Divide[1,2]*(HankelH1[\[Nu], z]+ HankelH2[\[Nu], z])
Successful Successful - Successful [Tested: 70]
10.4#Ex8 Y ν ⁡ ( z ) = 1 2 ⁢ i ⁢ ( H ν ( 1 ) ⁡ ( z ) - H ν ( 2 ) ⁡ ( z ) ) Bessel-Y-Weber 𝜈 𝑧 1 2 𝑖 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{1}{2i}\left({H^{(1)}_% {\nu}}\left(z\right)-{H^{(2)}_{\nu}}\left(z\right)\right)}}
\BesselY{\nu}@{z} = \frac{1}{2i}\left(\HankelH{1}{\nu}@{z}-\HankelH{2}{\nu}@{z}\right)		 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}} 
BesselY(nu, z) = (1)/(2*I)*(HankelH1(nu, z)- HankelH2(nu, z))

BesselY[\[Nu], z] == Divide[1,2*I]*(HankelH1[\[Nu], z]- HankelH2[\[Nu], z])
Successful Successful - Successful [Tested: 70]
10.4.E5 J ν ⁡ ( z ) = csc ⁡ ( ν ⁢ π ) ⁢ ( Y - ν ⁡ ( z ) - Y ν ⁡ ( z ) ⁢ cos ⁡ ( ν ⁢ π ) ) Bessel-J 𝜈 𝑧 𝜈 𝜋 Bessel-Y-Weber 𝜈 𝑧 Bessel-Y-Weber 𝜈 𝑧 𝜈 𝜋 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\csc\left(\nu\pi\right)% \left(Y_{-\nu}\left(z\right)-Y_{\nu}\left(z\right)\cos\left(\nu\pi\right)% \right)}}
\BesselJ{\nu}@{z} = \csc@{\nu\pi}\left(\BesselY{-\nu}@{z}-\BesselY{\nu}@{z}\cos@{\nu\pi}\right)		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( - ν ) ) + k + 1 ) > 0 formulae-sequence 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,\Re((-(-\nu))+k+1% )>0}} 
BesselJ(nu, z) = csc(nu*Pi)*(BesselY(- nu, z)- BesselY(nu, z)*cos(nu*Pi))

BesselJ[\[Nu], z] == Csc[\[Nu]*Pi]*(BesselY[- \[Nu], z]- BesselY[\[Nu], z]*Cos[\[Nu]*Pi])
Successful Successful -
Failed [14 / 70]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]}


Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]}

... skip entries to safe data
10.4#Ex9 H - ν ( 1 ) ⁡ ( z ) = e ν ⁢ π ⁢ i ⁢ H ν ( 1 ) ⁡ ( z ) Hankel-H-1-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{-\nu}}\left(z\right)=e^{\nu\pi i}{H^{(1)% }_{\nu}}\left(z\right)}}
\HankelH{1}{-\nu}@{z} = e^{\nu\pi i}\HankelH{1}{\nu}@{z}		 

HankelH1(- nu, z) = exp(nu*Pi*I)*HankelH1(nu, z)

HankelH1[- \[Nu], z] == Exp[\[Nu]*Pi*I]*HankelH1[\[Nu], z]
Successful Failure - Successful [Tested: 70]
10.4#Ex10 H - ν ( 2 ) ⁡ ( z ) = e - ν ⁢ π ⁢ i ⁢ H ν ( 2 ) ⁡ ( z ) Hankel-H-2-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{-\nu}}\left(z\right)=e^{-\nu\pi i}{H^{(2% )}_{\nu}}\left(z\right)}}
\HankelH{2}{-\nu}@{z} = e^{-\nu\pi i}\HankelH{2}{\nu}@{z}		 

HankelH2(- nu, z) = exp(- nu*Pi*I)*HankelH2(nu, z)

HankelH2[- \[Nu], z] == Exp[- \[Nu]*Pi*I]*HankelH2[\[Nu], z]
Successful Failure - Successful [Tested: 70]
10.4.E7 H ν ( 1 ) ⁡ ( z ) = i ⁢ csc ⁡ ( ν ⁢ π ) ⁢ ( e - ν ⁢ π ⁢ i ⁢ J ν ⁡ ( z ) - J - ν ⁡ ( z ) ) Hankel-H-1-Bessel-third-kind 𝜈 𝑧 𝑖 𝜈 𝜋 superscript 𝑒 𝜈 𝜋 𝑖 Bessel-J 𝜈 𝑧 Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=i\csc\left(\nu\pi% \right)\left(e^{-\nu\pi i}J_{\nu}\left(z\right)-J_{-\nu}\left(z\right)\right)}}
\HankelH{1}{\nu}@{z} = i\csc@{\nu\pi}\left(e^{-\nu\pi i}\BesselJ{\nu}@{z}-\BesselJ{-\nu}@{z}\right)		 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}} 
HankelH1(nu, z) = I*csc(nu*Pi)*(exp(- nu*Pi*I)*BesselJ(nu, z)- BesselJ(- nu, z))

HankelH1[\[Nu], z] == I*Csc[\[Nu]*Pi]*(Exp[- \[Nu]*Pi*I]*BesselJ[\[Nu], z]- BesselJ[- \[Nu], z])
Successful Successful -
Failed [14 / 70]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]}


Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]}

... skip entries to safe data
10.4.E7 i ⁢ csc ⁡ ( ν ⁢ π ) ⁢ ( e - ν ⁢ π ⁢ i ⁢ J ν ⁡ ( z ) - J - ν ⁡ ( z ) ) = csc ⁡ ( ν ⁢ π ) ⁢ ( Y - ν ⁡ ( z ) - e - ν ⁢ π ⁢ i ⁢ Y ν ⁡ ( z ) ) 𝑖 𝜈 𝜋 superscript 𝑒 𝜈 𝜋 𝑖 Bessel-J 𝜈 𝑧 Bessel-J 𝜈 𝑧 𝜈 𝜋 Bessel-Y-Weber 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle i\csc\left(\nu\pi\right)\left(e^{-\nu\pi i}J_{\nu% }\left(z\right)-J_{-\nu}\left(z\right)\right)=\csc\left(\nu\pi\right)\left(Y_{% -\nu}\left(z\right)-e^{-\nu\pi i}Y_{\nu}\left(z\right)\right)}}
i\csc@{\nu\pi}\left(e^{-\nu\pi i}\BesselJ{\nu}@{z}-\BesselJ{-\nu}@{z}\right) = \csc@{\nu\pi}\left(\BesselY{-\nu}@{z}-e^{-\nu\pi i}\BesselY{\nu}@{z}\right)		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( - ν ) ) + k + 1 ) > 0 formulae-sequence 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,\Re((-(-\nu))+k+1% )>0}} 
I*csc(nu*Pi)*(exp(- nu*Pi*I)*BesselJ(nu, z)- BesselJ(- nu, z)) = csc(nu*Pi)*(BesselY(- nu, z)- exp(- nu*Pi*I)*BesselY(nu, z))

I*Csc[\[Nu]*Pi]*(Exp[- \[Nu]*Pi*I]*BesselJ[\[Nu], z]- BesselJ[- \[Nu], z]) == Csc[\[Nu]*Pi]*(BesselY[- \[Nu], z]- Exp[- \[Nu]*Pi*I]*BesselY[\[Nu], z])
Successful Successful -
Failed [14 / 70]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]}


Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]}

... skip entries to safe data
10.4.E8 H ν ( 2 ) ⁡ ( z ) = i ⁢ csc ⁡ ( ν ⁢ π ) ⁢ ( J - ν ⁡ ( z ) - e ν ⁢ π ⁢ i ⁢ J ν ⁡ ( z ) ) Hankel-H-2-Bessel-third-kind 𝜈 𝑧 𝑖 𝜈 𝜋 Bessel-J 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=i\csc\left(\nu\pi% \right)\left(J_{-\nu}\left(z\right)-e^{\nu\pi i}J_{\nu}\left(z\right)\right)}}
\HankelH{2}{\nu}@{z} = i\csc@{\nu\pi}\left(\BesselJ{-\nu}@{z}-e^{\nu\pi i}\BesselJ{\nu}@{z}\right)		 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}} 
HankelH2(nu, z) = I*csc(nu*Pi)*(BesselJ(- nu, z)- exp(nu*Pi*I)*BesselJ(nu, z))

HankelH2[\[Nu], z] == I*Csc[\[Nu]*Pi]*(BesselJ[- \[Nu], z]- Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], z])
Successful Successful -
Failed [14 / 70]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]}


Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]}

... skip entries to safe data
10.4.E8 i ⁢ csc ⁡ ( ν ⁢ π ) ⁢ ( J - ν ⁡ ( z ) - e ν ⁢ π ⁢ i ⁢ J ν ⁡ ( z ) ) = csc ⁡ ( ν ⁢ π ) ⁢ ( Y - ν ⁡ ( z ) - e ν ⁢ π ⁢ i ⁢ Y ν ⁡ ( z ) ) 𝑖 𝜈 𝜋 Bessel-J 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 Bessel-J 𝜈 𝑧 𝜈 𝜋 Bessel-Y-Weber 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle i\csc\left(\nu\pi\right)\left(J_{-\nu}\left(z% \right)-e^{\nu\pi i}J_{\nu}\left(z\right)\right)=\csc\left(\nu\pi\right)\left(% Y_{-\nu}\left(z\right)-e^{\nu\pi i}Y_{\nu}\left(z\right)\right)}}
i\csc@{\nu\pi}\left(\BesselJ{-\nu}@{z}-e^{\nu\pi i}\BesselJ{\nu}@{z}\right) = \csc@{\nu\pi}\left(\BesselY{-\nu}@{z}-e^{\nu\pi i}\BesselY{\nu}@{z}\right)		 ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ( - ν ) ) + k + 1 ) > 0 formulae-sequence 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,\Re((-(-\nu))+k+1% )>0}} 
I*csc(nu*Pi)*(BesselJ(- nu, z)- exp(nu*Pi*I)*BesselJ(nu, z)) = csc(nu*Pi)*(BesselY(- nu, z)- exp(nu*Pi*I)*BesselY(nu, z))

I*Csc[\[Nu]*Pi]*(BesselJ[- \[Nu], z]- Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], z]) == Csc[\[Nu]*Pi]*(BesselY[- \[Nu], z]- Exp[\[Nu]*Pi*I]*BesselY[\[Nu], z])
Successful Successful -
Failed [14 / 70]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]}


Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]}

... skip entries to safe data
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