Bessel Functions - 10.11 Analytic Continuation

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.11.E1 J ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = e m ⁢ ν ⁢ π ⁢ i ⁢ J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 𝑒 𝑚 𝜈 𝜋 𝑖 Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\nu}\left(ze^{m\pi i}\right)=e^{m\nu\pi i}J_{% \nu}\left(z\right)}}
\BesselJ{\nu}@{ze^{m\pi i}} = e^{m\nu\pi i}\BesselJ{\nu}@{z}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselJ(nu, z*exp(m*Pi*I)) = exp(m*nu*Pi*I)*BesselJ(nu, z)

BesselJ[\[Nu], z*Exp[m*Pi*I]] == Exp[m*\[Nu]*Pi*I]*BesselJ[\[Nu], z]
Failure Failure
Failed [132 / 210]
Result: -1.978604450-.5916012221*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}


Result: .4256613630-.5580360922e-1*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2}

... skip entries to safe data
Failed [120 / 210]
Result: Complex[-1.9786044502778974, -0.5916012230349773]
Test Values: {Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.42566136315461117, -0.05580360945599949]
Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.11.E2 Y ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = e - m ⁢ ν ⁢ π ⁢ i ⁢ Y ν ⁡ ( z ) + 2 ⁢ i ⁢ sin ⁡ ( m ⁢ ν ⁢ π ) ⁢ cot ⁡ ( ν ⁢ π ) ⁢ J ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 𝑒 𝑚 𝜈 𝜋 𝑖 Bessel-Y-Weber 𝜈 𝑧 2 𝑖 𝑚 𝜈 𝜋 𝜈 𝜋 Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\nu}\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}Y_{% \nu}\left(z\right)+2i\sin\left(m\nu\pi\right)\cot\left(\nu\pi\right)J_{\nu}% \left(z\right)}}
\BesselY{\nu}@{ze^{m\pi i}} = e^{-m\nu\pi i}\BesselY{\nu}@{z}+2i\sin@{m\nu\pi}\cot@{\nu\pi}\BesselJ{\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}} 
BesselY(nu, z*exp(m*Pi*I)) = exp(- m*nu*Pi*I)*BesselY(nu, z)+ 2*I*sin(m*nu*Pi)*cot(nu*Pi)*BesselJ(nu, z)

BesselY[\[Nu], z*Exp[m*Pi*I]] == Exp[- m*\[Nu]*Pi*I]*BesselY[\[Nu], z]+ 2*I*Sin[m*\[Nu]*Pi]*Cot[\[Nu]*Pi]*BesselJ[\[Nu], z]
Failure Failure
Failed [170 / 210]
Result: -4.492502702+3.271310776*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}


Result: 19.72399963+2.416868418*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2}

... skip entries to safe data
Failed [162 / 210]
Result: Complex[-4.49250270148862, 3.2713107749000305]
Test Values: {Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[19.723999620348792, 2.416868461226219]
Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.11.E3 sin ⁡ ( ν ⁢ π ) ⁢ H ν ( 1 ) ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = - sin ⁡ ( ( m - 1 ) ⁢ ν ⁢ π ) ⁢ H ν ( 1 ) ⁡ ( z ) - e - ν ⁢ π ⁢ i ⁢ sin ⁡ ( m ⁢ ν ⁢ π ) ⁢ H ν ( 2 ) ⁡ ( z ) 𝜈 𝜋 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 𝑚 1 𝜈 𝜋 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 𝑚 𝜈 𝜋 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle\sin\left(\nu\pi\right){H^{(1)}_{\nu}}\left(ze^{m% \pi i}\right)=-\sin\left((m-1)\nu\pi\right){H^{(1)}_{\nu}}\left(z\right)-e^{-% \nu\pi i}\sin\left(m\nu\pi\right){H^{(2)}_{\nu}}\left(z\right)}}
\sin@{\nu\pi}\HankelH{1}{\nu}@{ze^{m\pi i}} = -\sin@{(m-1)\nu\pi}\HankelH{1}{\nu}@{z}-e^{-\nu\pi i}\sin@{m\nu\pi}\HankelH{2}{\nu}@{z}		 

sin(nu*Pi)*HankelH1(nu, z*exp(m*Pi*I)) = - sin((m - 1)*nu*Pi)*HankelH1(nu, z)- exp(- nu*Pi*I)*sin(m*nu*Pi)*HankelH2(nu, z)

Sin[\[Nu]*Pi]*HankelH1[\[Nu], z*Exp[m*Pi*I]] == - Sin[(m - 1)*\[Nu]*Pi]*HankelH1[\[Nu], z]- Exp[- \[Nu]*Pi*I]*Sin[m*\[Nu]*Pi]*HankelH2[\[Nu], z]
Failure Failure
Failed [132 / 210]
Result: -16.06107638+5.815014709*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}


Result: 39.27071892+24.34608468*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2}

... skip entries to safe data
Failed [120 / 210]
Result: Complex[-16.061076381218605, 5.815014694873561]
Test Values: {Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[39.27071883811536, 24.346084784539414]
Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.11.E4 sin ⁡ ( ν ⁢ π ) ⁢ H ν ( 2 ) ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = e ν ⁢ π ⁢ i ⁢ sin ⁡ ( m ⁢ ν ⁢ π ) ⁢ H ν ( 1 ) ⁡ ( z ) + sin ⁡ ( ( m + 1 ) ⁢ ν ⁢ π ) ⁢ H ν ( 2 ) ⁡ ( z ) 𝜈 𝜋 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 𝑒 𝜈 𝜋 𝑖 𝑚 𝜈 𝜋 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 𝑚 1 𝜈 𝜋 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle\sin\left(\nu\pi\right){H^{(2)}_{\nu}}\left(ze^{m% \pi i}\right)=e^{\nu\pi i}\sin\left(m\nu\pi\right){H^{(1)}_{\nu}}\left(z\right% )+\sin\left((m+1)\nu\pi\right){H^{(2)}_{\nu}}\left(z\right)}}
\sin@{\nu\pi}\HankelH{2}{\nu}@{ze^{m\pi i}} = e^{\nu\pi i}\sin@{m\nu\pi}\HankelH{1}{\nu}@{z}+\sin@{(m+1)\nu\pi}\HankelH{2}{\nu}@{z}		 

sin(nu*Pi)*HankelH2(nu, z*exp(m*Pi*I)) = exp(nu*Pi*I)*sin(m*nu*Pi)*HankelH1(nu, z)+ sin((m + 1)*nu*Pi)*HankelH2(nu, z)

Sin[\[Nu]*Pi]*HankelH2[\[Nu], z*Exp[m*Pi*I]] == Exp[\[Nu]*Pi*I]*Sin[m*\[Nu]*Pi]*HankelH1[\[Nu], z]+ Sin[(m + 1)*\[Nu]*Pi]*HankelH2[\[Nu], z]
Failure Failure
Failed [132 / 210]
Result: 9.518923666+1.283901315*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}


Result: -38.63237633-26.24866521*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2}

... skip entries to safe data
Failed [120 / 210]
Result: Complex[9.518923662743454, 1.2839013369012835]
Test Values: {Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-38.63237622058036, -26.24866530437453]
Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.11#Ex1 H ν ( 1 ) ⁡ ( z ⁢ e π ⁢ i ) = - e - ν ⁢ π ⁢ i ⁢ H ν ( 2 ) ⁡ ( z ) Hankel-H-1-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 superscript 𝑒 𝜈 𝜋 𝑖 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(ze^{\pi i}\right)=-e^{-\nu\pi i% }{H^{(2)}_{\nu}}\left(z\right)}}
\HankelH{1}{\nu}@{ze^{\pi i}} = -e^{-\nu\pi i}\HankelH{2}{\nu}@{z}		 

HankelH1(nu, z*exp(Pi*I)) = - exp(- nu*Pi*I)*HankelH2(nu, z)

HankelH1[\[Nu], z*Exp[Pi*I]] == - Exp[- \[Nu]*Pi*I]*HankelH2[\[Nu], z]
Failure Failure
Failed [20 / 70]
Result: -5.249915228-5.084103922*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -3.129030441-5.176244122*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [20 / 70]
Result: Complex[-5.2499152251779275, -5.084103924523598]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.4609763579335797, 35.01102127779514]
Test Values: {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
10.11#Ex2 H ν ( 2 ) ⁡ ( z ⁢ e - π ⁢ i ) = - e ν ⁢ π ⁢ i ⁢ H ν ( 1 ) ⁡ ( z ) Hankel-H-2-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 superscript 𝑒 𝜈 𝜋 𝑖 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(ze^{-\pi i}\right)=-e^{\nu\pi i% }{H^{(1)}_{\nu}}\left(z\right)}}
\HankelH{2}{\nu}@{ze^{-\pi i}} = -e^{\nu\pi i}\HankelH{1}{\nu}@{z}		 

HankelH2(nu, z*exp(- Pi*I)) = - exp(nu*Pi*I)*HankelH1(nu, z)

HankelH2[\[Nu], z*Exp[- Pi*I]] == - Exp[\[Nu]*Pi*I]*HankelH1[\[Nu], z]
Failure Failure
Failed [50 / 70]
Result: 1.033334476+.7163604616*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}


Result: 1.427918302+.5187414665*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [50 / 70]
Result: Complex[1.0333344760783634, 0.7163604618419928]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.538721989873022, -0.29666827540401164]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.11.E6 Y n ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = ( - 1 ) m ⁢ n ⁢ ( Y n ⁡ ( z ) + 2 ⁢ i ⁢ m ⁢ J n ⁡ ( z ) ) Bessel-Y-Weber 𝑛 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 1 𝑚 𝑛 Bessel-Y-Weber 𝑛 𝑧 2 𝑖 𝑚 Bessel-J 𝑛 𝑧 {\displaystyle{\displaystyle Y_{n}\left(ze^{m\pi i}\right)=(-1)^{mn}(Y_{n}% \left(z\right)+2imJ_{n}\left(z\right))}}
\BesselY{n}@{ze^{m\pi i}} = (-1)^{mn}(\BesselY{n}@{z}+2im\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}} 
BesselY(n, z*exp(m*Pi*I)) = (- 1)^(m*n)*(BesselY(n, z)+ 2*I*m*BesselJ(n, z))

BesselY[n, z*Exp[m*Pi*I]] == (- 1)^(m*n)*(BesselY[n, z]+ 2*I*m*BesselJ[n, z])
Failure Failure
Failed [57 / 63]
Result: -.7553141392+1.723217630*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, n = 1}


Result: .3969469092-.2695422112*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, n = 2}

... skip entries to safe data
Failed [48 / 63]
Result: Complex[-0.7553141389736522, 1.7232176296930342]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.39694690825884216, -0.26954221211204654]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.11.E7 H n ( 1 ) ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = ( - 1 ) m ⁢ n - 1 ⁢ ( ( m - 1 ) ⁢ H n ( 1 ) ⁡ ( z ) + m ⁢ H n ( 2 ) ⁡ ( z ) ) Hankel-H-1-Bessel-third-kind 𝑛 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 1 𝑚 𝑛 1 𝑚 1 Hankel-H-1-Bessel-third-kind 𝑛 𝑧 𝑚 Hankel-H-2-Bessel-third-kind 𝑛 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{n}}\left(ze^{m\pi i}\right)=(-1)^{mn-1}(% (m-1){H^{(1)}_{n}}\left(z\right)+m{H^{(2)}_{n}}\left(z\right))}}
\HankelH{1}{n}@{ze^{m\pi i}} = (-1)^{mn-1}((m-1)\HankelH{1}{n}@{z}+m\HankelH{2}{n}@{z})		 

HankelH1(n, z*exp(m*Pi*I)) = (- 1)^(m*n - 1)*((m - 1)*HankelH1(n, z)+ m*HankelH2(n, z))

HankelH1[n, z*Exp[m*Pi*I]] == (- 1)^(m*n - 1)*((m - 1)*HankelH1[n, z]+ m*HankelH2[n, z])
Failure Failure
Failed [57 / 63]
Result: -1.723217630-.7553141394*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, n = 1}


Result: .2695422111+.3969469092*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, n = 2}

... skip entries to safe data
Failed [48 / 63]
Result: Complex[-1.7232176296930342, -0.7553141389736522]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.26954221211204654, 0.39694690825884216]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.11.E8 H n ( 2 ) ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = ( - 1 ) m ⁢ n ⁢ ( m ⁢ H n ( 1 ) ⁡ ( z ) + ( m + 1 ) ⁢ H n ( 2 ) ⁡ ( z ) ) Hankel-H-2-Bessel-third-kind 𝑛 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 1 𝑚 𝑛 𝑚 Hankel-H-1-Bessel-third-kind 𝑛 𝑧 𝑚 1 Hankel-H-2-Bessel-third-kind 𝑛 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{n}}\left(ze^{m\pi i}\right)=(-1)^{mn}(m{% H^{(1)}_{n}}\left(z\right)+(m+1){H^{(2)}_{n}}\left(z\right))}}
\HankelH{2}{n}@{ze^{m\pi i}} = (-1)^{mn}(m\HankelH{1}{n}@{z}+(m+1)\HankelH{2}{n}@{z})		 

HankelH2(n, z*exp(m*Pi*I)) = (- 1)^(m*n)*(m*HankelH1(n, z)+(m + 1)*HankelH2(n, z))

HankelH2[n, z*Exp[m*Pi*I]] == (- 1)^(m*n)*(m*HankelH1[n, z]+(m + 1)*HankelH2[n, z])
Failure Failure
Failed [57 / 63]
Result: 1.723217630+.755314139*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, n = 1}


Result: -.269542211-.396946909*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, n = 2}

... skip entries to safe data
Failed [48 / 63]
Result: Complex[1.7232176296930342, 0.7553141389736524]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.26954221211204654, -0.39694690825884216]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.11#E9X J ν ⁡ ( z ¯ ) = J ν ⁡ ( z ) ¯ Bessel-J 𝜈 𝑧 Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle\displaystyle J_{\nu}\left(\overline{z}\right)=% \overline{J_{\nu}\left(z\right)}}}
\displaystyle\BesselJ{\nu}@{\conj{z}} = \conj{\BesselJ{\nu}@{z}}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselJ(nu, conjugate(z)) = conjugate(BesselJ(nu, z))
 
BesselJ[\[Nu], Conjugate[z]] == Conjugate[BesselJ[\[Nu], z]]
 Skipped - no semantic math Skipped - no semantic math - -
10.11#E9X Y ν ⁡ ( z ¯ ) = Y ν ⁡ ( z ) ¯ Bessel-Y-Weber 𝜈 𝑧 Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle\displaystyle Y_{\nu}\left(\overline{z}\right)=% \overline{Y_{\nu}\left(z\right)}}}
\displaystyle\BesselY{\nu}@{\conj{z}} = \conj{\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}} 
BesselY(nu, conjugate(z)) = conjugate(BesselY(nu, z))
 
BesselY[\[Nu], Conjugate[z]] == Conjugate[BesselY[\[Nu], z]]
 Skipped - no semantic math Skipped - no semantic math - -
10.11#E9Xa H ν ( 1 ) ⁡ ( z ¯ ) = H ν ( 2 ) ⁡ ( z ) ¯ Hankel-H-1-Bessel-third-kind 𝜈 𝑧 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle\displaystyle{H^{(1)}_{\nu}}\left(\overline{z}% \right)=\overline{{H^{(2)}_{\nu}}\left(z\right)}}}
\displaystyle\HankelH{1}{\nu}@{\conj{z}} = \conj{\HankelH{2}{\nu}@{z}}		 

HankelH1(nu, conjugate(z)) = conjugate(HankelH2(nu, z))
 
HankelH1[\[Nu], Conjugate[z]] == Conjugate[HankelH2[\[Nu], z]]
 Skipped - no semantic math Skipped - no semantic math - -
10.11#E9Xa H ν ( 2 ) ⁡ ( z ¯ ) = H ν ( 1 ) ⁡ ( z ) ¯ Hankel-H-2-Bessel-third-kind 𝜈 𝑧 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle\displaystyle{H^{(2)}_{\nu}}\left(\overline{z}% \right)=\overline{{H^{(1)}_{\nu}}\left(z\right)}}}
\displaystyle\HankelH{2}{\nu}@{\conj{z}} = \conj{\HankelH{1}{\nu}@{z}}		 

HankelH2(nu, conjugate(z)) = conjugate(HankelH1(nu, z))
 
HankelH2[\[Nu], Conjugate[z]] == Conjugate[HankelH1[\[Nu], z]]
 Skipped - no semantic math Skipped - no semantic math - -
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