Bessel Functions - 10.34 Analytic Continuation

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

BesselI[\[Nu], z*Exp[m*Pi*I]] == Exp[m*\[Nu]*Pi*I]*BesselI[\[Nu], z]
Failure Failure
Failed [132 / 210]
Result: -2.206479866-1.131319388*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}


Result: .5147384726+.2724622562e-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[-2.206479866313521, -1.1313193889480602]
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[0.5147384728800724, 0.02724622519878004]
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
10.34.E2 K ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = e - m ⁢ ν ⁢ π ⁢ i ⁢ K ν ⁡ ( z ) - π ⁢ i ⁢ sin ⁡ ( m ⁢ ν ⁢ π ) ⁢ csc ⁡ ( ν ⁢ π ) ⁢ I ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 𝑒 𝑚 𝜈 𝜋 𝑖 modified-Bessel-second-kind 𝜈 𝑧 𝜋 𝑖 𝑚 𝜈 𝜋 𝜈 𝜋 modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\nu}\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}K_{% \nu}\left(z\right)-\pi i\sin\left(m\nu\pi\right)\csc\left(\nu\pi\right)I_{\nu}% \left(z\right)}}
\modBesselK{\nu}@{ze^{m\pi i}} = e^{-m\nu\pi i}\modBesselK{\nu}@{z}-\pi i\sin@{m\nu\pi}\csc@{\nu\pi}\modBesselI{\nu}@{z}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselK(nu, z*exp(m*Pi*I)) = exp(- m*nu*Pi*I)*BesselK(nu, z)- Pi*I*sin(m*nu*Pi)*csc(nu*Pi)*BesselI(nu, z)

BesselK[\[Nu], z*Exp[m*Pi*I]] == Exp[- m*\[Nu]*Pi*I]*BesselK[\[Nu], z]- Pi*I*Sin[m*\[Nu]*Pi]*Csc[\[Nu]*Pi]*BesselI[\[Nu], z]
Failure Failure
Failed [170 / 210]
Result: 2.965939338+3.157233720*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}


Result: -10.37113928-12.75980866*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[2.965939340334436, 3.157233721966529]
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[-10.371139260352992, -12.75980869099896]
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
10.34.E3 I ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = ( i / π ) ⁢ ( + e m ⁢ ν ⁢ π ⁢ i ⁢ K ν ⁡ ( z ⁢ e + π ⁢ i ) - e ( m - 1 ) ⁢ ν ⁢ π ⁢ i ⁢ K ν ⁡ ( z ) ) modified-Bessel-first-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 𝑖 𝜋 superscript 𝑒 𝑚 𝜈 𝜋 𝑖 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 superscript 𝑒 𝑚 1 𝜈 𝜋 𝑖 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(ze^{m\pi i}\right)=(i/\pi)\left(+e^{% m\nu\pi i}K_{\nu}\left(ze^{+\pi i}\right)-e^{(m-1)\nu\pi i}K_{\nu}\left(z% \right)\right)}}
\modBesselI{\nu}@{ze^{m\pi i}} = (i/\pi)\left(+ e^{m\nu\pi i}\modBesselK{\nu}@{ze^{+\pi i}}- e^{(m- 1)\nu\pi i}\modBesselK{\nu}@{z}\right)		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselI(nu, z*exp(m*Pi*I)) = (I/Pi)*(+ exp(m*nu*Pi*I)*BesselK(nu, z*exp(+ Pi*I))- exp((m - 1)*nu*Pi*I)*BesselK(nu, z))

BesselI[\[Nu], z*Exp[m*Pi*I]] == (I/Pi)*(+ Exp[m*\[Nu]*Pi*I]*BesselK[\[Nu], z*Exp[+ Pi*I]]- Exp[(m - 1)*\[Nu]*Pi*I]*BesselK[\[Nu], z])
Failure Failure
Failed [152 / 210]
Result: -2.316975457-.8668337446*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}


Result: .5132395470-.3232131754e-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 [140 / 210]
Result: Complex[-2.3169754573845194, -0.8668337451474188]
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[0.5132395471581521, -0.03232131806579792]
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
10.34.E3 I ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = ( i / π ) ⁢ ( - e m ⁢ ν ⁢ π ⁢ i ⁢ K ν ⁡ ( z ⁢ e - π ⁢ i ) + e ( m + 1 ) ⁢ ν ⁢ π ⁢ i ⁢ K ν ⁡ ( z ) ) modified-Bessel-first-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 𝑖 𝜋 superscript 𝑒 𝑚 𝜈 𝜋 𝑖 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 superscript 𝑒 𝑚 1 𝜈 𝜋 𝑖 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(ze^{m\pi i}\right)=(i/\pi)\left(-e^{% m\nu\pi i}K_{\nu}\left(ze^{-\pi i}\right)+e^{(m+1)\nu\pi i}K_{\nu}\left(z% \right)\right)}}
\modBesselI{\nu}@{ze^{m\pi i}} = (i/\pi)\left(- e^{m\nu\pi i}\modBesselK{\nu}@{ze^{-\pi i}}+ e^{(m+ 1)\nu\pi i}\modBesselK{\nu}@{z}\right)		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselI(nu, z*exp(m*Pi*I)) = (I/Pi)*(- exp(m*nu*Pi*I)*BesselK(nu, z*exp(- Pi*I))+ exp((m + 1)*nu*Pi*I)*BesselK(nu, z))

BesselI[\[Nu], z*Exp[m*Pi*I]] == (I/Pi)*(- Exp[m*\[Nu]*Pi*I]*BesselK[\[Nu], z*Exp[- Pi*I]]+ Exp[(m + 1)*\[Nu]*Pi*I]*BesselK[\[Nu], z])
Failure Failure
Failed [190 / 210]
Result: -2.206479866-1.131319388*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}


Result: .5147384726+.2724622561e-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 [190 / 210]
Result: Complex[-2.206479866313521, -1.1313193889480602]
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[0.5147384728800724, 0.027246225198780036]
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
10.34.E4 K ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = csc ⁡ ( ν ⁢ π ) ⁢ ( + sin ⁡ ( m ⁢ ν ⁢ π ) ⁢ K ν ⁡ ( z ⁢ e + π ⁢ i ) - sin ⁡ ( ( m - 1 ) ⁢ ν ⁢ π ) ⁢ K ν ⁡ ( z ) ) modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 𝜈 𝜋 𝑚 𝜈 𝜋 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 𝑚 1 𝜈 𝜋 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\nu}\left(ze^{m\pi i}\right)=\csc\left(\nu\pi% \right)\left(+\sin\left(m\nu\pi\right)K_{\nu}\left(ze^{+\pi i}\right)-\sin% \left((m-1)\nu\pi\right)K_{\nu}\left(z\right)\right)}}
\modBesselK{\nu}@{ze^{m\pi i}} = \csc@{\nu\pi}\left(+\sin@{m\nu\pi}\modBesselK{\nu}@{ze^{+\pi i}}-\sin@{(m- 1)\nu\pi}\modBesselK{\nu}@{z}\right)		 

BesselK(nu, z*exp(m*Pi*I)) = csc(nu*Pi)*(+ sin(m*nu*Pi)*BesselK(nu, z*exp(+ Pi*I))- sin((m - 1)*nu*Pi)*BesselK(nu, z))

BesselK[\[Nu], z*Exp[m*Pi*I]] == Csc[\[Nu]*Pi]*(+ Sin[m*\[Nu]*Pi]*BesselK[\[Nu], z*Exp[+ Pi*I]]- Sin[(m - 1)*\[Nu]*Pi]*BesselK[\[Nu], z])
Failure Failure
Failed [158 / 210]
Result: -2.723238516+7.278993081*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2}


Result: 29.12762958-25.06220737*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 3}

... skip entries to safe data
Failed [154 / 210]
Result: Complex[-2.7232385256388585, 7.278993075467058]
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]]]}


Result: Complex[29.127629620508102, -25.062207299552764]
Test Values: {Rule[m, 3], 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
10.34.E4 K ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = csc ⁡ ( ν ⁢ π ) ⁢ ( - sin ⁡ ( m ⁢ ν ⁢ π ) ⁢ K ν ⁡ ( z ⁢ e - π ⁢ i ) + sin ⁡ ( ( m + 1 ) ⁢ ν ⁢ π ) ⁢ K ν ⁡ ( z ) ) modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 𝜈 𝜋 𝑚 𝜈 𝜋 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 𝑚 1 𝜈 𝜋 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\nu}\left(ze^{m\pi i}\right)=\csc\left(\nu\pi% \right)\left(-\sin\left(m\nu\pi\right)K_{\nu}\left(ze^{-\pi i}\right)+\sin% \left((m+1)\nu\pi\right)K_{\nu}\left(z\right)\right)}}
\modBesselK{\nu}@{ze^{m\pi i}} = \csc@{\nu\pi}\left(-\sin@{m\nu\pi}\modBesselK{\nu}@{ze^{-\pi i}}+\sin@{(m+ 1)\nu\pi}\modBesselK{\nu}@{z}\right)		 

BesselK(nu, z*exp(m*Pi*I)) = csc(nu*Pi)*(- sin(m*nu*Pi)*BesselK(nu, z*exp(- Pi*I))+ sin((m + 1)*nu*Pi)*BesselK(nu, z))

BesselK[\[Nu], z*Exp[m*Pi*I]] == Csc[\[Nu]*Pi]*(- Sin[m*\[Nu]*Pi]*BesselK[\[Nu], z*Exp[- Pi*I]]+ Sin[(m + 1)*\[Nu]*Pi]*BesselK[\[Nu], z])
Failure Failure
Failed [170 / 210]
Result: 2.965939338+3.157233717*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}


Result: -10.37113929-12.75980866*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 [182 / 210]
Result: Complex[2.9659393403344363, 3.1572337219665294]
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[-10.371139260352981, -12.759808690998973]
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
10.34.E5 K n ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = ( - 1 ) m ⁢ n ⁢ K n ⁡ ( z ) + ( - 1 ) n ⁢ ( m - 1 ) - 1 ⁢ m ⁢ π ⁢ i ⁢ I n ⁡ ( z ) modified-Bessel-second-kind 𝑛 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 1 𝑚 𝑛 modified-Bessel-second-kind 𝑛 𝑧 superscript 1 𝑛 𝑚 1 1 𝑚 𝜋 𝑖 modified-Bessel-first-kind 𝑛 𝑧 {\displaystyle{\displaystyle K_{n}\left(ze^{m\pi i}\right)=(-1)^{mn}K_{n}\left% (z\right)+(-1)^{n(m-1)-1}m\pi iI_{n}\left(z\right)}}
\modBesselK{n}@{ze^{m\pi i}} = (-1)^{mn}\modBesselK{n}@{z}+(-1)^{n(m-1)-1}m\pi i\modBesselI{n}@{z}		 ℜ ⁡ ( n + k + 1 ) > 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0}} 
BesselK(n, z*exp(m*Pi*I)) = (- 1)^(m*n)* BesselK(n, z)+(- 1)^(n*(m - 1)- 1)* m*Pi*I*BesselI(n, z)

BesselK[n, z*Exp[m*Pi*I]] == (- 1)^(m*n)* BesselK[n, z]+(- 1)^(n*(m - 1)- 1)* m*Pi*I*BesselI[n, z]
Failure Failure
Failed [57 / 63]
Result: -1.971501919+2.706233555*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, n = 1}


Result: -.7368261646+.3579119854*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[-1.9715019183470535, 2.7062335550125516]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.736826162742255, 0.3579119863626685]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.34.E6 K n ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = + ( - 1 ) n ⁢ ( m - 1 ) ⁢ m ⁢ K n ⁡ ( z ⁢ e + π ⁢ i ) - ( - 1 ) n ⁢ m ⁢ ( m - 1 ) ⁢ K n ⁡ ( z ) modified-Bessel-second-kind 𝑛 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 1 𝑛 𝑚 1 𝑚 modified-Bessel-second-kind 𝑛 𝑧 superscript 𝑒 𝜋 𝑖 superscript 1 𝑛 𝑚 𝑚 1 modified-Bessel-second-kind 𝑛 𝑧 {\displaystyle{\displaystyle K_{n}\left(ze^{m\pi i}\right)=+(-1)^{n(m-1)}mK_{n% }\left(ze^{+\pi i}\right)-(-1)^{nm}(m-1)K_{n}\left(z\right)}}
\modBesselK{n}@{ze^{m\pi i}} = +(-1)^{n(m-1)}m\modBesselK{n}@{ze^{+\pi i}}-(-1)^{nm}(m- 1)\modBesselK{n}@{z}		 

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

BesselK[n, z*Exp[m*Pi*I]] == +(- 1)^(n*(m - 1))* m*BesselK[n, z*Exp[+ Pi*I]]-(- 1)^(n*m)*(m - 1)*BesselK[n, z]
Failure Failure
Failed [51 / 63]
Result: -1.971501920+2.706233556*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 2, n = 1}


Result: .7368261602-.357911988*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 2, n = 2}

... skip entries to safe data
Failed [42 / 63]
Result: Complex[-1.9715019183470535, 2.7062335550125516]
Test Values: {Rule[m, 2], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.736826162742255, -0.3579119863626685]
Test Values: {Rule[m, 2], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.34.E6 K n ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = - ( - 1 ) n ⁢ ( m - 1 ) ⁢ m ⁢ K n ⁡ ( z ⁢ e - π ⁢ i ) + ( - 1 ) n ⁢ m ⁢ ( m + 1 ) ⁢ K n ⁡ ( z ) modified-Bessel-second-kind 𝑛 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 1 𝑛 𝑚 1 𝑚 modified-Bessel-second-kind 𝑛 𝑧 superscript 𝑒 𝜋 𝑖 superscript 1 𝑛 𝑚 𝑚 1 modified-Bessel-second-kind 𝑛 𝑧 {\displaystyle{\displaystyle K_{n}\left(ze^{m\pi i}\right)=-(-1)^{n(m-1)}mK_{n% }\left(ze^{-\pi i}\right)+(-1)^{nm}(m+1)K_{n}\left(z\right)}}
\modBesselK{n}@{ze^{m\pi i}} = -(-1)^{n(m-1)}m\modBesselK{n}@{ze^{-\pi i}}+(-1)^{nm}(m+ 1)\modBesselK{n}@{z}		 

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

BesselK[n, z*Exp[m*Pi*I]] == -(- 1)^(n*(m - 1))* m*BesselK[n, z*Exp[- Pi*I]]+(- 1)^(n*m)*(m + 1)*BesselK[n, z]
Failure Failure
Failed [54 / 63]
Result: -1.971501919+2.706233556*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, n = 1}


Result: -.7368261645+.357911985*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, n = 2}

... skip entries to safe data
Failed [63 / 63]
Result: Complex[-1.9715019183470535, 2.7062335550125516]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.736826162742255, 0.3579119863626685]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.34#Ex1 I ν ⁡ ( z ¯ ) = I ν ⁡ ( z ) ¯ modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(\overline{z}\right)=\overline{I_{\nu% }\left(z\right)}}}
\modBesselI{\nu}@{\conj{z}} = \conj{\modBesselI{\nu}@{z}}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselI(nu, conjugate(z)) = conjugate(BesselI(nu, z))

BesselI[\[Nu], Conjugate[z]] == Conjugate[BesselI[\[Nu], z]]
Failure Failure Skipped - Because timed out
Failed [28 / 70]
Result: Complex[-0.1457476573229447, -0.7449450592023206]
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[1.100244133383339, 1.2347828003590728]
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
10.34#Ex2 K ν ⁡ ( z ¯ ) = K ν ⁡ ( z ) ¯ modified-Bessel-second-kind 𝜈 𝑧 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\nu}\left(\overline{z}\right)=\overline{K_{\nu% }\left(z\right)}}}
\modBesselK{\nu}@{\conj{z}} = \conj{\modBesselK{\nu}@{z}}		 

BesselK(nu, conjugate(z)) = conjugate(BesselK(nu, z))

BesselK[\[Nu], Conjugate[z]] == Conjugate[BesselK[\[Nu], z]]
Failure Failure
Failed [28 / 70]
Result: -.3322466664+.1347267497*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: .8978926857-1.555608423*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 [28 / 70]
Result: Complex[-0.332246666369582, 0.13472674975137633]
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[0.23222824698313052, -0.12812607679285354]
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
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