Bessel Functions - 10.24 Functions of Imaginary Order

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.24.E1 x 2 ⁒ d 2 w d x 2 + x ⁒ d w d x + ( x 2 + Ξ½ 2 ) ⁒ w = 0 superscript π‘₯ 2 derivative 𝑀 π‘₯ 2 π‘₯ derivative 𝑀 π‘₯ superscript π‘₯ 2 superscript 𝜈 2 𝑀 0 {\displaystyle{\displaystyle x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+% x\frac{\mathrm{d}w}{\mathrm{d}x}+(x^{2}+\nu^{2})w=0}}
x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(x^{2}+\nu^{2})w = 0		 

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

(x)^(2)* D[w, {x, 2}]+ x*D[w, x]+((x)^(2)+ \[Nu]^(2))*w == 0
Failure Failure
Failed [300 / 300]
Result: 1.948557159+2.125000000*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: .2165063513+1.125000001*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[1.9485571585149875, 2.125]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.948557158514987, 0.12499999999999989]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24#Ex1 J ~ Ξ½ ⁑ ( x ) = sech ⁑ ( 1 2 ⁒ Ο€ ⁒ Ξ½ ) ⁒ β„œ ⁑ ( J i ⁒ Ξ½ ⁑ ( x ) ) Bessel-J-imaginary-order 𝜈 π‘₯ 1 2 πœ‹ 𝜈 Bessel-J 𝑖 𝜈 π‘₯ {\displaystyle{\displaystyle\widetilde{J}_{\nu}\left(x\right)=\operatorname{% sech}\left(\tfrac{1}{2}\pi\nu\right)\Re\left(J_{i\nu}\left(x\right)\right)}}
\BesselJimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselJ{i\nu}@{x}}		 β„œ ⁑ ( ( i ⁒ Ξ½ ) + k + 1 ) > 0 imaginary-unit 𝜈 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0}} 
sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselJ(I*nu, x))

Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselJ[I*\[Nu], x]]
Successful Successful - Successful [Tested: 30]
10.24#Ex2 Y ~ Ξ½ ⁑ ( x ) = sech ⁑ ( 1 2 ⁒ Ο€ ⁒ Ξ½ ) ⁒ β„œ ⁑ ( Y i ⁒ Ξ½ ⁑ ( x ) ) Bessel-Y-Weber-imaginary-order 𝜈 π‘₯ 1 2 πœ‹ 𝜈 Bessel-Y-Weber 𝑖 𝜈 π‘₯ {\displaystyle{\displaystyle\widetilde{Y}_{\nu}\left(x\right)=\operatorname{% sech}\left(\tfrac{1}{2}\pi\nu\right)\Re\left(Y_{i\nu}\left(x\right)\right)}}
\BesselYimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselY{i\nu}@{x}}		 β„œ ⁑ ( ( i ⁒ Ξ½ ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( i ⁒ Ξ½ ) ) + k + 1 ) > 0 formulae-sequence imaginary-unit 𝜈 π‘˜ 1 0 imaginary-unit 𝜈 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0,\Re((-(\mathrm{i}\nu))+% k+1)>0}} 
sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselY(I*nu, x))

Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselY[I*\[Nu], x]]
Successful Successful - Successful [Tested: 30]
10.24.E3 Ξ“ ⁑ ( 1 + i ⁒ Ξ½ ) = ( Ο€ ⁒ Ξ½ sinh ⁑ ( Ο€ ⁒ Ξ½ ) ) 1 2 ⁒ e i ⁒ Ξ³ Ξ½ Euler-Gamma 1 𝑖 𝜈 superscript πœ‹ 𝜈 πœ‹ 𝜈 1 2 superscript 𝑒 𝑖 subscript 𝛾 𝜈 {\displaystyle{\displaystyle\Gamma\left(1+i\nu\right)=\left(\frac{\pi\nu}{% \sinh\left(\pi\nu\right)}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}}}
\EulerGamma@{1+i\nu} = \left(\frac{\pi\nu}{\sinh@{\pi\nu}}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}		 β„œ ⁑ ( 1 + i ⁒ Ξ½ ) > 0 1 imaginary-unit 𝜈 0 {\displaystyle{\displaystyle\Re(1+\mathrm{i}\nu)>0}} 
GAMMA(1 + I*nu) = ((Pi*nu)/(sinh(Pi*nu)))^((1)/(2))* exp(I*gamma[nu])

Gamma[1 + I*\[Nu]] == (Divide[Pi*\[Nu],Sinh[Pi*\[Nu]]])^(Divide[1,2])* Exp[I*Subscript[\[Gamma], \[Nu]]]
Failure Failure
Failed [300 / 300]
Result: .131682196e-1-.6479738907*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = 1/2*3^(1/2)+1/2*I}


Result: .2393622021-.2867640040*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.013168219691258531, -0.6479738909120968]
Test Values: {Rule[Ξ³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ξ³, Ξ½], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.23936220222535412, -0.28676400411697583]
Test Values: {Rule[Ξ³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ξ³, Ξ½], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24#Ex3 J ~ - Ξ½ ⁑ ( x ) = J ~ Ξ½ ⁑ ( x ) Bessel-J-imaginary-order 𝜈 π‘₯ Bessel-J-imaginary-order 𝜈 π‘₯ {\displaystyle{\displaystyle\widetilde{J}_{-\nu}\left(x\right)=\widetilde{J}_{% \nu}\left(x\right)}}
\BesselJimag{-\nu}@{x} = \BesselJimag{\nu}@{x}		 

sech((1/2)*Pi*(- nu))*Re(BesselJ(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x))

Sech[1/2 Pi - \[Nu]] Re[BesselJ[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]]
Failure Failure
Failed [12 / 30]
Result: .1765981285-.1547836875*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -1.059084556+.9282601935*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-0.6353785354467336, 0.04153700144653363]
Test Values: {Rule[x, 1.5], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.2910880978413849, 0.681683596996288]
Test Values: {Rule[x, 1.5], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24#Ex4 Y ~ - Ξ½ ⁑ ( x ) = Y ~ Ξ½ ⁑ ( x ) Bessel-Y-Weber-imaginary-order 𝜈 π‘₯ Bessel-Y-Weber-imaginary-order 𝜈 π‘₯ {\displaystyle{\displaystyle\widetilde{Y}_{-\nu}\left(x\right)=\widetilde{Y}_{% \nu}\left(x\right)}}
\BesselYimag{-\nu}@{x} = \BesselYimag{\nu}@{x}		 β„œ ⁑ ( ( i ⁒ ( - Ξ½ ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( i ⁒ Ξ½ ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( i ⁒ ( - Ξ½ ) ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( i ⁒ Ξ½ ) ) + k + 1 ) > 0 formulae-sequence imaginary-unit 𝜈 π‘˜ 1 0 formulae-sequence imaginary-unit 𝜈 π‘˜ 1 0 formulae-sequence imaginary-unit 𝜈 π‘˜ 1 0 imaginary-unit 𝜈 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((\mathrm{i}(-\nu))+k+1)>0,\Re((\mathrm{i}\nu)+% k+1)>0,\Re((-(\mathrm{i}(-\nu)))+k+1)>0,\Re((-(\mathrm{i}\nu))+k+1)>0}} 
sech((1/2)*Pi*(- nu))*Re(BesselY(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))

Sech[1/2 Pi - \[Nu]] Re[BesselY[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]
Failure Failure
Failed [12 / 30]
Result: -.6730010946+.5898680353*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -.1980888923+.1736197856*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[0.16541121369118172, 0.7534126929509344]
Test Values: {Rule[x, 1.5], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.3242468905843751, -0.9796849117084342]
Test Values: {Rule[x, 1.5], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24.E5 𝒲 ⁑ { J ~ Ξ½ ⁑ ( x ) , Y ~ Ξ½ ⁑ ( x ) } = 2 / ( Ο€ ⁒ x ) Wronskian Bessel-J-imaginary-order 𝜈 π‘₯ Bessel-Y-Weber-imaginary-order 𝜈 π‘₯ 2 πœ‹ π‘₯ {\displaystyle{\displaystyle\mathscr{W}\left\{\widetilde{J}_{\nu}\left(x\right% ),\widetilde{Y}_{\nu}\left(x\right)\right\}=2/(\pi x)}}
\Wronskian@{\BesselJimag{\nu}@{x},\BesselYimag{\nu}@{x}} = 2/(\pi x)		 β„œ ⁑ ( ( i ⁒ Ξ½ ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( i ⁒ Ξ½ ) ) + k + 1 ) > 0 formulae-sequence imaginary-unit 𝜈 π‘˜ 1 0 imaginary-unit 𝜈 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0,\Re((-(\mathrm{i}\nu))+% k+1)>0}} 
(sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)))*diff(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)), x)-diff(sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)), x)*(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))) = 2/(Pi*x)

Wronskian[{Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]], Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]}, x] == 2/(Pi*x)
Failure Failure
Failed [12 / 30]
Result: -.3214564733-.7786157192*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -.6431025084-4.765445687*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [30 / 30]
Result: Plus[-0.4244131815783876, Times[Complex[0.017184424665049866, -0.12995814793225188], Plus[Times[Complex[5.94457417937745, -0.08806734388290616], Derivative[1][Re][Complex[0.5424102683642863, 1.3820413572565333]]], Times[Complex[0.04670634387761448, 2.0064149502593187], Derivative[1][Re][Complex[1.5013396639532606, -0.5145465005058608]]]]]]
Test Values: {Rule[x, 1.5], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[-0.4244131815783876, Times[Complex[-0.5062208144169521, 0.3689208146583662], Plus[Times[Complex[1.2690034139339206, -1.428145592425075], Derivative[1][Re][Complex[-0.5230512553281585, -0.7250724679588263]]], Times[Complex[0.9907135967899046, 0.5862869255257461], Derivative[1][Re][Complex[0.9118063408652576, -0.381897212811936]]]]]]
Test Values: {Rule[x, 1.5], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24.E9 Y ~ 0 ⁑ ( x ) = Y 0 ⁑ ( x ) Bessel-Y-Weber-imaginary-order 0 π‘₯ Bessel-Y-Weber 0 π‘₯ {\displaystyle{\displaystyle\widetilde{Y}_{0}\left(x\right)=Y_{0}\left(x\right% )}}
\BesselYimag{0}@{x} = \BesselY{0}@{x}		 β„œ ⁑ ( 0 + k + 1 ) > 0 , β„œ ⁑ ( ( - 0 ) + k + 1 ) > 0 , β„œ ⁑ ( ( i ⁒ 0 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( i ⁒ 0 ) ) + k + 1 ) > 0 formulae-sequence 0 π‘˜ 1 0 formulae-sequence 0 π‘˜ 1 0 formulae-sequence imaginary-unit 0 π‘˜ 1 0 imaginary-unit 0 π‘˜ 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((-0)+k+1)>0,\Re((\mathrm{i}0)+k+1% )>0,\Re((-(\mathrm{i}0))+k+1)>0}} 
sech((1/2)*Pi*(0))*Re(BesselY(I*(0), x)) = BesselY(0, x)

Sech[1/2 Pi 0] Re[BesselY[I 0, x]] == BesselY[0, x]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
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