Bessel Functions - 10.45 Functions of Imaginary Order

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.45.E1	${\displaystyle{\displaystyle x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+x\frac{\mathrm{d}w}{\mathrm{d}x}+(\nu^{2}-x^{2})w=0}}$ x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(\nu^{2}-x^{2})w = 0		(x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)+((nu)^(2)- (x)^(2))*w = 0	(x)^(2)* D[w, {x, 2}]+ x*D[w, x]+(\[Nu]^(2)- (x)^(2))*w == 0	Failure	Failure	Failed [240 / 300] Result: -1.948557159-.1249999996*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: -.2165063507+.8750000006*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 [240 / 300] Result: Complex[-1.948557158514987, -0.12499999999999989] 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[-1.9485571585149875, -2.125] 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
10.45.E2	${\displaystyle{\displaystyle\displaystyle\widetilde{I}_{\nu}\left(x\right)=\Re\left(I_{i\nu}\left(x\right)\right)}}$ \displaystyle\modBesselIimag{\nu}@{x} = \realpart@{\modBesselI{i\nu}@{x}}	${\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0}}$	Re(BesselI(I*(nu), x)) = Re(BesselI(I*nu, x))	Re[BesselI[I*\[Nu], x]] == Re[BesselI[I*\[Nu], x]]	Successful	Successful	-	Successful [Tested: 30]
10.45.E2	${\displaystyle{\displaystyle\displaystyle\widetilde{K}_{\nu}\left(x\right)=K_{i\nu}\left(x\right)}}$ \displaystyle\modBesselKimag{\nu}@{x} = \modBesselK{i\nu}@{x}		BesselK(I*(nu), x) = BesselK(I*nu, x)	BesselK[I*\[Nu], x] == BesselK[I*\[Nu], x]	Successful	Successful	-	Successful [Tested: 30]
10.45.E3	${\displaystyle{\displaystyle\displaystyle\widetilde{I}_{-\nu}\left(x\right)=\widetilde{I}_{\nu}\left(x\right)}}$ \displaystyle\modBesselIimag{-\nu}@{x} = \modBesselIimag{\nu}@{x}		Re(BesselI(I*(- nu), x)) = Re(BesselI(I*(nu), x))	Re[BesselI[I*- \[Nu], x]] == Re[BesselI[I*\[Nu], x]]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.45.E3	${\displaystyle{\displaystyle\displaystyle\widetilde{K}_{-\nu}\left(x\right)=\widetilde{K}_{\nu}\left(x\right)}}$ \displaystyle\modBesselKimag{-\nu}@{x} = \modBesselKimag{\nu}@{x}		BesselK(I*(- nu), x) = BesselK(I*(nu), x)	BesselK[I*- \[Nu], x] == BesselK[I*\[Nu], x]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.45.E4	${\displaystyle{\displaystyle\mathscr{W}\left\{\widetilde{K}_{\nu}\left(x\right),\widetilde{I}_{\nu}\left(x\right)\right\}=1/x}}$ \Wronskian@{\modBesselKimag{\nu}@{x},\modBesselIimag{\nu}@{x}} = 1/x		(BesselK(I*(nu), x))*diff(Re(BesselI(I*(nu), x)), x)-diff(BesselK(I*(nu), x), x)*(Re(BesselI(I*(nu), x))) = 1/x	Wronskian[{BesselK[I*\[Nu], x], Re[BesselI[I*\[Nu], x]]}, x] == 1/x	Failure	Failure	Error	Failed [30 / 30] Result: Plus[-0.6666666666666666, Times[0.5, Plus[Complex[1.0700115379721733, -0.3754447148158467], Times[Complex[0.1636629185333998, 0.09141848176750039], Derivative[1][Re][Complex[2.445786867824693, 0.6492150843755028]]]]]] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[-0.6666666666666666, Times[0.5, Plus[Complex[0.8415452902387464, 0.2726729041814867], Times[Complex[0.3412924192180222, 0.19179892830603273], Derivative[1][Re][Complex[1.3137906770541619, -0.7251169608509622]]]]]] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.45.E8	${\displaystyle{\displaystyle\widetilde{K}_{0}\left(x\right)=K_{0}\left(x\right)}}$ \modBesselKimag{0}@{x} = \modBesselK{0}@{x}		BesselK(I*(0), x) = BesselK(0, x)	BesselK[I*0, x] == BesselK[0, x]	Successful	Successful	-	Successful [Tested: 3]