Bessel Functions - 10.25 Definitions

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.25.E1	${\displaystyle{\displaystyle z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+z\frac{\mathrm{d}w}{\mathrm{d}z}-(z^{2}+\nu^{2})w=0}}$ z^{2}\deriv[2]{w}{z}+z\deriv{w}{z}-(z^{2}+\nu^{2})w = 0		(z)^(2)* diff(w, [z$(2)])+ z*diff(w, z)-((z)^(2)+ (nu)^(2))*w = 0	(z)^(2)* D[w, {z, 2}]+ z*D[w, z]-((z)^(2)+ \[Nu]^(2))*w == 0	Failure	Failure	Failed [220 / 300] Result: -.6467477718e-9-2.000000002*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: -.8660254040e-9-2.000000001*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [264 / 300] Result: Complex[0.0, -2.0] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0, -2.0] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
10.25.E2	${\displaystyle{\displaystyle I_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!\Gamma\left(\nu+k+1\right)}}}$ \modBesselI{\nu}@{z} = (\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!\EulerGamma@{\nu+k+1}}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	BesselI(nu, z) = ((1)/(2)*z)^(nu)* sum((((1)/(4)*(z)^(2))^(k))/(factorial(k)*GAMMA(nu + k + 1)), k = 0..infinity)	BesselI[\[Nu], z] == (Divide[1,2]*z)^\[Nu]* Sum[Divide[(Divide[1,4]*(z)^(2))^(k),(k)!*Gamma[\[Nu]+ k + 1]], {k, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 70]