Airy and Related Functions - 10.2 Definitions

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.2.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 [217 / 300] 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+1/2*I*3^(1/2)} 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 = 1/2-1/2*I*3^(1/2)} ... skip entries to safe data	Failed [240 / 300] Result: Complex[1.1102230246251565*^-16, 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[2, 3]], Pi]]]} Result: Complex[1.1102230246251565*^-16, 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, 3]], Pi]]]} ... skip entries to safe data
10.2.E2	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!\Gamma\left(\nu+k+1\right)}}}$ \BesselJ{\nu}@{z} = (\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!\EulerGamma@{\nu+k+1}}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	BesselJ(nu, z) = ((1)/(2)*z)^(nu)* sum((- 1)^(k)*(((1)/(4)*(z)^(2))^(k))/(factorial(k)*GAMMA(nu + k + 1)), k = 0..infinity)	BesselJ[\[Nu], z] == (Divide[1,2]*z)^\[Nu]* Sum[(- 1)^(k)*Divide[(Divide[1,4]*(z)^(2))^(k),(k)!*Gamma[\[Nu]+ k + 1]], {k, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 70]
10.2.E3	${\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{J_{\nu}\left(z\right)\cos\left(\nu\pi\right)-J_{-\nu}\left(z\right)}{\sin\left(\nu\pi\right)}}}$ \BesselY{\nu}@{z} = \frac{\BesselJ{\nu}@{z}\cos@{\nu\pi}-\BesselJ{-\nu}@{z}}{\sin@{\nu\pi}}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	BesselY(nu, z) = (BesselJ(nu, z)*cos(nu*Pi)- BesselJ(- nu, z))/(sin(nu*Pi))	BesselY[\[Nu], z] == Divide[BesselJ[\[Nu], z]*Cos[\[Nu]*Pi]- BesselJ[- \[Nu], z],Sin[\[Nu]*Pi]]	Successful	Successful	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data