Coulomb Functions - 33.5 Limiting Forms for Small , Small , or Large

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
33.5#Ex7	${\displaystyle{\displaystyle F_{\ell}\left(0,\rho\right)=(\pi\rho/2)^{1/2}J_{\ell+\frac{1}{2}}\left(\rho\right)}}$ \regCoulombF{\ell}@{0}{\rho} = (\pi\rho/2)^{1/2}\BesselJ{\ell+\frac{1}{2}}@{\rho}	${\displaystyle{\displaystyle\Re((\ell+\frac{1}{2})+k+1)>0}}$	CoulombF(ell, 0, rho) = (Pi*rho/2)^(1/2)* BesselJ(ell +(1)/(2), rho)	Error	Failure	Missing Macro Error	Error	-
33.5#Ex9	${\displaystyle{\displaystyle F_{0}\left(0,\rho\right)=\sin\rho}}$ \regCoulombF{0}@{0}{\rho} = \sin@@{\rho}		CoulombF(0, 0, rho) = sin(rho)	Error	Successful	Missing Macro Error	-	-
33.5.E6	${\displaystyle{\displaystyle\frac{2^{\ell}\ell!}{(2\ell+1)!}=\frac{1}{(2\ell+1)!!}}}$ \frac{2^{\ell}\ell!}{(2\ell+1)!} = \frac{1}{(2\ell+1)!!}	${\displaystyle{\displaystyle\Re(\ell+1+\mathrm{i}\eta)>0}}$	((2)^(ell)* factorial(ell))/(factorial(2*ell + 1)) = (1)/(doublefactorial(2*ell + 1))	Divide[(2)^\[ScriptL]* (\[ScriptL])!,(2*\[ScriptL]+ 1)!] == Divide[1,(2*\[ScriptL]+ 1)!!]	Failure	Failure	Error	Failed [1 / 1] Result: Plus[Times[Power[2.0, ℓ], Factorial[ℓ], Power[Factorial[Plus[1.0, Times[2.0, ℓ]]], -1]], Times[-1.0, Power[Factorial2[Plus[1.0, Times[2.0, ℓ]]], -1]]] Test Values: {}