Anger function `AngerJ[nu, z]` which equals `BesselJ` for integer orders.
AngerJ[ν, z]
AngerJ[ν, μ, z]
AngerJ[1 / 2, 5.0]
→ -0.28573350818808113Series[AngerJ[1 / 2, x], {x, 0, 4}]//FullSimplify
→ SeriesData[x, 0, {2/Pi, 4/(3*Pi), -8/(15*Pi), -16/(105*Pi), 32/(945*Pi)}, 0, 5, 1]ReImPlot[AngerJ[1 / 2, x], {x, -25, 25}, PlotRange -> All]
→ -Graphics-