It appears that Mathematica does not know this information about MarcumQ. It also seems not to be able to compute a power series expansion about mu=0
, oddly just returning the MarcumQ function it self:
In[1]:= Series[MarcumQ[1, 3, \[Mu]], {\[Mu], 0, 2}]
Out[1]= MarcumQ[1, 3, \[Mu]]
This is so, even though Mathematica knows how to compute derivatives, e.g.:
In[2]:= D[MarcumQ[1, 3, \[Mu]], {\[Mu], 3}]
Out[2]=
3 E^(1/2 (-9 - \[Mu]^2)) \[Mu] Hypergeometric0F1Regularized[1, (
9 \[Mu]^2)/4] -
E^(1/2 (-9 - \[Mu]^2)) \[Mu]^3 Hypergeometric0F1Regularized[1, (
9 \[Mu]^2)/4] -
27/2 E^(1/2 (-9 - \[Mu]^2)) \[Mu] Hypergeometric0F1Regularized[2, (
9 \[Mu]^2)/4] +
9 E^(1/2 (-9 - \[Mu]^2)) \[Mu]^3 Hypergeometric0F1Regularized[2, (
9 \[Mu]^2)/4] -
81/4 E^(1/2 (-9 - \[Mu]^2)) \[Mu]^3 Hypergeometric0F1Regularized[3, (
9 \[Mu]^2)/4]