[?] Why does this series have imaginary terms?

Questions
1

I am trying to construct a series around infinity and I get the following result:

In[14]:= Normal@Series[ Sqrt[2 Pi x] BesselI[3, x]/Exp[x], {x, \[Infinity], 3}, Assumptions -> (x > 0)]
Out[15]=(-I + E^(2*x)*(1 + 945/(128*x^2) - 35/(8*x)) - (945*I)/(128*x^2) - (35*I)/(8*x))/E^(2*x)

Notice that x is assumed real which means that the result should not not have any imaginary terms but it does. What is going on here? And what is the best way to get rid of these terms?

2

Try:

ComplexExpand@Re@Normal@Series[Sqrt[2 Pi x] BesselI[3, x]/Exp[x], {x, \[Infinity], 3}, Assumptions -> {x > 0}]
3

OK that gets rid of the imaginary terms.

Much Thanks!

Anyone have any ideas as to why there are imaginary terms to begin with?

4

The asymptotic expansions for modified Bessel functions are inherently complex functions. See asymptotic expansion for I_nu(z) in

?? ?

5

Agreed. But the series is done with the assumption that x is real and under this assumption the imaginary part is zero. But what Mathematica shows is a non-zero imaginary part. This should not be.