BesselJ function on negative real numbers

Question

I have evaluated Bessel$J_v(x)$ with some real $v$ and negative real $x$ in MATHEMATICA. I cannot understand how the result is complex (non-real). I look at the series definition of BesselJ and I cannot see where an imaginary number comes in. What am I missing?

Answer 1

The complex values are a consequence of the complex power function. Look at the power series of $J_{\nu}$ with the term in front of the sum: $$J_\nu(x) = (\tfrac{1}{2}x)^{\nu} \sum\limits_{k=0}^{\infty}(-1)^k \frac{(\tfrac{1}{4}x^2)^k}{k!\Gamma(\nu+k+1)!}\;,$$ If you take e.g. $x=-2$ and $\nu=2.2$ it has the value $$(-2)^{2.2}\approx3.7172660+2.7007518i$$ The situation is the same for $I_{\nu}$ because these functions also have the factor $(\tfrac{1}{2}x)^{\nu}.$