= Bessel functions of order one half
{c}
{title2=$J_{1/2}(x)=\sqrt{2/(\pi x)}\sin x,\quad J_{-1/2}(x)=\sqrt{2/(\pi x)}\cos x$}
For $x>0$, the <Bessel differential equation> of order $1/2$ has independent solutions $x^{-1/2}\sin x$ and $x^{-1/2}\cos x$. Substituting $y=x^{-1/2}u$ reduces it to $u''+u=0$. The normalization of the <Bessel function of the first kind> gives the displayed formulas for orders $1/2$ and $-1/2$. The <Frobenius method> obtains the same pair from exponents $1/2$ and $-1/2$: their even-coefficient series reduce to the sine and cosine <power series>. In the negative-exponent recurrence, the free coefficient at the next exponent can add the positive-exponent solution, an example of an <undetermined coefficient at Frobenius resonance>.
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