Bessel functions of order one half ($J_{1/2}(x)=\sqrt{2/(\pi x)}\sin x,J_{-1/2}(x)=\sqrt{2/(\pi x)}\cos x$)

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 $u^{\prime \prime }+u=0$. 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.