= Bessel function of the first kind
{c}
{title2=$J_\nu(z)$}
= Bessel functions of the first kind
{c}
{synonym}
The <Bessel function of the first kind> is normalized by the Frobenius series
$$
J_\nu(z)=\sum_{j=0}^\infty\frac{(-1)^j(z/2)^{2j+\nu}}{j!\,\Gamma(j+\nu+1)}.
$$
The reciprocal of the <gamma function> is interpreted by its <entire function> continuation, which vanishes at $0,-1,-2,\ldots$. For noninteger order, the factor $z^\nu$ requires a specified <branch of a multivalued function>. It need not be regular at zero: negative noninteger order can give a divergent leading power. For nonnegative integer order, the <power series> is regular and single valued, and negative integer orders are covered by the <Integer-order Bessel reflection identity>.<NIST Digital Library of Mathematical Functions>[§10.2(ii), equation 10.2.2 and branch/entire-function discussion]{ref}
For $\operatorname{Re}\nu>-1/2$, the Poisson <integral representation> is
$$
J_\nu(z)=\frac{(z/2)^\nu}{\Gamma(\nu+1/2)\sqrt\pi}\int_{-1}^1 e^{izt}(1-t^2)^{\nu-1/2}\,dt.
$$
The endpoint condition on $\nu$ makes the weight integrable. The zero-order series begins $J_0(z)=1-z^2/4+z^4/64-\cdots$, giving a concrete check of normalization and the origin behavior.
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