= Nonunique weights in a closed Bessel integral representation
{c}
{title2=$\oint_{|\zeta|=R}(1+\zeta^2)^{-1/2}e^{z\zeta}d\zeta=2\pi iJ_0(z),\quad R>1$}
Choose the square-root branch asymptotic to $\zeta$ outside the cut $[-i,i]$. The Laurent series of its reciprocal gives the displayed <integral> by termwise residue calculation. Adding any <entire function> to the weight leaves the <integral> unchanged, since its product with $e^{z\zeta}$ is entire. Thus a nonzero solution represented by a <contour> <integral> does not by itself determine the weight uniquely; an inference from the <differential equation> to an equation for the weight needs an injective representation or an explicit construction assumption.
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