Discussion
The integral \(\frac{1}{2\pi i}\oint\frac{\exp[\frac{z}{2}(t - \frac{1}{t})]}{t^{n+1}}dt\), where n is an integer and the closed contour is a circle enclosing the origin, can be used to :
Answer: B
The integral is the contour-integral form of the Bessel functions of the first kind, \(J_{n}(z)\).
The generating function of the Bessel functions is
\[ \begin{aligned} &\exp\left[\frac{z}{2}\left(t - \frac{1}{t}\right)\right] \\ &= \sum_{n=-\infty}^{\infty} J_{n}(z)\,t^{n} \end{aligned} \]
Dividing by \(t^{n+1}\) and integrating round the origin keeps only the \(1/t\) term, so by the residue theorem the integral equals \(J_{n}(z)\), which is the standard contour-integral representation of \(J_{n}\).
The other families have different generating functions: \(e^{2xt - t^{2}}\) for the Hermite polynomials and \((1 - 2xt + t^{2})^{-1/2}\) for the Legendre polynomials, from which the associated Legendre functions are obtained by differentiation.
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