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21: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►This representation has poles with residues ${\left|\widehat{f}({\lambda }_n)\right|}^2$ at the discrete eigenvalues and a branch cut along $[0,\mathrm{\infty })$ with discontinuity, from below to above the cut, $2\pi \mathrm{i}{\left|\widehat{f}(\lambda )\right|}^2$, as in (1.18.53), see Newton (2002, §7.1.1). ►Note that the integral in (1.18.66) is not singular if approached separately from above, or below, the real axis: in fact analytic continuation from the upper half of the complex plane, across the cut, and onto higher Riemann Sheets can access complex poles with singularities at discrete energies ${\lambda }_{\mathrm{res}}-\mathrm{i}{\mathrm{\Gamma }}_{\mathrm{res}}/2$ corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to $1/{\mathrm{\Gamma }}_{\mathrm{res}}$. …

22: 23.10 Addition Theorems and Other Identities

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23: 2.10 Sums and Sequences

… ►The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5. …

24: 23.6 Relations to Other Functions

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25: 25.11 Hurwitz Zeta Function

… ► $\zeta (s,a)$ has a meromorphic continuation in the $s$-plane, its only singularity in $ℂ$ being a simple pole at $s=1$ with residue $1$. …