analytically continued

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41: 9.11 Products

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9.11.4 $${\mathrm{Ai}}^2\left(z\right)+{\mathrm{Bi}}^2\left(z\right)=\frac{1}{{\pi }^{3/2}}{\int }_0^{\mathrm{\infty }}\exp \left(zt-\frac{1}{12}{t}^3\right)t^{-1/2}dt.$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral and $z$: complex variable

Source:

Muldoon (1977, p. 32, extended to complex $z$ by analytic continuation)

Referenced by:

§9.5(ii)

Permalink:

http://dlmf.nist.gov/9.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(iii), §9.11 and Ch.9

… ►For any continuously-differentiable function $f$ …

42: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►For $f(x)$ piecewise continuously differentiable on $[0,\mathrm{\infty })$ … ►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 $1/{\mathrm{\Gamma }}_{\mathrm{res}}$. For this latter see Simon (1973), and Reinhardt (1982); wherein advantage is taken of the fact that although branch points are actual singularities of an analytic function, the location of the branch cuts are often at our disposal, as they are not singularities of the function, but simply arbitrary lines to keep a function single valued, and thus only singularities of a specific representation of that analytic function. …This dilatation transformation, which does require analyticity of $q(x)$ in (1.18.28), or an analytic approximation thereto, leaves the poles, corresponding to the discrete spectrum, invariant, as they are, as is the branch point, actual singularities of $⟨{\left(z-T\right)}^{-1}f,f⟩$. …