Geodesic
Analytic continuation with respect to a parameter of the Green's functions of exterior boundary value problems for the two-dimensional Helmholtz equation. I
Sbornik. Mathematics, Tome 26 (1975) no. 3, pp. 373-402 
L. A. Muravei. Analytic continuation with respect to a parameter of the Green's functions of exterior boundary value problems for the two-dimensional Helmholtz equation. I. Sbornik. Mathematics, Tome 26 (1975) no. 3, pp. 373-402. http://geodesic.mathdoc.fr/item/SM_1975_26_3_a5/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the first part of the paper one studies the distribution in the half-plane $\{\nu:|{\arg\nu}|<\pi/2\}$ of the roots of the functions $H_\nu'(k)$ and $H_\nu'(k)+igH_\nu(k)$ and of the variable $\nu$ for arbitrary fixed complex $k$ from the region $K(\delta,\varkappa)=\{k:-\delta<\arg k<\pi/2-\delta,\ \varkappa<|k|\}$ for some $\delta\in(0,\pi/2)$ and $\varkappa>0$, where $H_\nu(k)$ is the first Hankel function, $H_\nu'(k)$ is its derivative with respect to $k$, and $g$ is an arbitrary nonnegative number. Figures: 4. Bibliography: 10 titles.

[1] L. A. Muravei, “Ob analiticheskom prodolzhenii funktsii Grina vneshnikh kraevykh zadach dlya dvumernogo uravneniya Gelmgoltsa”, DAN SSSR, 220:1 (1975), 35–37 | MR | Zbl

[2] F. I. W. Ursell, “Short-wave asymptotic theory of the wave equation $(\nabla^2+k^2)=0$”, Proc. Cambridge Phil. Soc., 53 (1957), 115–133 | DOI | MR | Zbl

[3] V. M. Babich, “O korotkovolnovoi asimptotike funktsii Grina dlya uravneniya Gelmgoltsa”, Matem. sb., 65(107) (1964), 576–630 | Zbl

[4] R. Grimshaw, “High-frequency scattering by finite convex region”, Comm. Pure and Appl. Math., 19 (1966), 167–198 | DOI | MR | Zbl

[5] L. A. Muravei, “Asimptoticheskoe povedenie pri bolshikh znacheniyakh vremeni reshenii vtoroi i tretei vneshnikh kraevykh zadach dlya volnovogo uravneniya s dvumya prostranstvennymi peremennymi”, Trudy Matem. in-ta im. V. A. Steklova, CXXVI (1973), 73–144 | MR

[6] F. W. J. Olver, “The asymptotic solution of linear differential equation of the second order for large values of a parameter”, Phil. Trans. Roy. Soc. London, 247 (1954), 307–369 | DOI | MR

[7] G. N. Vatson, Teoriya besselevykh funktsii, IL, Moskva, 1949

[8] V. P. Mikhailov, “O kornyakh funktsii Makdonalda”, Trudy Matem. in-ta im. V. A. Steklova, CIII (1968), 162–171

[9] A. Erdelyi, W. O. Kermack, “Note on the equation $f(z)K_n'(z)-g(z)K_n(z)$”, Proc. Cambridge Phil. Soc., 41 (1954), 74–75 | DOI | MR

[10] L. A. Muravei, “O kornyakh funktsii $Ai'(z)-\sigma Ai(z)$”, Diff. uravneniya, 11:6 (1975), 1054–1077 | MR | Zbl