M. Katsav and E. Heyman, "Phase space Gaussian beam summation analysis of half plane diffraction," IEEE Trans. Antennas Propag. 55, 1535-1545 (2007).

[CrossRef]

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.

[CrossRef]

R. Borghi, "Evaluation of diffraction catastrophes by using Weniger transformation," Opt. Lett. 32, 226-228 (2007).

[CrossRef]
[PubMed]

E. J. Weniger, "Mathematical properties of a new Levin-type sequence transformation introduced by Cizek, Zamastil, and Skala. I. Algebraic theory," J. Math. Phys. 45, 1209-1246 (2004).

[CrossRef]

J. Cizek, J. Zamastil, and L. Skála, "New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field," J. Math. Phys. 44, 962-968 (2003).

[CrossRef]

R. Borghi and M. Santarsiero, "Summing Lax series for nonparaxial propagation," Opt. Lett. 28, 774-776 (2003).

[CrossRef]
[PubMed]

J. H. Hannay and A. Thain, "Exact scattering theory for any straight reflectors in two dimensions," J. Phys. A 36, 4063-4080 (2003).

[CrossRef]

V. Nesterenko, G. Lambiase, and G. Scarpetta, "Casimir effect for a perfectly conducting wedge in terms of local zeta function," Ann. Phys. (N.Y.) 298, 403-420 (2002).

[CrossRef]

M. V. Vesnik, "Analytical solution for electromagnetic diffraction on 2-D perfectly conducting scatterers of arbitrary shape," IEEE Trans. Antennas Propag. 49, 1638-1644 (2001).

[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

M. Alvarez, "Calculation of the Aharonov-Bohm wave function," Phys. Rev. A 54, 1128-1135 (1996).

[CrossRef]
[PubMed]

H. Bruusy and N. Whelan, "Edge diffraction, trace formulae and the cardioid billiard," Nonlinearity 9, 1023-1047 (1996).

[CrossRef]

N. Pavloff and C. Schimt, "Diffractive orbits in quantum billiards," Phys. Rev. Lett. 75, 61-64 (1995).

[CrossRef]
[PubMed]

G. H. Hardy, Divergent Series (AMS, 1991).

E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10, 189-371 (1989).

[CrossRef]

F. Gori, "Diffraction from a half-plane. A new derivation of the Sommerfeld solution," Opt. Commun. 48, 67-70 (1983).

[CrossRef]

R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, 1973).

M.Abramowitz and I.Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).

A. Sommerfeld, Lectures on Theoretical Physics. Optics (Academic, 1964).

P. C. Clemmow, "Some extensions to the method of integration by steepest descents," Q. J. Mech. Appl. Math. 3, 241-256 (1950).

[CrossRef]

W. Pauli, "On asymptotic series for functions in the theory of diffraction of light," Phys. Rev. 54, 924-931 (1938).

[CrossRef]

Darboux, "Mémoire sur l'approximation de fonctions de très grande nombres et sur une classe étendue de développments en séries," J. Math. Pures Appl. 4, 5-56 (1878).

M. Alvarez, "Calculation of the Aharonov-Bohm wave function," Phys. Rev. A 54, 1128-1135 (1996).

[CrossRef]
[PubMed]

M. V. Berry, "Universal oscillations of high derivatives," Proc. R. Soc. London, Ser. A 461, 1735-1751 (2005).

[CrossRef]

R. Borghi, "Evaluation of diffraction catastrophes by using Weniger transformation," Opt. Lett. 32, 226-228 (2007).

[CrossRef]
[PubMed]

M. A. Alonso and R. Borghi, "Complete far-field asymptotic series for free fields," Opt. Lett. 31, 3028-3030 (2006).

[CrossRef]
[PubMed]

R. Borghi and M. Santarsiero, "Summing Lax series for nonparaxial propagation," Opt. Lett. 28, 774-776 (2003).

[CrossRef]
[PubMed]

R. Borghi, "Optical asymptotics via Weniger transformation," arXiv.org e-Print archieve, physics/0706.3573, 25 June 2007, http://arxiv.org/abs/0706.3573.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

H. Bruusy and N. Whelan, "Edge diffraction, trace formulae and the cardioid billiard," Nonlinearity 9, 1023-1047 (1996).

[CrossRef]

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.

[CrossRef]

J. Cizek, J. Zamastil, and L. Skála, "New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field," J. Math. Phys. 44, 962-968 (2003).

[CrossRef]

P. C. Clemmow, "Some extensions to the method of integration by steepest descents," Q. J. Mech. Appl. Math. 3, 241-256 (1950).

[CrossRef]

R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, 1973).

F. Gori, "Diffraction from a half-plane. A new derivation of the Sommerfeld solution," Opt. Commun. 48, 67-70 (1983).

[CrossRef]

J. H. Hannay and A. Thain, "Exact scattering theory for any straight reflectors in two dimensions," J. Phys. A 36, 4063-4080 (2003).

[CrossRef]

G. H. Hardy, Divergent Series (AMS, 1991).

M. Katsav and E. Heyman, "Phase space Gaussian beam summation analysis of half plane diffraction," IEEE Trans. Antennas Propag. 55, 1535-1545 (2007).

[CrossRef]

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.

[CrossRef]

M. Katsav and E. Heyman, "Phase space Gaussian beam summation analysis of half plane diffraction," IEEE Trans. Antennas Propag. 55, 1535-1545 (2007).

[CrossRef]

V. Nesterenko, G. Lambiase, and G. Scarpetta, "Casimir effect for a perfectly conducting wedge in terms of local zeta function," Ann. Phys. (N.Y.) 298, 403-420 (2002).

[CrossRef]

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.

[CrossRef]

V. Nesterenko, G. Lambiase, and G. Scarpetta, "Casimir effect for a perfectly conducting wedge in terms of local zeta function," Ann. Phys. (N.Y.) 298, 403-420 (2002).

[CrossRef]

W. Pauli, "On asymptotic series for functions in the theory of diffraction of light," Phys. Rev. 54, 924-931 (1938).

[CrossRef]

N. Pavloff and C. Schimt, "Diffractive orbits in quantum billiards," Phys. Rev. Lett. 75, 61-64 (1995).

[CrossRef]
[PubMed]

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.

[CrossRef]

V. Nesterenko, G. Lambiase, and G. Scarpetta, "Casimir effect for a perfectly conducting wedge in terms of local zeta function," Ann. Phys. (N.Y.) 298, 403-420 (2002).

[CrossRef]

N. Pavloff and C. Schimt, "Diffractive orbits in quantum billiards," Phys. Rev. Lett. 75, 61-64 (1995).

[CrossRef]
[PubMed]

J. Cizek, J. Zamastil, and L. Skála, "New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field," J. Math. Phys. 44, 962-968 (2003).

[CrossRef]

A. Sommerfeld, Lectures on Theoretical Physics. Optics (Academic, 1964).

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.

[CrossRef]

J. H. Hannay and A. Thain, "Exact scattering theory for any straight reflectors in two dimensions," J. Phys. A 36, 4063-4080 (2003).

[CrossRef]

M. V. Vesnik, "Analytical solution for electromagnetic diffraction on 2-D perfectly conducting scatterers of arbitrary shape," IEEE Trans. Antennas Propag. 49, 1638-1644 (2001).

[CrossRef]

E. J. Weniger, "Mathematical properties of a new Levin-type sequence transformation introduced by Cizek, Zamastil, and Skala. I. Algebraic theory," J. Math. Phys. 45, 1209-1246 (2004).

[CrossRef]

E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10, 189-371 (1989).

[CrossRef]

E. J. Weniger, "Asymptotic approximations to truncation errors of series representations for special functions," arXiv.org e-Print archive, math.CA/0511074v1, 11 March 2005, http://arxiv.org/abs/math/0511074v1.

H. Bruusy and N. Whelan, "Edge diffraction, trace formulae and the cardioid billiard," Nonlinearity 9, 1023-1047 (1996).

[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

J. Cizek, J. Zamastil, and L. Skála, "New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field," J. Math. Phys. 44, 962-968 (2003).

[CrossRef]

V. Nesterenko, G. Lambiase, and G. Scarpetta, "Casimir effect for a perfectly conducting wedge in terms of local zeta function," Ann. Phys. (N.Y.) 298, 403-420 (2002).

[CrossRef]

E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10, 189-371 (1989).

[CrossRef]

M. Katsav and E. Heyman, "Phase space Gaussian beam summation analysis of half plane diffraction," IEEE Trans. Antennas Propag. 55, 1535-1545 (2007).

[CrossRef]

M. V. Vesnik, "Analytical solution for electromagnetic diffraction on 2-D perfectly conducting scatterers of arbitrary shape," IEEE Trans. Antennas Propag. 49, 1638-1644 (2001).

[CrossRef]

J. Cizek, J. Zamastil, and L. Skála, "New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field," J. Math. Phys. 44, 962-968 (2003).

[CrossRef]

E. J. Weniger, "Mathematical properties of a new Levin-type sequence transformation introduced by Cizek, Zamastil, and Skala. I. Algebraic theory," J. Math. Phys. 45, 1209-1246 (2004).

[CrossRef]

Darboux, "Mémoire sur l'approximation de fonctions de très grande nombres et sur une classe étendue de développments en séries," J. Math. Pures Appl. 4, 5-56 (1878).

J. H. Hannay and A. Thain, "Exact scattering theory for any straight reflectors in two dimensions," J. Phys. A 36, 4063-4080 (2003).

[CrossRef]

H. Bruusy and N. Whelan, "Edge diffraction, trace formulae and the cardioid billiard," Nonlinearity 9, 1023-1047 (1996).

[CrossRef]

F. Gori, "Diffraction from a half-plane. A new derivation of the Sommerfeld solution," Opt. Commun. 48, 67-70 (1983).

[CrossRef]

Y. Umul, "Modified theory of the physical-optics approach to the impedance wedge problem," Opt. Lett. 31, 401-403 (2006).

[CrossRef]
[PubMed]

M. A. Alonso and R. Borghi, "Complete far-field asymptotic series for free fields," Opt. Lett. 31, 3028-3030 (2006).

[CrossRef]
[PubMed]

R. Borghi, "Evaluation of diffraction catastrophes by using Weniger transformation," Opt. Lett. 32, 226-228 (2007).

[CrossRef]
[PubMed]

R. Borghi and M. Santarsiero, "Summing Lax series for nonparaxial propagation," Opt. Lett. 28, 774-776 (2003).

[CrossRef]
[PubMed]

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.

[CrossRef]

W. Pauli, "On asymptotic series for functions in the theory of diffraction of light," Phys. Rev. 54, 924-931 (1938).

[CrossRef]

M. Alvarez, "Calculation of the Aharonov-Bohm wave function," Phys. Rev. A 54, 1128-1135 (1996).

[CrossRef]
[PubMed]

N. Pavloff and C. Schimt, "Diffractive orbits in quantum billiards," Phys. Rev. Lett. 75, 61-64 (1995).

[CrossRef]
[PubMed]

M. V. Berry, "Universal oscillations of high derivatives," Proc. R. Soc. London, Ser. A 461, 1735-1751 (2005).

[CrossRef]

P. C. Clemmow, "Some extensions to the method of integration by steepest descents," Q. J. Mech. Appl. Math. 3, 241-256 (1950).

[CrossRef]

It should be noted that in the original paper, Pauli used the opposite convention; i. e., the temporal factor had the form exp(+iωt). For this reason, the subsequent formulas are the complex conjugate of those written in .

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

E. J. Weniger, "Asymptotic approximations to truncation errors of series representations for special functions," arXiv.org e-Print archive, math.CA/0511074v1, 11 March 2005, http://arxiv.org/abs/math/0511074v1.

R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, 1973).

G. H. Hardy, Divergent Series (AMS, 1991).

When z is a real negative, the integral in Eq. must be intended in the Cauchy principal value sense. This is related to the fact that the line arg z=π represents a branch cut for the exponential integral function .

A. Sommerfeld, Lectures on Theoretical Physics. Optics (Academic, 1964).

R. Borghi, "Optical asymptotics via Weniger transformation," arXiv.org e-Print archieve, physics/0706.3573, 25 June 2007, http://arxiv.org/abs/0706.3573.

We use, as far as possible, a notation similar to the original one adopted in .

If, on the contrary, the illumination produces no geometrical shadow, to obtain a regular representation also at the second reflection, it is sufficient to change φ into φ−2πn in all the following equations.

M.Abramowitz and I.Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).