*Chun Wa Wong*

- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199641390
- eISBN:
- 9780191747786
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199641390.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Functions of a complex variable are shown to be more complete and rigid than functions of a real variable. Analytic functions with well defined derivatives satisfy two Cauch–Riemann conditions. ...
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Functions of a complex variable are shown to be more complete and rigid than functions of a real variable. Analytic functions with well defined derivatives satisfy two Cauch–Riemann conditions. Multivalued functions can be made single-valued on a multi-sheet Riemann surface. The values of an analytic function in a region of the complex plane are completely defined by the knowledge of their values on a closed boundary of the region. Important properties and techniques of complex analysis are described. These include Taylor and Laurent expansions, contour integration and residue calculus, Green's functions, Laplace transforms and Bromwich integrals, dispersion relations and asymptotic expansions. Analytic functions are defined by their properties at the locations called singularities (poles and branch cuts) where they cease to be analytic. This feature makes analytic functions of particular interest in the construction of physical theories.Less

Functions of a complex variable are shown to be more complete and rigid than functions of a real variable. Analytic functions with well defined derivatives satisfy two Cauch–Riemann conditions. Multivalued functions can be made single-valued on a multi-sheet Riemann surface. The values of an analytic function in a region of the complex plane are completely defined by the knowledge of their values on a closed boundary of the region. Important properties and techniques of complex analysis are described. These include Taylor and Laurent expansions, contour integration and residue calculus, Green's functions, Laplace transforms and Bromwich integrals, dispersion relations and asymptotic expansions. Analytic functions are defined by their properties at the locations called singularities (poles and branch cuts) where they cease to be analytic. This feature makes analytic functions of particular interest in the construction of physical theories.

*John A. Adam*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691148373
- eISBN:
- 9781400885404
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148373.003.0007
- Subject:
- Mathematics, Applied Mathematics

This chapter deals with the WKB(J) approximation, commonly used in applied mathematics and mathematical physics to find approximate solutions of linear ordinary differential equations (of any order ...
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This chapter deals with the WKB(J) approximation, commonly used in applied mathematics and mathematical physics to find approximate solutions of linear ordinary differential equations (of any order in principle) with spatially varying coefficients. The WKB(J) approximation is closely related to the semiclassical approach in quantum mechanics in which the wavefunction is characterized by a slowly varying amplitude and/or phase. The chapter first introduces an inhomogeneous differential equation, from which the first derivative term is eliminated, before discussing the Liouville transformation and the one-dimensional Schrödinger equation. It then presents a physical interpretation of the WKB(J) approximation and its application to a potential well. It also considers the “patching region” in which the Airy function solution (the local turning point) is valid, the relation between Airy functions and Bessel functions, Airy integral and related topics, and related integrals.Less

This chapter deals with the WKB(J) approximation, commonly used in applied mathematics and mathematical physics to find approximate solutions of linear ordinary differential equations (of any order in principle) with spatially varying coefficients. The WKB(J) approximation is closely related to the semiclassical approach in quantum mechanics in which the wavefunction is characterized by a slowly varying amplitude and/or phase. The chapter first introduces an inhomogeneous differential equation, from which the first derivative term is eliminated, before discussing the Liouville transformation and the one-dimensional Schrödinger equation. It then presents a physical interpretation of the WKB(J) approximation and its application to a potential well. It also considers the “patching region” in which the Airy function solution (the local turning point) is valid, the relation between Airy functions and Bessel functions, Airy integral and related topics, and related integrals.

*John A. Adam*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691148373
- eISBN:
- 9781400885404
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148373.003.0028
- Subject:
- Mathematics, Applied Mathematics

This chapter returns to the subject of rainbows, offering some reflections based on the author's review of the book The Rainbow Bridge: Rainbows in Art, Myth, and Science by Raymond L. Lee, Jr. and ...
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This chapter returns to the subject of rainbows, offering some reflections based on the author's review of the book The Rainbow Bridge: Rainbows in Art, Myth, and Science by Raymond L. Lee, Jr. and Alistair B. Frase. In particular, it discusses various topics related to the rainbow, including historical descriptions of the rainbow, some common misperceptions about rainbows, theories of the rainbow, angular momentum, rainbow ray, and Airy functions. The chapter also considers ray optics, with emphasis on Luneberg inversion and gravitational lensing, Abel's integral equation, and the Luneberg lens. Finally, it explains the rainbow's connection with classical scattering and gravitational lensing, focusing on weak gravitational fields and the black hole lens.Less

This chapter returns to the subject of rainbows, offering some reflections based on the author's review of the book *The Rainbow Bridge: Rainbows in Art, Myth, and Science* by Raymond L. Lee, Jr. and Alistair B. Frase. In particular, it discusses various topics related to the rainbow, including historical descriptions of the rainbow, some common misperceptions about rainbows, theories of the rainbow, angular momentum, rainbow ray, and Airy functions. The chapter also considers ray optics, with emphasis on Luneberg inversion and gravitational lensing, Abel's integral equation, and the Luneberg lens. Finally, it explains the rainbow's connection with classical scattering and gravitational lensing, focusing on weak gravitational fields and the black hole lens.

*Efstratios Manousakis*

- Published in print:
- 2015
- Published Online:
- December 2015
- ISBN:
- 9780198749349
- eISBN:
- 9780191813474
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198749349.003.0012
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter presents a simple way to derive the semiclassical approximation of Wenzel, Kramers, and Brillouin (WKB). This approximation is valid in the limit where the external potential varies ...
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This chapter presents a simple way to derive the semiclassical approximation of Wenzel, Kramers, and Brillouin (WKB). This approximation is valid in the limit where the external potential varies smoothly over a length scale much larger than the particle local de Broglie wavelength. The chapter discusses the concept of asymptotic matching the WKB solution with the exact solution very close to the classical turning points.Less

This chapter presents a simple way to derive the semiclassical approximation of Wenzel, Kramers, and Brillouin (WKB). This approximation is valid in the limit where the external potential varies smoothly over a length scale much larger than the particle local de Broglie wavelength. The chapter discusses the concept of asymptotic matching the WKB solution with the exact solution very close to the classical turning points.