On Kahan's Rules for Determining Branch Cuts

In computer algebra there are different ways of approaching the mathematical concept of functions, one of which is by defining them as solutions of differential equations. We compare different such approaches and discuss the occurring problems. The main focus is on the question of determining possible branch cuts. We explore the extent to which the treatment of branch cuts can be rendered (more) algorithmic, by adapting Kahan's rules to the differential equation setting.Comment: SYNASC 2011. 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing. (2011

Higher Derivatives of the Tangent and Inverse Tangent Functions and Chebyshev Polynomials

The higher derivatives of the tangent and hyperbolic tangent functions are determined. Formulas for the higher derivatives of the inverse tangent and inverse hyperbolic tangent functions as polynomials are stated and proved. Using another formula for the higher derivatives of the inverse tangent function from literature, two known formulas for the Chebyshev polynomials of the first and second kind are proved. From these formulas the higher derivatives of the inverse tangent and inverse hyperbolic tangent functions in terms of the Chebyshev polynomial of the second kind are provided

Point Canonical Transformation versus Deformed Shape Invariance for Position-Dependent Mass Schr\"odinger Equations

On using the known equivalence between the presence of a position-dependent mass (PDM) in the Schr\"odinger equation and a deformation of the canonical commutation relations, a method based on deformed shape invariance has recently been devised for generating pairs of potential and PDM for which the Schr\"odinger equation is exactly solvable. This approach has provided the bound-state energy spectrum, as well as the ground-state and the first few excited-state wavefunctions. The general wavefunctions have however remained unknown in explicit form because for their determination one would need the solutions of a rather tricky differential-difference equation. Here we show that solving this equation may be avoided by combining the deformed shape invariance technique with the point canonical transformation method in a novel way. It consists in employing our previous knowledge of the PDM problem energy spectrum to construct a constant-mass Schr\"odinger equation with similar characteristics and in deducing the PDM wavefunctions from the known constant-mass ones. Finally, the equivalence of the wavefunctions coming from both approaches is checked

Perfect imaging: they don't do it with mirrors

Imaging with a spherical mirror in empty space is compared with the case when the mirror is filled with the medium of Maxwell's fish eye. Exact time-dependent solutions of Maxwell's equations show that perfect imaging is not achievable with an electrical ideal mirror on its own, but with Maxwell's fish eye in the regime when it implements a curved geometry for full electromagnetic waves