The most general region of this kind is the curvilinear quadrangle whose boundary consists of two elliptic arcs and two arcs of hyperbolas which are confocal with the former. The determination of the eigen values leads to complicated transcendental equations which we cannot discuss here.
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For the full ellipse we have, in addition to the boundary condition u = 0, a condition of continuity for ξ = 0 (focal line) and the condition of periodicity for η = ± π. The solution ξ = const yields the ellipses which belong to the family of curves η = const yields the hyperbolas of the family. The wave equation (1) written in elliptic coordinates ξ, η can be separated (see v.II exercise IV.3) and leads to a so-called Mathieu equation in each coordinate. c)Įllipse and elliptic-hyperbolic curvilinear quadrangle. The most general region which can be treated in this manner is the circular ring sector b ≤ r ≤ a, 0 ≤ φ ≤ α, which is bounded by two circular arcs and two radii. Infinitely many eigen values k = k nm exist the problem is not degenerate. (5) u = I u ( k undefined r ) sin u undefined φ, μ = m π ∝ ,where the k are determined by the condition I μ( ka) = 0. In order to understand fully these general considerations, it is recommended to redo the simple example of a semiinfinite linear atomic chain, see Ref. The reference system is defined here as that constituted by all corresponding infinite subsystems. The procedure defined above for a finite composite system has to be used in order to get the eigen vector corresponding to a localized state falling outside the bulk bands of the reference system. ( 36) may be used in order to obtain all transmitted and reflected waves induced by this action. 3.5.2 Infinite Systemįor an infinite system, when the action is applied inside one homogeneous part i of the system and for E equal to an eigen value of this subsystem i, U( D) has to be constructed with the help of the corresponding bulk eigenvalue inside this subsystem i and zeros inside all other subsystems.
In order to understand fully these general considerations, it is recommended to redo the simple example of a finite atomic chain, see Ref. An appropriate normalization factor can be calculated afterwards. Which provides the unnormalized corresponding eigen vector. Through all of this, Eigenmath can help to resolve algebra and calculus problems and is an excellent option of schools and universities, as it is open and free.Ī graphic calculator such as Eigenmath may not be understandable for Users who have come here by accident, but if you love maths then you are sure to see the advantage that you can take of this tool, one of the most sought after in the field of computer algebra.(39) u ( D ) = U ( M ) det Δ ( M M ) − 1 A ( M D ) , It is also open source, so if you can handle C language you can adapt and improve the programme according to your needs.Įigenmath is much more than just a simple calculator, and it deals with far more advanced operations, such as routines, derivatives, integrals, simplification of expressions, solutions of lineal equasions, defined and undefined integrals of mutliple dimension, product matrix operations, inverses, determiners, etc.
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