Since the condition of being an intertwiner can be written as a system of linear equations on components of (cid:8) with polynomial coe(cid:14)cents, solution can be written as a trigonometric rational function.
Representation-theoretic proof of the inner product and symmetry identities for MacDonald's polynomials
The appropriate conditioned evolution can be described by averaging all the trigonometric functions of φ in the evolution equations (2.4) except where they are multiplied by Itˆo increments.
State determination in continuous measurement
Equation (6.3) has the form of an oscillator equation, hence its solutions are trigonometric or hyperbolic functions, depending on whether K is positive or negative.
Weak Gravitational Lensing
But because we only use algebraic (or trigonometric) functions all of the functions we actually use can be continued to complex variables.
Algebra versus analysis in statistical mechanics and quantum field theory
The structure of the multiplier simpliﬁes by the addition theorem of trigonometric functions.
Solution Representations for a Wave Equation with Weak Dissipation
For all even values of µ it is possible to represent the occuring Bessel functions by trigonometric functions, but only for µ = 2 the representation simpliﬁes signiﬁcantly, for the reason cf. A.2.
Solution Representations for a Wave Equation with Weak Dissipation
In the general case an addition theorem for Bessel functions like that for trigonometric ones is not available.
Solution Representations for a Wave Equation with Weak Dissipation
Finally, for the speciﬁc treatment of the wave equation, we need that the trigonometric functions cos and sin should be present.
Some calculus with extensive quantities: wave equation
B (z ) are the combinations of trigonometric functions and Airy functions and their derivatives deﬁned by (160).
Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles
The modiﬁed P¨oschl-Teller potentials which can be obtained by replacing the trigonometric functions by their hyperbolic counterparts [12, 14], (iv) The Rosen-Morse potential which is the symmetric modiﬁed P¨oschl-Teller potentials .
Coherent states \`a la Klauder-Perelomov for the P\"oschl-Teller potentials
Wigner functions the quantum number α appears next to µ, and the quantity γ − 2π/3 in the trigonometric functions is obtained from γ − 2πk/3 for k = 1, since in the present case the pro jection α along the body-ﬁxed ˆx′ -axis is used.
Z(5): Critical point symmetry for the prolate to oblate nuclear shape phase transition
Both for position and spacetime translation representations, there arise the trigonometric functions, as seen, e.g., in t 7−→ (cos µt, sin µt) as matrix elements of selfdual time representations.
The Hilbert spaces for stable and unstable particles
To provide this property generating reﬁnable functions should be compactly supported and wavelet masks should be trigonometric polynomials.
Multivariate Wavelet Frames
This algebra is generated by polynomial functions on H , the group W , and trigonometric Dunkl operators.
Reducibility of the polynomial representation of the degenerate double affine Hecke algebra
However, if c is not a constant function, the answer in the trigonometric case may differ from the rational case, as explained below.
Reducibility of the polynomial representation of the degenerate double affine Hecke algebra
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