fundamental property

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1: 4.37 Inverse Hyperbolic Functions

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§4.37(v) Fundamental Property

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2: 4.23 Inverse Trigonometric Functions

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§4.23(v) Fundamental Property

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3: 27.2 Functions

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer $n>1$ can be represented uniquely as a product of prime powers, …

4: 28.2 Definitions and Basic Properties

§28.2 Definitions and Basic Properties

… ►(28.2.1) possesses a fundamental pair of solutions $w_{\text{I}}(z;a,q),w_{\text{II}}(z;a,q)$ called basic solutions with …Other properties are as follows. … ►A solution with the pseudoperiodic property (28.2.14) is called a Floquet solution with respect to $\nu$. … ►

Change of Sign of $q$

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5: 3.11 Approximation Techniques

… ►They enjoy an orthogonal property with respect to integrals: …as well as an orthogonal property with respect to sums, as follows. … ►For these and further properties of Chebyshev polynomials, see Chapter 18, Gil et al. (2007a, Chapter 3), and Mason and Handscomb (2003). … ►The property …is of fundamental importance in the FFT algorithm. …

6: 28.29 Definitions and Basic Properties

§28.29 Definitions and Basic Properties

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28.29.4 $$w_{\text{I}}(z+\pi ,\lambda )=w_{\text{I}}(\pi ,\lambda )w_{\text{I}}(z,\lambda )+w_{\text{I}}^{\prime }(\pi ,\lambda )w_{\text{II}}(z,\lambda ),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $w(z)$: Mathieu’s equation solution, $w_{\text{I}}(z;a,q)$: fundamental solution, $w_{\text{II}}(z;a,q)$: fundamental solution and $\lambda$: parameter

Permalink:

http://dlmf.nist.gov/28.29.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(ii), §28.29 and Ch.28

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28.29.5 $$w_{\text{II}}(z+\pi ,\lambda )=w_{\text{II}}(\pi ,\lambda )w_{\text{I}}(z,\lambda )+w_{\text{II}}^{\prime }(\pi ,\lambda )w_{\text{II}}(z,\lambda ).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $w(z)$: Mathieu’s equation solution, $w_{\text{I}}(z;a,q)$: fundamental solution, $w_{\text{II}}(z;a,q)$: fundamental solution and $\lambda$: parameter

Permalink:

http://dlmf.nist.gov/28.29.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(ii), §28.29 and Ch.28

… ►Then (28.29.1) has a nontrivial solution $w(z)$ with the pseudoperiodic property … ►If $\nu$ $(\ne 0,1)$ is a solution of (28.29.9), then $F_{\nu }(z)$, $F_{-\nu }(z)$ comprise a fundamental pair of solutions of Hill’s equation. …

7: 18.38 Mathematical Applications

… ►The monic Chebyshev polynomial $2^{1-n}{T}_n\left(x\right)$, $n\ge 1$, enjoys the ‘minimax’ property on the interval $[-1,1]$, that is, $|2^{1-n}{T}_n\left(x\right)|$ has the least maximum value among all monic polynomials of degree $n$. … ►Classical OP’s play a fundamental role in Gaussian quadrature. …

8: 18.39 Applications in the Physical Sciences

… ►The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form … ►The properties of $V(x)$ determine whether the spectrum, this being the set of eigenvalues of $\mathscr{H}$, is discrete, continuous, or mixed, see §1.18. … ►As the Coulomb–Pollaczek OP’s are members of the Nevai-Blumenthal class, this is, for , a physical example of the properties of the zeros of such OP’s, and their possible accumulation at $x=-1$, as discussed in §18.2(xi). …

9: Bibliography G

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Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.

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Notes:

Translated from the Russian by Eugene Saletan. A paperbound reprinting by the same publisher appeared in 1977

External Links:

MathReview, ZentralBlatt

Referenced by:

§2.6(i).

Encodings:

BibTeX

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Generalized Fermi-Dirac functions and derivatives: Properties and evaluation. Comput. Phys. Comm. 136 (3), pp. 294–309.

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Notes:

Double-precision Fortran, maximum accuracy 20D

External Links:

Document, Code, ZentralBlatt

Referenced by:

7th item.

Encodings:

BibTeX

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Quantum mechanics: fundamentals. Second edition, Springer-Verlag, New York.

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External Links:

ISBN 0-387-22023-2, MathReview (Howard E. Brandt), ZentralBlatt

Referenced by:

§18.39(i).

Encodings:

BibTeX

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Do integrable mappings have the Painlevé property?. Phys. Rev. Lett. 67 (14), pp. 1825–1828.

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External Links:

ISSN 0031-9007, Document, MathReview, ZentralBlatt

Referenced by:

§32.16.

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BibTeX

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A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.

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External Links:

ISSN 0019-3577, Document, MathReview (Sumitra Purkayastha), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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10: 20.2 Definitions and Periodic Properties

§20.2 Definitions and Periodic Properties

… ►The four points $(0,\pi ,\pi +\tau \pi ,\tau \pi )$ are the vertices of the fundamental parallelogram in the $z$-plane; see Figure 20.2.1. … ►

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Figure 20.2.1: $z$-plane. Fundamental parallelogram. …

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