fundamental parallelogram

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1: 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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2: Howard S. Cohl

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …

3: Leonard C. Maximon

… ►Maximon published numerous papers on the fundamental processes of quantum electrodynamics and on the special functions of mathematical physics. …

4: 28.29 Definitions and Basic Properties

… ►

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

… ►

28.29.8

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w(z)$: 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.E8

Encodings:

pMML, png, TeX

See also:

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

… ►

28.29.9 $$2\cos \left(\pi \nu \right)=w_{\text{I}}(\pi ,\lambda )+w_{\text{II}}^{\prime }(\pi ,\lambda ).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\nu$: complex parameter, $w_{\text{I}}(z;a,q)$: fundamental solution, $w_{\text{II}}(z;a,q)$: fundamental solution and $\lambda$: parameter

Referenced by:

§28.29(ii), §28.29(ii), §28.29(ii)

Permalink:

http://dlmf.nist.gov/28.29.E9

Encodings:

pMML, png, TeX

See also:

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

… ►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. …

5: 23.5 Special Lattices

… ►Then $\mathrm{\Delta }>0$ and the parallelogram with vertices at $0$, $2{\omega }_1$, $2{\omega }_1+2{\omega }_3$, $2{\omega }_3$ is a rectangle. … ►The parallelogram $0$, $2{\omega }_1$, $2{\omega }_1+2{\omega }_3$, $2{\omega }_3$ is a square, and … ►The parallelogram $0$, $2{\omega }_1-2{\omega }_3$, $2{\omega }_1$, $2{\omega }_3$, is a rhombus: see Figure 23.5.1. …

6: 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 ►

28.2.5

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Symbols:

$q=h^2$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution, $w_{\text{I}}(z;a,q)$: fundamental solution and $w_{\text{II}}(z;a,q)$: fundamental solution

A&S Ref:

20.3.9 (in different notation)

Referenced by:

§28.29(ii), item (a), item (c)

Permalink:

http://dlmf.nist.gov/28.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(ii), §28.2 and Ch.28

… ►

28.2.6 $$𝒲\left\{w_{\text{I}},w_{\text{II}}\right\}=1,$$

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Symbols:

$𝒲$: Wronskian, $w_{\text{I}}(z;a,q)$: fundamental solution and $w_{\text{II}}(z;a,q)$: fundamental solution

Referenced by:

§28.29(ii)

Permalink:

http://dlmf.nist.gov/28.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(ii), §28.2 and Ch.28

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28.2.7 $$w_{\text{I}}(z\pm \pi ;a,q)=w_{\text{I}}(\pi ;a,q)w_{\text{I}}(z;a,q)\pm w_{\text{I}}^{\prime }(\pi ;a,q)w_{\text{II}}(z;a,q),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $q=h^2$: parameter, $z$: complex variable, $a$: parameter, $w(z)$: Mathieu’s equation solution, $w_{\text{I}}(z;a,q)$: fundamental solution and $w_{\text{II}}(z;a,q)$: fundamental solution

Referenced by:

§28.2(iv)

Permalink:

http://dlmf.nist.gov/28.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(ii), §28.2 and Ch.28

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28.2.8 $$w_{\text{II}}(z\pm \pi ;a,q)=\pm w_{\text{II}}(\pi ;a,q)w_{\text{I}}(z;a,q)+w_{\text{II}}^{\prime }(\pi ;a,q)w_{\text{II}}(z;a,q),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $q=h^2$: parameter, $z$: complex variable, $a$: parameter, $w(z)$: Mathieu’s equation solution, $w_{\text{I}}(z;a,q)$: fundamental solution and $w_{\text{II}}(z;a,q)$: fundamental solution

Permalink:

http://dlmf.nist.gov/28.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(ii), §28.2 and Ch.28

…

7: 3.12 Mathematical Constants

… ►The fundamental constant …

8: 8.24 Physical Applications

… ►With more general values of $p$, $E_p\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).

9: 16.21 Differential Equation

… ►A fundamental set of solutions of (16.21.1) is given by …For other fundamental sets see Erdélyi et al. (1953a, §5.4) and Marichev (1984).

10: 15.10 Hypergeometric Differential Equation

… ►

§15.10(i) Fundamental Solutions

… ►When none of the exponent pairs differ by an integer, that is, when none of $c$, $c-a-b$, $a-b$ is an integer, we have the following pairs $f_1(z)$, $f_2(z)$ of fundamental solutions. … ►(a) If $c$ equals $n=1,2,3,\mathrm{\dots }$, and $a=1,2,\mathrm{\dots },n-1$, then fundamental solutions in the neighborhood of $z=0$ are given by (15.10.2) with the interpretation (15.2.5) for $f_2(z)$. … ► … ►The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions. …