fundamental parallelogram

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11: 27.2 Functions

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§27.2(i) Definitions

►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, …

12: 1.6 Vectors and Vector-Valued Functions

… ►Area of parallelogram with vectors $𝐚$ and $𝐛$ as sides $=\Vert 𝐚\times 𝐛\Vert$. …

13: 22.4 Periods, Poles, and Zeros

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§22.4(ii) Graphical Interpretation via Glaisher’s Notation

►Figure 22.4.2 depicts the fundamental unit cell in the $z$-plane, with vertices $\text{s}=0$, $\text{c}=K$, $\text{d}=K+\mathrm{i}{K}^{\prime }$, $\text{n}=\mathrm{i}{K}^{\prime }$. The set of points $z=mK+n\mathrm{i}{K}^{\prime }$, $m,n\in \mathbb{Z}$, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by $mK+n\mathrm{i}{K}^{\prime }$, where again $m,n\in \mathbb{Z}$. ►

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Figure 22.4.2: $z$-plane. Fundamental unit cell. Magnify

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14: 1.13 Differential Equations

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Fundamental Pair

►Two solutions $w_1(z)$ and $w_2(z)$ are called a fundamental pair if any other solution $w(z)$ is expressible as …A fundamental pair can be obtained, for example, by taking any $z_0\in D$ and requiring that … ►The following three statements are equivalent: $w_1(z)$ and $w_2(z)$ comprise a fundamental pair in $D$; $𝒲\left\{w_1(z),w_2(z)\right\}$ does not vanish in $D$; $w_1(z)$ and $w_2(z)$ are linearly independent, that is, the only constants $A$ and $B$ such that … ►If $w_0(z)$ is any one solution, and $w_1(z)$, $w_2(z)$ are a fundamental pair of solutions of the corresponding homogeneous equation (1.13.1), then every solution of (1.13.8) can be expressed as …

15: 16.8 Differential Equations

… ►When no $b_j$ is an integer, and no two $b_j$ differ by an integer, a fundamental set of solutions of (16.8.3) is given by … ►When $p=q+1$, and no two $a_j$ differ by an integer, another fundamental set of solutions of (16.8.3) is given by … ►In this reference it is also explained that in general when $q>1$ no simple representations in terms of generalized hypergeometric functions are available for the fundamental solutions near $z=1$. …

16: 13.2 Definitions and Basic Properties

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§13.2(v) Numerically Satisfactory Solutions

►Fundamental pairs of solutions of (13.2.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are … ►A fundamental pair of solutions that is numerically satisfactory near the origin is … ►When $b=n+1=1,2,3,\mathrm{\dots }$, a fundamental pair that is numerically satisfactory near the origin is $M(a,n+1,z)$ and … ► …

17: 4.37 Inverse Hyperbolic Functions

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§4.37(v) Fundamental Property

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18: 8.22 Mathematical Applications

… ►plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon. …

19: 29.7 Asymptotic Expansions

… ►Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (29.2.1) (and (29.11.1)) as $\nu \to \mathrm{\infty }$, one in terms of Jacobian elliptic functions and two in terms of Hermite polynomials. …

20: 27.4 Euler Products and Dirichlet Series

… ►The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product. …