fundamental theorem of calculus

SearchAdvancedHelp

(0.001 seconds)

31—40 of 146 matching pages

31: 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. … ►

…

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

…

32: 19.35 Other Applications

… ►

§19.35(i) Mathematical

►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). …

33: 22.4 Periods, Poles, and Zeros

… ►

§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 $m,n\in \mathbb{Z}$. ►

►►

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

…

34: Bibliography S

… ►

Sturm oscillation and comparison theorems. In Sturm-Liouville theory, pp. 29–43.

ⓘ

External Links:

Document, Link, MathReview Entry

Referenced by:

§18.36(vi), §18.39(i).

Encodings:

BibTeX

►

Szegő’s Theorem and Its Descendants. Spectral Theory for $L^2$ Perturbations of Orthogonal Polynomials. M. B. Porter Lectures, Princeton University Press, Princeton, NJ.

ⓘ

External Links:

ISBN 978-0-691-14704-8, MathReview (Harry Dym)

Referenced by:

§18.2(xi).

Encodings:

BibTeX

… ►

Relations among the fundamental solutions of the generalized hypergeometric equation when $p=q+1$. II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.

ⓘ

External Links:

Document, MathReview (C. S. Meijer), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

…

35: 1.13 Differential Equations

… ►

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 …

36: 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 $z=1$. …

37: 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 … ►An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with $L^2$ eigenfunctions vanishing at the end points, in this case $\pm \mathrm{\infty }$ see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem. …Both satisfy Sturm’s theorem. … ►

►►

Figure 18.39.1: Graphs of the first and fourth excited state eigenfunctions of the harmonic oscillator, for $\mathrm{\hslash }=k=m=1$, of (18.39.13), in ${\psi }_1(x)$, ${\psi }_4(x)$ and those of the rational potential of (18.39.19), in ${\widehat{\psi }}_3(x)$, ${\widehat{\psi }}_6(x)$. Both sets satisfy the Sturm oscillation theorem. Magnify

…

38: 13.2 Definitions and Basic Properties

… ►

§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 … ► …

39: 4.37 Inverse Hyperbolic Functions

… ►

§4.37(v) Fundamental Property

…

40: 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. …