integrals with respect to degree

SearchAdvancedHelp

(0.005 seconds)

11—20 of 26 matching pages

11: 22.18 Mathematical Applications

… ►With $k\in [0,1]$ the mapping $z\to w=\mathrm{sn}(z,k)$ gives a conformal map of the closed rectangle $[-K,K]\times [0,K^{\prime }]$ onto the half-plane $\mathrm{\Im }w\ge 0$, with $0,\pm K,\pm K+\mathrm{i}{K}^{\prime },\mathrm{i}{K}^{\prime }$ mapping to $0,\pm 1,\pm k^{-2},\mathrm{\infty }$ respectively. The half-open rectangle $(-K,K)\times [-K^{\prime },K^{\prime }]$ maps onto $ℂ$ cut along the intervals $(-\mathrm{\infty },-1]$ and $[1,\mathrm{\infty })$. … ►Algebraic curves of the form $y^2=P(x)$, where $P$ is a nonsingular polynomial of degree 3 or 4 (see McKean and Moll (1999, §1.10)), are elliptic curves, which are also considered in §23.20(ii). …a construction due to Abel; see Whittaker and Watson (1927, pp. 442, 496–497). …The existence of this group structure is connected to the Jacobian elliptic functions via the differential equation (22.13.1). …

12: 18.27 $q$-Hahn Class

… ►These families depend on further parameters, in addition to $q$. … ►Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval. … ►

From Big $q$-Jacobi to Jacobi

… ►

From Big $q$-Jacobi to Little $q$-Jacobi

… ►

From Little $q$-Jacobi to Jacobi

…

13: 18.30 Associated OP’s

… ►However, if the recurrence coefficients are polynomial, or rational, functions of $n$, polynomials of degree $n$ may be well defined for $c\in ℝ$ provided that $A_{n+c}{B}_{n+c}\ne 0,n=0,1,\mathrm{\dots }$ Askey and Wimp (1984). … ►These constraints guarantee that the orthogonality only involves the integral $x\in [0,\mathrm{\infty })$, as above. ►For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real $x$-axis each multiplied by the polynomial product evaluated at the corresponding values of $x$, as in (18.2.3). … ►The corresponding results for $c=0$ appear as (18.21.12) and (18.21.13), respectively. … ►In the monic case, the monic associated polynomials ${\widehat{p}}_n(x;c)$ of order $c$ with respect to the ${\widehat{p}}_n(x)$ are obtained by simply changing the initialization and recursions, respectively, of (18.30.2) and (18.30.3) to …

14: 29.12 Definitions

… ►

§29.12(i) Elliptic-Function Form

… ►The superscript $m$ on the left-hand sides of (29.12.1)–(29.12.8) agrees with the number of $z$-zeros of each Lamé polynomial in the interval $(0,K)$, while $n-m$ is the number of $z$-zeros in the open line segment from $K$ to $K+\mathrm{i}{K}^{\prime }$. … ►In the fourth column the variable $z$ and modulus $k$ of the Jacobian elliptic functions have been suppressed, and $P({\mathrm{sn}}^2)$ denotes a polynomial of degree $n$ in ${\mathrm{sn}}^2(z,k)$ (different for each type). … ►The polynomial $P(\xi )$ is of degree $n$ and has $m$ zeros (all simple) in $(0,1)$ and $n-m$ zeros (all simple) in $(1,k^{-2})$. … ►This result admits the following electrostatic interpretation: Given three point masses fixed at $t=0$, $t=1$, and $t=k^{-2}$ with positive charges $\rho +\frac{1}{4}$, $\sigma +\frac{1}{4}$, and $\tau +\frac{1}{4}$, respectively, and $n$ movable point masses at $t_1,t_2,\mathrm{\dots },t_n$ arranged according to (29.12.12) with unit positive charges, the equilibrium position is attained when $t_j={\xi }_j$ for $j=1,2,\mathrm{\dots },n$.

15: 2.8 Differential Equations with a Parameter

… ►dots denoting differentiations with respect to $\xi$. Then … ►The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to $\xi$. … ►The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to $\xi$. … ►The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to $\xi$. …

16: 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. When $n>0$ and $0\le j\le n$, $0\le k\le n$, … ► … ►Splines are defined piecewise and usually by low-degree polynomials. …

17: 21.7 Riemann Surfaces

… ►

§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces

… ►Although there are other ways to represent Riemann surfaces (see e. …To accomplish this we write (21.7.1) in terms of homogeneous coordinates: … ►is a Riemann matrix and it is used to define the corresponding Riemann theta function. … ►where $Q(\lambda )$ is a polynomial in $\lambda$ of odd degree $2g+1$ $(\ge 5)$. …

18: 19.36 Methods of Computation

… ►For $R_F$ the polynomial of degree 7, for example, is … ►For both $R_D$ and $R_J$ the factor ${(r/4)}^{-1/6}$ in Carlson (1995, (2.18)) is changed to ${(r/5)}^{-1/8}$ when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms: … ►Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. … ►As $n\to \mathrm{\infty }$, $c_n$, $a_n$, and $t_n$ converge quadratically to limits $0$, $M$, and $T$, respectively; hence … ►Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). …

19: Bibliography S

… ►

A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.

ⓘ

External Links:

ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

… ►

On the derivative of the Legendre function of the first kind with respect to its degree. J. Phys. A 39 (49), pp. 15147–15172.

ⓘ

External Links:

ISSN 0305-4470, Document, FullText, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

►

On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.

ⓘ

External Links:

ISSN 0259-9791, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

►

On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.

ⓘ

External Links:

ISSN 0259-9791, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

►

On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Anal. Appl. 386 (1), pp. 332–342.

ⓘ

External Links:

ISSN 0022-247X, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

20: 18.28 Askey–Wilson Class

… ►The Askey–Wilson polynomials form a system of OP’s $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. The $q$-Racah polynomials form a system of OP’s $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots },N$, that are orthogonal with respect to a weight function on a sequence $\{q^{-y}+c{q}^{y+1}\}$, $y=0,1,\mathrm{\dots },N$, with $c$ a constant. … ►Specialization to continuous $q$-ultraspherical: … ►

From Askey–Wilson to Wilson

… ►

From $q$-Racah to Racah

…