# integrals with respect to degree

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## 11—20 of 22 matching pages

##### 11: 22.18 Mathematical Applications

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►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 $\u2102$ cut along the intervals $(-\mathrm{\infty},-1]$ and $[1,\mathrm{\infty})$.
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►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: 2.8 Differential Equations with a Parameter

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►dots denoting differentiations with respect to
$\xi $.
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►The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to
$\xi $.
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►The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to
$\xi $.
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►The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to
$\xi $.
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►The expansions (2.8.29) and (2.8.30) are both uniform and differentiable with respect to
$\xi $.
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##### 13: 29.12 Definitions

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###### §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 $j=1,2,\mathrm{\dots},n$.##### 14: 3.11 Approximation Techniques

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►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 $0\le j\le n$, $0\le k\le n$,
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►Splines are defined piecewise and usually by low-degree polynomials.
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##### 15: 21.7 Riemann Surfaces

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###### §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 $2g+1$ $(\ge 5)$. …##### 16: 19.36 Methods of Computation

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►For ${R}_{F}$ the polynomial of degree 7, for example, is
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►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:
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►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.
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►As $n\to \mathrm{\infty}$, ${c}_{n}$, ${a}_{n}$, and ${t}_{n}$ converge quadratically to limits $0$, $M$, and $T$, respectively; hence
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►Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)).
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##### 17: Bibliography S

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Complements to asymptotic development of sine cosine integrals, and auxiliary functions.
Univ. Beograd. Publ. Elecktrotehn. Fak., Ser. Mat. Fiz. 461–497, pp. 185–191.
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On the derivative of the Legendre function of the first kind with respect to its degree.
J. Phys. A 39 (49), pp. 15147–15172.
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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.
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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.
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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.

##### 18: 18.28 Askey–Wilson Class

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►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.
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###### §18.28(ii) Askey–Wilson Polynomials

… ►###### §18.28(viii) $q$-Racah Polynomials

… ►For ${\omega}_{y}$ and ${h}_{n}$ see Koekoek et al. (2010, Eq. (14.2.2)).##### 19: 22.19 Physical Applications

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►This formulation gives the bounded and unbounded solutions from the same formula (22.19.3), for $k\ge 1$ and $k\le 1$, respectively.
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►Its dynamics for purely imaginary time is connected to the theory of instantons (Itzykson and Zuber (1980, p. 572), Schäfer and Shuryak (1998)), to WKB theory, and to large-order perturbation theory (Bender and Wu (1973), Simon (1982)).
►For $\beta $ real and positive, three of the four possible combinations of signs give rise to bounded oscillatory motions.
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###### §22.19(iv) Tops

… ►Hyperelliptic functions $u(z)$ are solutions of the equation $z={\int}_{0}^{u}{(f(x))}^{-1/2}dx$, where $f(x)$ is a polynomial of degree higher than 4. …##### 20: 31.15 Stieltjes Polynomials

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►where $\mathrm{\Phi}(z)$ is a polynomial of degree not exceeding $N-2$.
There exist at most $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ polynomials $V(z)$ of degree not exceeding $N-2$ such that for $\mathrm{\Phi}(z)=V(z)$, (31.15.1) has a polynomial solution $w=S(z)$ of degree
$n$.
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►If ${z}_{1},{z}_{2},\mathrm{\dots},{z}_{n}$ are the zeros of an $n$th degree Stieltjes polynomial $S(z)$, then every zero ${z}_{k}$ is either one of the parameters ${a}_{j}$ or a solution of the system of equations
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►If ${t}_{k}$ is a zero of the Van Vleck polynomial $V(z)$, corresponding to an $n$th degree Stieltjes polynomial $S(z)$, and ${z}_{1}^{\prime},{z}_{2}^{\prime},\mathrm{\dots},{z}_{n-1}^{\prime}$ are the zeros of ${S}^{\prime}(z)$ (the derivative of $S(z)$), then ${t}_{k}$ is either a zero of ${S}^{\prime}(z)$ or a solution of the equation
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►with respect to the inner product
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