# integrals with respect to degree

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## 1—10 of 26 matching pages

##### 2: 7.23 Tables
• Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\operatorname{erf}z$, $x\in[0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of $\int_{x}^{\infty}e^{\pm\mathrm{i}t^{2}}\,\mathrm{d}t$, $(1/\sqrt{\pi})e^{\mp\mathrm{i}(x^{2}+(\pi/4))}\int_{x}^{\infty}e^{\pm\mathrm{i% }t^{2}}\,\mathrm{d}t$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

• ##### 3: Bibliography C
• H. S. Cohl (2010) Derivatives with respect to the degree and order of associated Legendre functions for $|z|>1$ using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.
• ##### 4: 18.33 Polynomials Orthogonal on the Unit Circle
A system of polynomials $\{\phi_{n}(z)\}$, $n=0,1,\dots$, where $\phi_{n}(z)$ is of proper degree $n$, is orthonormal on the unit circle with respect to the weight function $w(z)$ ($\geq 0$) if … Let $\{p_{n}(x)\}$ and $\{q_{n}(x)\}$, $n=0,1,\dots$, be OP’s with weight functions $w_{1}(x)$ and $w_{2}(x)$, respectively, on $(-1,1)$. … See Askey (1982a) and Pastro (1985) for special cases extending (18.33.13)–(18.33.14) and (18.33.15)–(18.33.16), respectively. … A system of monic polynomials $\{\Phi_{n}(z)\}$, $n=0,1,\dots$, where $\Phi_{n}(x)$ is of proper degree $n$, is orthogonal on the unit circle with respect to the measure $\mu$ if … with complex coefficients $c_{k}$ and of a certain degree $n$ define the reversed polynomial $p^{*}(z)$ by …
##### 5: 18.2 General Orthogonal Polynomials
A system (or set) of polynomials $\{p_{n}(x)\}$, $n=0,1,2,\ldots$, where $p_{n}(x)$ has degree $n$ as in §18.1(i), is said to be orthogonal on $(a,b)$ with respect to the weight function $w(x)$ ($\geq 0$) ifFor OP’s $\{p_{n}(x)\}$ on $\mathbb{R}$ with respect to an even weight function $w(x)$ we have … As a slight variant let $\{p_{n}(x)\}$ be OP’s with respect to an even weight function $w(x)$ on $(-1,1)$. … The monic OP’s $p_{n}(x)$ with respect to the measure $\,\mathrm{d}\mu(x)$ can be expressed in terms of the moments by …
##### 6: 12.10 Uniform Asymptotic Expansions for Large Parameter
In this section we give asymptotic expansions of PCFs for large values of the parameter $a$ that are uniform with respect to the variable $z$, when both $a$ and $z$ $(=x)$ are real. … where $u_{s}(t)$ and $v_{s}(t)$ are polynomials in $t$ of degree $3s$, ($s$ odd), $3s-2$ ($s$ even, $s\geq 2$). … and the coefficients $\mathsf{A}_{s}(\tau)$ are the product of $\tau^{s}$ and a polynomial in $\tau$ of degree $2s$. … The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when $\mu\to\infty$ uniformly with respect to $t\in[1+\delta,\infty)$. … where $\xi,\eta$ are given by (12.10.7), (12.10.23), respectively, and …
##### 7: 8.20 Asymptotic Expansions of $E_{p}\left(z\right)$
###### §8.20(i) Large $z$
Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). …
###### §8.20(ii) Large $p$
so that $A_{k}(\lambda)$ is a polynomial in $\lambda$ of degree $k-1$ when $k\geq 1$. …
Let $\{p_{n}\}$ denote the set of monic polynomials $p_{n}$ of degree $n$ (coefficient of $x^{n}$ equal to $1$) that are orthogonal with respect to a positive weight function $w$ on a finite or infinite interval $(a,b)$; compare §18.2(i). …As a consequence, the rule is exact for any polynomial $f(x)$ of degree $\leq 2n-1$, that is, …In particular, with $h_{m}=\int_{a}^{b}p_{m}(x)^{2}w(x)\,\mathrm{d}x$, we have a finite system of orthogonal polynomials $p_{m}(x)$ ($m=0,1,\ldots,n-1$) on $\{x_{1},x_{2},\ldots,x_{n}\}$ with respect to the weights $w_{k}$: … The standard Monte Carlo method samples points uniformly from the integration region to estimate the integral and its error. In more advanced methods points are sampled from a probability distribution, so that they are concentrated in regions that make the largest contribution to the integral. …
14.6.1 $\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{% \mathrm{d}}^{m}\mathsf{P}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}},$
14.6.2 $\mathsf{Q}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{% \mathrm{d}}^{m}\mathsf{Q}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}.$
14.6.3 $P^{m}_{\nu}\left(x\right)=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}P_{% \nu}\left(x\right)}{{\mathrm{d}x}^{m}},$
14.6.4 $Q^{m}_{\nu}\left(x\right)=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}Q_{% \nu}\left(x\right)}{{\mathrm{d}x}^{m}},$
14.6.5 ${\left(\nu+1\right)_{m}}\boldsymbol{Q}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(x% ^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}\boldsymbol{Q}_{\nu}\left(x\right)}{{% \mathrm{d}x}^{m}}.$
For fixed $\gamma$, as $z\to\infty$ in the sector $|\operatorname{ph}z|\leq\pi-\delta$ ($<\pi$), …
30.11.10 $K_{n}^{m}(\gamma)=\frac{\sqrt{\pi}}{2}\left(\frac{\gamma}{2}\right)^{m}\frac{(% -1)^{m}a_{n,\frac{1}{2}(m-n)}^{-m}(\gamma^{2})}{\Gamma\left(\frac{3}{2}+m% \right)A_{n}^{-m}(\gamma^{2})\mathsf{Ps}^{m}_{n}\left(0,\gamma^{2}\right)},$ $n-m$ even,
30.11.11 $K_{n}^{m}(\gamma)=\frac{\sqrt{\pi}}{2}\left(\frac{\gamma}{2}\right)^{m+1}\*% \frac{(-1)^{m}a_{n,\frac{1}{2}(m-n+1)}^{-m}(\gamma^{2})}{\Gamma\left(\frac{5}{% 2}+m\right)A_{n}^{-m}(\gamma^{2})(\left.\ifrac{\mathrm{d}\mathsf{Ps}^{m}_{n}(z% ,\gamma^{2})}{\mathrm{d}z}\right|_{z=0})},$ $n-m$ odd.
30.11.12 $A_{n}^{-m}(\gamma^{2})S^{m(1)}_{n}\left(z,\gamma\right)=\frac{1}{2}{\mathrm{i}% }^{m+n}\gamma^{m}\frac{(n-m)!}{(n+m)!}z^{m}(1-z^{-2})^{\frac{1}{2}m}\*\int_{-1% }^{1}e^{-\mathrm{i}\gamma zt}(1-t^{2})^{\frac{1}{2}m}\mathsf{Ps}^{m}_{n}\left(% t,\gamma^{2}\right)\,\mathrm{d}t.$