integrals with respect to degree

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21: 22.19 Physical Applications

… ►The period is $4K\left(\sin \left(\frac{1}{2}\alpha \right)\right)$. … ►This formulation gives the bounded and unbounded solutions from the same formula (22.19.3), for $k\ge 1$ and $k\le 1$, respectively. … ►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)). … ►

§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. …

22: 31.15 Stieltjes Polynomials

… ►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 $n$. … ►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 … ►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 … ►with respect to the inner product …

23: 4.13 Lambert $W$-Function

… ►The other branches $W_k\left(z\right)$ are single-valued analytic functions on $ℂ\setminus (-\mathrm{\infty },0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. … ►in which the $p_n(x)$ are polynomials of degree $n$ with … ►As $\left|z\right|\to \mathrm{\infty }$ …As $x\to 0-$ … ►For these and other integral representations of the Lambert $W$-function see Kheyfits (2004), Kalugin et al. (2012) and Mező (2020). …

24: 2.10 Sums and Sequences

… ►

On the strip $a\le \mathrm{\Re }z\le n$, $f(z)$ is analytic in its interior, $f^{(2m)}(z)$ is continuous on its closure, and $f(z)=o\left({\mathrm{e}}^{2\pi |\mathrm{\Im }z|}\right)$ as $\mathrm{\Im }z\to \pm \mathrm{\infty }$, uniformly with respect to $\mathrm{\Re }z\in [a,n]$.

… ►For an extension to integrals with Cauchy principal values see Elliott (1998). … ►This identity can be used to find asymptotic approximations for large $n$ when the factor $v_j$ changes slowly with $j$, and $u_j$ is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i). … ►(5.11.7) shows that the integrals around the large quarter circles vanish as $n\to \mathrm{\infty }$. … ►

Example

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25: 10.41 Asymptotic Expansions for Large Order

… ►As $\nu \to \mathrm{\infty }$ through positive real values, …Also, $U_k(p)$ and $V_k(p)$ are polynomials in $p$ of degree $3k$, given by $U_0(p)=V_0(p)=1$, and … ►To establish (10.41.12) we substitute into (10.34.3), with $m=0$ and $z$ replaced by $\nu z$, by means of (10.41.13) observing that when $|z|$ is large the effect of replacing $z$ by $z{\mathrm{e}}^{\pm \pi \mathrm{i}}$ is to replace $\eta$, ${(1+z^2)}^{\frac{1}{4}}$, and $p$ by $-\eta$, $\pm \mathrm{i}{(1+z^2)}^{\frac{1}{4}}$, and $-p$, respectively. … ►We first prove that for the expansions (10.20.6) for the Hankel functions $H_{\nu }^{(1)}\left(\nu z\right)$ and $H_{\nu }^{(2)}\left(\nu z\right)$ the $z$-asymptotic property applies when $z\to \pm \mathrm{i}\mathrm{\infty }$, respectively. … ►It needs to be noted that the results (10.41.14) and (10.41.15) do not apply when $z\to 0+$ or equivalently $\zeta \to +\mathrm{\infty }$. …

26: 18.39 Applications in the Physical Sciences

… ►For further details about the Schrödinger equation, including applications in physics and chemistry, see Gottfried and Yan (2004) and Pauling and Wilson (1985), respectively, among many others. … ►Kuijlaars and Milson (2015, §1) refer to these, in this case complex zeros, as exceptional, as opposed to regular, zeros of the EOP’s, these latter belonging to the (real) orthogonality integration range. … ►Analogous to (18.39.7) the 3D Schrödinger operator is … ►Orthogonality and normalization of eigenfunctions of this form is respect to the measure $r^{2}dr\sin \theta d\theta d\varphi$. … ►As this follows from the three term recursion of (18.39.46) it is referred to as the J-Matrix approach, see (3.5.31), to single and multi-channel scattering numerics. …