change of order of integration

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11: 2.11 Remainder Terms; Stokes Phenomenon

… ►By integration by parts (§2.3(i)) … ►That the change in their forms is discontinuous, even though the function being approximated is analytic, is an example of the Stokes phenomenon. Where should the change-over take place? Can it be accomplished smoothly? … ►For higher-order Stokes phenomena see Olde Daalhuis (2004b) and Howls et al. (2004). … ►For higher-order differential equations, see Olde Daalhuis (1998a, b). …

12: 19.25 Relations to Other Functions

… ►then the five nontrivial permutations of $x,y,z$ that leave $R_F$ invariant change $k^2$ ($=(z-y)/(z-x)$) into $1/k^2$, $k_{}^{\prime }{}_{}{}^2$, $1/k_{}^{\prime }{}_{}{}^2$, $-k^2/k_{}^{\prime }{}_{}{}^2$, $-k_{}^{\prime }{}_{}{}^2/k^2$, and $\sin \varphi$ ($=\sqrt{(z-x)/z}$) into $k\sin \varphi$, $-\mathrm{i}\tan \varphi$, $-\mathrm{i}{k}^{\prime }\tan \varphi$, $(k^{\prime }\sin \varphi )/\sqrt{1-k^2{\sin }^2\varphi }$, $-\mathrm{i}k\sin \varphi /\sqrt{1-k^2{\sin }^2\varphi }$. … ►The three changes of parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ in §19.7(iii) are unified in (19.21.12) by way of (19.25.14). … ►The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which , for some $j$. … ►The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which , for some $j$. …

13: 3.5 Quadrature

§3.5 Quadrature

… ►

§3.5(iii) Romberg Integration

►Further refinements are achieved by Romberg integration. … ►For these cases the integration path may need to be deformed; see §3.5(ix). … ►With $N$ function values, the Monte Carlo method aims at an error of order $1/\sqrt{N}$, independently of the dimension of the domain of integration. …

14: 3.7 Ordinary Differential Equations

… ►Consideration will be limited to ordinary linear second-order differential equations … ►($𝐈$ and $\mathrm{𝟎}$ being the identity and zero matrices of order $2\times 2$.) … ►

First-Order Equations

►For $w^{\prime }=f(z,w)$ the standard fourth-order rule reads … ►

Second-Order Equations

…

15: Bibliography B

… ►

Algebro-geometric Approach to Nonlinear Integrable Problems. Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-50265-3, ZentralBlatt

Referenced by:

§21.1, 1st item, §21.6(ii), §21.7(i), §21.7(i), §21.9, Ch.21, Profile Alexander I. Bobenko, Profile Alexander A. Its, Notations T.

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A general program to calculate atomic continuum processes using the R-matrix method. Comput. Phys. Comm. 8 (3), pp. 149–198.

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External Links:

Document, Code

Referenced by:

2nd item.

Encodings:

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Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and Procrustes analysis. J. Multivariate Anal. 41 (2), pp. 314–337.

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External Links:

ISSN 0047-259X, Document, MathReview (A. L. Rukhin), ZentralBlatt

Referenced by:

§35.10, §35.9.

Encodings:

BibTeX

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Constant mean curvature surfaces and integrable equations. Uspekhi Mat. Nauk 46 (4(280)), pp. 3–42, 192 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 46(1991), no. 4, pp. 1–45

External Links:

ISSN 0042-1316, MathReview (P. E. Markov), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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Integrable Hamiltonian systems and the Painlevé property. Phys. Rev. A (3) 25 (3), pp. 1257–1264.

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External Links:

ISSN 0556-2791, Document, MathReview

Referenced by:

§32.16.

Encodings:

BibTeX

…

16: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

§1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►We integrate by parts twice giving: … ►

§1.18(iv) Formally Self-adjoint Linear Second Order Differential Operators

… ►In general, operators $T$ being formally self-adjoint second order differential operators of the form (1.18.28), with $X$ unbounded, will have both a continuous and a point spectrum, and thus, correspondingly, $non-L^2\left(X\right)$ eigenfunctions as in §1.18(vi) and $L^2\left(X\right)$ eigenfunctions as in §1.18(v). … ►For a formally self-adjoint second order differential operator $ℒ$, such as that of (1.18.28), the space $𝒟({ℒ}^{\ast })$ can be seen to consist of all $f\in L^2\left(X\right)$ such that the distribution $ℒf$ can be identified with a function in $L^2\left(X\right)$, which is the function ${ℒ}^{\ast }f$. …

17: 2.8 Differential Equations with a Parameter

… ►For example, $u$ can be the order of a Bessel function or degree of an orthogonal polynomial. … ►This introduces new variables $W$ and $\xi$, related by … ►(the constants of integration being arbitrary). … ►

§2.8(iv) Case III: Simple Pole

… ►Lastly, for an example of a fourth-order differential equation, see Wong and Zhang (2007). …

18: 1.10 Functions of a Complex Variable

… ►and the integration contour is described once in the positive sense. … ►If $-n$ is the first negative integer (counting from $-\mathrm{\infty }$) with $a_{-n}\ne 0$, then $z_0$ is a pole of order (or multiplicity) $n$. … ►where $N$ and $P$ are respectively the numbers of zeros and poles, counting multiplicity, of $f$ within $C$, and ${\mathrm{\Delta }}_C(\mathrm{ph}f(z))$ is the change in any continuous branch of $\mathrm{ph}\left(f(z)\right)$ as $z$ passes once around $C$ in the positive sense. … ►(For example, when $\mu$ is an integer $f(z)-f(z_0)$ has a zero of order $\mu$ at $z_0$.) … ►is analytic in $D$ and its derivatives of all orders can be found by differentiating under the sign of integration. …

19: 25.11 Hurwitz Zeta Function

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25.11.7 $$\zeta (s,a)=\frac{1}{a^s}+\frac{1}{{(1+a)}^s}\left(\frac{1}{2}+\frac{1+a}{s-1}\right)+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}\frac{1}{{(1+a)}^{s+2k-1}}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{{(x+a)}^{s+2n+1}}dx,$$ $s\ne 1$, $a>0$, $n=1,2,3,\mathrm{\dots }$, $\mathrm{\Re }s>-2n$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Proof sketch:

Derivable from (25.11.5) by setting $N=1$ and repeatedly integrating by parts using (1.2.6), (24.2.2), (24.2.4), (24.2.11), (24.2.12), (24.4.34).

Permalink:

http://dlmf.nist.gov/25.11.E7

Encodings:

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See also:

Annotations for §25.11(iii), §25.11 and Ch.25

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25.11.15 $$\zeta (s,ka)=k^{-s}\sum _{n=0}^{k-1}\zeta (s,a+\frac{n}{k}),$$ $s\ne 1$, $k=1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable

Keywords:

special value

Proof sketch:

Derivable by starting with (25.11.1), replacing $a\mapsto ka$, pulling out a factor of $k^{-s}$ and performing a change of sum index $n=km+l$ with $m\ge 0$ and $0\le l\le k-1$, which through Euclidean division converts the single sum into a double sum of the correct form.

Referenced by:

(25.11.24)

Permalink:

http://dlmf.nist.gov/25.11.E15

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See also:

Annotations for §25.11(v), §25.11 and Ch.25

… ►where the integration contour is a loop around the negative real axis as described for (25.5.20). … ►

25.11.31 $$\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{2\cosh x}dx=4^{-s}\left(\zeta (s,\frac{1}{4}+\frac{1}{4}a)-\zeta (s,\frac{3}{4}+\frac{1}{4}a)\right),$$ $\mathrm{\Re }s>0$, $\mathrm{\Re }a>-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, integral representation

Proof sketch:

The improper integral can be evaluated by using $2\cosh x={\mathrm{e}}^x(1+{\mathrm{e}}^{-2x})$, changing variables $2x=t$, and then applying (25.11.35).

Permalink:

http://dlmf.nist.gov/25.11.E31

Encodings:

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See also:

Annotations for §25.11(viii), §25.11 and Ch.25

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25.11.41 $$\zeta (s,a+1)=\zeta \left(s\right)-s\zeta \left(s+1\right)a+O\left(a^2\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $a$: real or complex parameter and $s$: complex variable

Keywords:

asymptotic approximation

Source:

Apostol (1952, (15), p. 6)

Permalink:

http://dlmf.nist.gov/25.11.E41

Encodings:

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See also:

Annotations for §25.11(xii), §25.11 and Ch.25

…

20: 1.8 Fourier Series

… ►where $f(x)$ is square-integrable on $[-\pi ,\pi ]$ and $a_n,b_n,c_n$ are given by (1.8.2), (1.8.4). If $g(x)$ is also square-integrable with Fourier coefficients $a_n^{\prime },b_n^{\prime }$ or $c_n^{\prime }$ then … ►Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. … ►

§1.8(iii) Integration and Differentiation

… ►Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $\left|f^{\prime \prime }(x)\right|$ are integrable over $(-\mathrm{\infty },\mathrm{\infty })$. …