with respect to integration

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41: 32.10 Special Function Solutions

… ►

32.10.8 $$w(z;n+\frac{1}{2})=\frac{d}{dz}\left(\ln \left(\frac{{\tau }_n(z)}{{\tau }_{n+1}(z)}\right)\right),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function, $n$: integer, $z$: real and ${\tau }_n(z)$: determinant

Permalink:

http://dlmf.nist.gov/32.10.E8

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Annotations for §32.10(ii), §32.10 and Ch.32

… ►When $a+\frac{1}{2}$ is zero or a negative integer the $U$ parabolic cylinder functions reduce to Hermite polynomials (§18.3) times an exponential function; thus … ►

32.10.22

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $z$: real, $\epsilon =\pm 1$ and $C$: constant

Permalink:

http://dlmf.nist.gov/32.10.E22

Encodings:

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Annotations for §32.10(iv), §32.10 and Ch.32

… ►

32.10.33 $$z(1-z)\frac{d^2\varphi }{{dz}^2}+(1-2z)\frac{d\varphi }{dz}-\frac{1}{4}\varphi =0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: real and $\varphi (z)$: function

Permalink:

http://dlmf.nist.gov/32.10.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §32.10(vi), §32.10 and Ch.32

… ►The solution (32.10.34) is an essentially transcendental function of both constants of integration since ${\text{P}}_{\text{VI}}$ with $\alpha =\beta =\gamma =0$ and $\delta =\frac{1}{2}$ does not admit an algebraic first integral of the form $P(z,w,w^{\prime },C)=0$, with $C$ a constant. …

42: 3.7 Ordinary Differential Equations

… ►Consideration will be limited to ordinary linear second-order differential equations …For applications to special functions $f$, $g$, and $h$ are often simple rational functions. … ►Assume that we wish to integrate (3.7.1) along a finite path $𝒫$ from $z=a$ to $z=b$ in a domain $D$. … ►The larger the absolute values of the eigenvalues ${\lambda }_k$ that are being sought, the smaller the integration steps $\left|{\tau }_j\right|$ need to be. … ►The Runge–Kutta method applies to linear or nonlinear differential equations. …

43: 36.12 Uniform Approximation of Integrals

… ►In the cuspoid case (one integration variable) … ►Define a mapping $u(t;𝐲)$ by relating $f(u;𝐲)$ to the normal form (36.2.1) of ${\mathrm{\Phi }}_K(t;𝐱)$ in the following way: … ►This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes ${\mathrm{\Psi }}_K(𝐱;k)$ in (36.2.10) away from $𝐱=\mathrm{𝟎}$, in terms of canonical integrals ${\mathrm{\Psi }}_J\left(\xi (𝐱;k)\right)$ for . For example, the diffraction catastrophe ${\mathrm{\Psi }}_2(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function ${\mathrm{\Psi }}_1\left(\xi (x,y;k)\right)$ when $k$ is large, provided that $x$ and $y$ are not small. … ►Also, ${\mathrm{\Delta }}^{1/4}/\sqrt{f_{+}^{\prime \prime }}$ and ${\mathrm{\Delta }}^{1/4}/\sqrt{-f_{-}^{\prime \prime }}$ are chosen to be positive real when $y$ is such that both critical points are real, and by analytic continuation otherwise. …

44: 2.10 Sums and Sequences

… ►Assume that $a,m$, and $n$ are integers such that $n>a$, $m>0$, and $f^{(2m)}(x)$ is absolutely integrable over $[a,n]$. Then … ►

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 $\mathrm{\Re }z\in [a,n]$.

… ►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). … ►In these circumstances the integrals in (2.10.28) are integrable by parts $m$ times, yielding …

45: 1.6 Vectors and Vector-Valued Functions

… ►where $𝐧$ is the unit vector normal to $𝐚$ and $𝐛$ whose direction is determined by the right-hand rule; see Figure 1.6.1. … ►If $h(a)=a^{\prime }$ and $h(b)=b^{\prime }$, then the reparametrization is called orientation-preserving, and …If $h(a)=b^{\prime }$ and $h(b)=a^{\prime }$, then the reparametrization is orientation-reversing and … ►are tangent to the surface at $𝚽(u_0,v_0)$. … ►If ${𝚽}_1$ and ${𝚽}_2$ are both orientation preserving or both orientation reversing parametrizations of $S$ defined on open sets $D_1$ and $D_2$ respectively, then …

46: 21.7 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: … ►Note that for the purposes of integrating these holomorphic differentials, all cycles on the surface are a linear combination of the cycles $a_j$, $b_j$, $j=1,2,\mathrm{\dots },g$. … ►where $P_1$ and $P_2$ are points on $\mathrm{\Gamma }$, $𝝎=({\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_g)$, and the path of integration on $\mathrm{\Gamma }$ from $P_1$ to $P_2$ is identical for all components. … ►where again all integration paths are identical for all components. …

47: 10.43 Integrals

… ►

10.43.15 $${\mathrm{Ki}}_{-n}\left(x\right)={(-1)}^n\frac{d^n}{{dx}^n}{K}_0\left(x\right),$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

${\mathrm{Ki}}_{\alpha }\left(x\right)$: Bickley function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $n$: integer and $x$: real variable

A&S Ref:

11.2.9

Permalink:

http://dlmf.nist.gov/10.43.E15

Encodings:

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See also:

Annotations for §10.43(iii), §10.43(iii), §10.43 and Ch.10

►

10.43.16 $${\mathrm{Ki}}_{\alpha }\left(0\right)=\frac{\sqrt{\pi }\mathrm{\Gamma }\left(\frac{1}{2}\alpha \right)}{2\mathrm{\Gamma }\left(\frac{1}{2}\alpha +\frac{1}{2}\right)},$$ $\alpha \ne 0,-2,-4,\mathrm{\dots }$.

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Symbols:

${\mathrm{Ki}}_{\alpha }\left(x\right)$: Bickley function, $\mathrm{\Gamma }\left(z\right)$: gamma function and $\pi$: the ratio of the circumference of a circle to its diameter

A&S Ref:

11.2.12, 11.2.13 (with less restrictive conditions)

Referenced by:

§10.43(iii)

Permalink:

http://dlmf.nist.gov/10.43.E16

Encodings:

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Annotations for §10.43(iii), §10.43(iii), §10.43 and Ch.10

… ►

10.43.18 $${\int }_0^{\mathrm{\infty }}{K}_{\nu }\left(t\right)dt=\frac{1}{2}\pi \sec \left(\frac{1}{2}\pi \nu \right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $\sec z$: secant function and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.43.E18

Encodings:

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See also:

Annotations for §10.43(iv), §10.43 and Ch.10

… ►

On the interval , $x^{-1}g(x)$ is continuously differentiable and each of $xg(x)$ and $xd(x^{-1}g(x))/dx$ is absolutely integrable.

… ►For collections of integrals of the functions $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$, including integrals with respect to the order, see Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §§5.5, 6.5–6.7), Gröbner and Hofreiter (1950, pp. 197–203), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1972), Oberhettinger (1974, §§1.11 and 2.7), Oberhettinger (1990, §§1.17–1.20 and 2.17–2.20), Oberhettinger and Badii (1973, §§1.15 and 2.13), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.11–1.12, 2.15–2.16, 3.2.8–3.2.10, and 3.4.1), Prudnikov et al. (1992a, §§3.15, 3.16), Prudnikov et al. (1992b, §§3.15, 3.16), Watson (1944, Chapter 13), and Wheelon (1968).

48: 2.1 Definitions and Elementary Properties

… ►

§2.1(ii) Integration and Differentiation

►Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. … ►The asymptotic property may also hold uniformly with respect to parameters. …as $x\to \mathrm{\infty }$ in $𝐗$, uniformly with respect to $u\in 𝐔$. … ►As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. …

49: 9.13 Generalized Airy Functions

… ►are used in approximating solutions to differential equations with multiple turning points; see §2.8(v). … ►When $n=1$, $A_n\left(z\right)$ and $B_n\left(z\right)$ become $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$, respectively. … ►As $z\to \mathrm{\infty }$ … ►(The overbar has nothing to do with complex conjugates.) … ►The integration paths ${ℒ}_0$, ${ℒ}_1$, ${ℒ}_2$, ${ℒ}_3$ are depicted in Figure 9.13.1. …

50: 10.22 Integrals

… ►In this subsection ${𝒞}_{\nu }\left(z\right)$ and ${𝒟}_{\mu }(z)$ denote cylinder functions(§10.2(ii)) of orders $\nu$ and $\mu$, respectively, not necessarily distinct. … ►

10.22.15 $${\int }_0^{\pi }{J}_{2\nu }\left(2z\sin \theta \right)\sin \left(2\mu \theta \right)d\theta =\pi \sin \left(\mu \pi \right)J_{\nu +\mu }\left(z\right)J_{\nu -\mu }\left(z\right),$$ $\mathrm{\Re }\nu >-1$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.22(ii)

Permalink:

http://dlmf.nist.gov/10.22.E15

Encodings:

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See also:

Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

►

10.22.16 $${\int }_0^{\frac{1}{2}\pi }{J}_0\left(2z\sin \theta \right)\cos \left(2n\theta \right)d\theta =\frac{1}{2}\pi J_n^2\left(z\right),$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $n$: integer and $z$: complex variable

Referenced by:

§10.22(ii)

Permalink:

http://dlmf.nist.gov/10.22.E16

Encodings:

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See also:

Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

… ►For the Ferrers function $𝖯$ and the associated Legendre function $Q$, see §§14.3(i) and 14.3(ii), respectively. … ►For collections of integrals of the functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, and $H_{\nu }^{(2)}\left(z\right)$, including integrals with respect to the order, see Andrews et al. (1999, pp. 216–225), Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §§5.5 and 6.5–6.7), Gröbner and Hofreiter (1950, pp. 196–204), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1974, §§1.10 and 2.7), Oberhettinger (1990, §§1.13–1.16 and 2.13–2.16), Oberhettinger and Badii (1973, §§1.14 and 2.12), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.8–1.10, 2.12–2.14, 3.2.4–3.2.7, 3.3.2, and 3.4.1), Prudnikov et al. (1992a, §§3.12–3.14), Prudnikov et al. (1992b, §§3.12–3.14), Watson (1944, Chapters 5, 12, 13, and 14), and Wheelon (1968).