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31: 2.4 Contour Integrals

… ►Except that $\lambda$ is now permitted to be complex, with $\mathrm{\Re }\lambda >0$, we assume the same conditions on $q(t)$ and also that the Laplace transform in (2.3.8) converges for all sufficiently large values of $\mathrm{\Re }z$. …

32: 13.2 Definitions and Basic Properties

… ►except that $M(a,b,z)$ does not exist when $b$ is a nonpositive integer. … ► … ►Except when $z=0$ each branch of $U(a,b,z)$ is entire in $a$ and $b$. Unless specified otherwise, however, $U(a,b,z)$ is assumed to have its principal value. … ►Except when $a=0,-1,\mathrm{\dots }$ (polynomial cases), …

33: 6.18 Methods of Computation

… ►For small or moderate values of $x$ and $|z|$, the expansion in power series (§6.6) or in series of spherical Bessel functions (§6.10(ii)) can be used. …However, this problem is less severe for the series of spherical Bessel functions because of their more rapid rate of convergence, and also (except in the case of (6.10.6)) absence of cancellation when $z=x$ ($>0$). … ► $A_0$, $B_0$, and $C_0$ can be computed by Miller’s algorithm (§3.6(iii)), starting with initial values $(A_N,B_N,C_N)=(1,0,0)$, say, where $N$ is an arbitrary large integer, and normalizing via $C_0=1/z$. … ►Zeros of $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations. …

34: 1.5 Calculus of Two or More Variables

… ►

1.5.2

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Symbols:

$\left|x\right|$: absolute value of $x$

Referenced by:

§1.9(ii)

Permalink:

http://dlmf.nist.gov/1.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(i), §1.5 and Ch.1

… ►

1.5.23

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $ϵ$: positive number and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.5.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(iv), §1.5(iv), §1.5 and Ch.1

… ►

1.5.42 $${\iint }_{D}f(x,y)dxdy={\iint }_{D^{\ast }}f(x(u,v),y(u,v))\left|\frac{\partial (x,y)}{\partial (u,v)}\right|dudv,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.5.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(vi), §1.5(vi), §1.5 and Ch.1

►where $D$ is the image of $D^{\ast }$ under a mapping $(u,v)\to (x(u,v),y(u,v))$ which is one-to-one except perhaps for a set of points of area zero. …Again the mapping is one-to-one except perhaps for a set of points of volume zero. …

35: 27.3 Multiplicative Properties

… ►Except for $\nu \left(n\right)$, $\mathrm{\Lambda }\left(n\right)$, $p_n$, and $\pi \left(x\right)$, the functions in §27.2 are multiplicative, which means $f(1)=1$ and … ►If $f$ is multiplicative, then the values $f(n)$ for $n>1$ are determined by the values at the prime powers. …

36: 1.6 Vectors and Vector-Valued Functions

§1.6 Vectors and Vector-Valued Functions

… ►

§1.6(iii) Vector-Valued Functions

… ►The geometrical image $C$ of a path $𝐜$ is called a simple closed curve if $𝐜$ is one-to-one, with the exception $𝐜(a)=𝐜(b)$. … ►

Stokes’s Theorem

… ►

37: 25.10 Zeros

… ►Except for the trivial zeros, $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s\le 0$. … ►Calculations relating to the zeros on the critical line make use of the real-valued function …is chosen to make $Z(t)$ real, and $\mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)$ assumes its principal value. …

38: 18.15 Asymptotic Approximations

… ►With the exception of the penultimate paragraph, we assume throughout this subsection that $\alpha$, $\beta$, and $M$ ($=0,1,2,\mathrm{\dots }$) are all fixed. …When $\alpha ,\beta \in (-\frac{1}{2},\frac{1}{2})$, the error term in (18.15.1) is less than twice the first neglected term in absolute value, in which one has to take $\cos {\theta }_{n,m,\mathrm{\ell }}=1$. …

39: 15.19 Methods of Computation

… ►For $z\in ℂ$ it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. … ►Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. … ►Initial values for moderate values of $|a|$ and $|b|$ can be obtained by the methods of §15.19(i), and for large values of $|a|$, $|b|$, or $|c|$ via the asymptotic expansions of §§15.12(ii) and 15.12(iii). …

40: 13.14 Definitions and Basic Properties

… ►except that $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$. … ► … ►Except when $z=0$, each branch of the functions $M_{\kappa ,\mu }\left(z\right)/\mathrm{\Gamma }\left(2\mu +1\right)$ and $W_{\kappa ,\mu }\left(z\right)$ is entire in $\kappa$ and $\mu$. Also, unless specified otherwise $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are assumed to have their principal values. … ►Except when $\mu -\kappa =-\frac{1}{2},-\frac{3}{2},\mathrm{\dots }$ (polynomial cases), …