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11: 1.5 Calculus of Two or More Variables

… ►that is, for every arbitrarily small positive constant $ϵ$ there exists $\delta$ ($>0$) such that … ►Suppose also that ${\int }_c^{d}f(x,y)dy$ converges and ${\int }_c^d(\partial f/\partial x)dy$ converges uniformly on $a\le x\le b$, that is, given any positive number $ϵ$, however small, we can find a number $c_0\in [c,d)$ that is independent of $x$ and is such that … ►let $({\xi }_j,{\eta }_k)$ denote any point in the rectangle $[x_j,x_{j+1}]\times [y_k,y_{k+1}]$, $j=0,\mathrm{\dots },n-1$, $k=0,\mathrm{\dots },m-1$. … ►

1.5.27 $${\iint }_{R}f(x,y)dA=lim\sum _{j,k}f({\xi }_j,{\eta }_k)(x_{j+1}-x_j)(y_{k+1}-y_k)$$

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Symbols:

$dx$: differential of $x$ , $j$: integer, $k$: integer and $R$: closed rectangle

Permalink:

http://dlmf.nist.gov/1.5.E27

Encodings:

pMML, png, TeX

See also:

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

…

12: 12.14 The Function $W(a,x)$

… ►In the following expansions, obtained from Olver (1959), $\mu$ is large and positive, and $\delta$ is again an arbitrary small positive constant. … ►

12.14.31 $$W(\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{l(\mu ){\mathrm{e}}^{{\mu }^2\eta }}{2^{\frac{1}{2}}{\mathrm{e}}^{\frac{1}{4}\pi {\mu }^2}{(1-t^2)}^{\frac{1}{4}}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\widetilde{𝒜}}_s(t)}{{\mu }^{2s}},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $W(a,x)$: parabolic cylinder function, $s$: nonnegative integer, $\eta$ , ${\widetilde{𝒜}}_s(t)$: coefficients and $l(\mu )$

Referenced by:

§12.14(ix)

Permalink:

http://dlmf.nist.gov/12.14.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

►uniformly for $t\in [-1+\delta ,1-\delta ]$, with $\eta$ given by (12.10.23) and ${\widetilde{𝒜}}_s(t)$ given by (12.10.24). …