limits of functions

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21: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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§33.5(i) Small $\rho$

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§33.5(iii) Small $|\eta |$

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§33.5(iv) Large $\mathrm{\ell }$

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22: 1.5 Calculus of Two or More Variables

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1.5.1 $$\underset{(x,y)\to (a,b)}{lim}f(x,y)=f(a,b),$$

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Symbols:

$(a,b)$: open interval

Referenced by:

§1.9(ii)

Permalink:

http://dlmf.nist.gov/1.5.E1

Encodings:

pMML, png, TeX

See also:

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

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23: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

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§33.10(i) Large $\rho$

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§33.10(ii) Large Positive $\eta$

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§33.10(iii) Large Negative $\eta$

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24: 10.45 Functions of Imaginary Order

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$${\tilde{K}}_{\nu }\left(x\right)={(\pi /(2x))}^{\frac{1}{2}}{\mathrm{e}}^{-x}\left(1+O\left(x^{-1}\right)\right).$$

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25: 7.21 Physical Applications

§7.21 Physical Applications

►The error functions, Fresnel integrals, and related functions occur in a variety of physical applications. … ►Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$. Fried and Conte (1961) mentions the role of $w\left(z\right)$ in the theory of linearized waves or oscillations in a hot plasma; $w\left(z\right)$ is called the plasma dispersion function or Faddeeva (or Faddeyeva) function; see Faddeeva and Terent’ev (1954). … ►

26: 9.14 Incomplete Airy Functions

… ►Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …

27: 30.8 Expansions in Series of Ferrers Functions

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30.8.1 $${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)=\sum _{k=-R}^{\mathrm{\infty }}{(-1)}^k{a}_{n,k}^m({\gamma }^2){𝖯}_{n+2k}^m\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $R=\lfloor (n-m)/2\rfloor$: limit and $a_{n,k}^m({\gamma }^2)$: coefficients

A&S Ref:

21.7.1 (in different form)

Referenced by:

§30.16(ii), §30.16(ii)

Permalink:

http://dlmf.nist.gov/30.8.E1

Encodings:

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See also:

Annotations for §30.8(i), §30.8 and Ch.30

►where ${𝖯}_{n+2k}^m\left(x\right)$ is the Ferrers function of the first kind (§14.3(i)), $R=\lfloor \frac{1}{2}(n-m)\rfloor$, and the coefficients $a_{n,k}^m({\gamma }^2)$ are given by … ►

30.8.9 $${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)=\sum _{k=-\mathrm{\infty }}^{-N-1}{(-1)}^{k}a_{}^{\prime }{}_{n,k}{}^m({\gamma }^2){𝖯}_{n+2k}^m\left(x\right)+\sum _{k=-N}^{\mathrm{\infty }}{(-1)}^k{a}_{n,k}^m({\gamma }^2){𝖰}_{n+2k}^m\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $a_{n,k}^m({\gamma }^2)$: coefficients and $N$: upper limit

A&S Ref:

21.7.21 (in different form)

Permalink:

http://dlmf.nist.gov/30.8.E9

Encodings:

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See also:

Annotations for §30.8(ii), §30.8 and Ch.30

►where ${𝖯}_n^m$ and ${𝖰}_n^m$ are again the Ferrers functions and $N=\lfloor \frac{1}{2}(n+m)\rfloor$. …

28: 7.2 Definitions

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$\underset{z\to \mathrm{\infty }}{lim}\mathrm{erf}z=1,$

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$\underset{z\to \mathrm{\infty }}{lim}\mathrm{erfc}z=0$ , .

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29: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

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30: 8.22 Mathematical Applications

… ►so that ${lim}_{x\to \mathrm{\infty }}{\zeta }_x(s)=\zeta \left(s\right)$, then …