limits of functions

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11: 22.5 Special Values

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§22.5(ii) Limiting Values of $k$

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Table 22.5.3: Limiting forms of Jacobian elliptic functions as $k\to 0$.

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$\mathrm{sn}(z,k)$ $\to$	$\sin z$	$\mathrm{cd}(z,k)$ $\to$	$\cos z$	$\mathrm{dc}(z,k)$ $\to$	$\sec z$	$\mathrm{ns}(z,k)$ $\to$	$\csc z$
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Table 22.5.4: Limiting forms of Jacobian elliptic functions as $k\to 1$.

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$\mathrm{sn}(z,k)$ $\to$	$\tanh z$	$\mathrm{cd}(z,k)$ $\to$	$1$	$\mathrm{dc}(z,k)$ $\to$	$1$	$\mathrm{ns}(z,k)$ $\to$	$\coth z$
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12: 10.2 Definitions

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Bessel Functions of the Third Kind (Hankel Functions)

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13: 1.4 Calculus of One Variable

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1.4.1 $$f(c+)\equiv \underset{x\to c+}{lim}f(x)=f(c),$$

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Symbols:

$\equiv$: equals by definition

Referenced by:

§1.4(ii)

Permalink:

http://dlmf.nist.gov/1.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(ii), §1.4 and Ch.1

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14: 13.2 Definitions and Basic Properties

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13.2.5 $$\underset{b\to -n}{lim}\frac{M(a,b,z)}{\mathrm{\Gamma }\left(b\right)}=𝐌(a,-n,z)=\frac{{\left(a\right)}_{n+1}}{(n+1)!}{z}^{n+1}M(a+n+1,n+2,z).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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§13.2(iii) Limiting Forms as $z\to 0$

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§13.2(iv) Limiting Forms as $z\to \mathrm{\infty }$

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15: 13.14 Definitions and Basic Properties

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13.14.7 $$\underset{2\mu \to -n-1}{lim}\frac{M_{\kappa ,\mu }\left(z\right)}{\mathrm{\Gamma }\left(2\mu +1\right)}=\frac{{\left(-\frac{1}{2}n-\kappa \right)}_{n+1}}{(n+1)!}{M}_{\kappa ,\frac{1}{2}(n+1)}\left(z\right)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{-\frac{1}{2}n}\sum _{s=n+1}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{2}n-\kappa \right)}_s}{\mathrm{\Gamma }\left(s-n\right)s!}{z}^s.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

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§13.14(iii) Limiting Forms as $z\to 0$

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§13.14(iv) Limiting Forms as $z\to \mathrm{\infty }$

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16: 19.6 Special Cases

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§19.6(v) $R_C(x,y)$

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17: 33.18 Limiting Forms for Large $\mathrm{\ell }$

§33.18 Limiting Forms for Large $\mathrm{\ell }$

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18: 10.24 Functions of Imaginary Order

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$${\tilde{Y}}_{\nu }\left(x\right)=\sqrt{2/(\pi x)}\sin \left(x-\frac{1}{4}\pi \right)+O\left(x^{-\frac{3}{2}}\right).$$

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19: 2.2 Transcendental Equations

… ►Let $f(x)$ be continuous and strictly increasing when and …

20: 1.9 Calculus of a Complex Variable

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Continuity

►A function $f(z)$ is continuous at a point $z_0$ if $\underset{z\to z_0}{lim}f(z)=f(z_0)$. That is, given any positive number $ϵ$, however small, we can find a positive number $\delta$ such that for all $z$ in the open disk . ►A function of two complex variables $f(z,w)$ is continuous at $(z_0,w_0)$ if $\underset{(z,w)\to (z_0,w_0)}{lim}f(z,w)=f(z_0,w_0)$; compare (1.5.1) and (1.5.2). … ►A function $f(z)$ is complex differentiable at a point $z$ if the following limit exists: …