Poincaré type

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1: 12.9 Asymptotic Expansions for Large Variable

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§12.9(i) Poincaré-Type Expansions

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2: 13.7 Asymptotic Expansions for Large Argument

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§13.7(i) Poincaré-Type Expansions

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3: 5.11 Asymptotic Expansions

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§5.11(i) Poincaré-Type Expansions

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4: 2.1 Definitions and Elementary Properties

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2.1.13 $$f(x)=\sum _{s=0}^{n-1}{a}_s{x}^{-s}+O\left(x^{-n}\right)$$

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Symbols:

$O\left(x\right)$: order not exceeding, $f(x)$: function, $n$: nonnegative integer and $a_s$: coefficients

Referenced by:

§2.1(iii), §2.1(iv)

Permalink:

http://dlmf.nist.gov/2.1.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §2.1(iii), §2.1 and Ch.2

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5: 9.7 Asymptotic Expansions

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§9.7(ii) Poincaré-Type Expansions

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6: 2.3 Integrals of a Real Variable

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2.3.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}q(t)dt\sim \sum _{s=0}^{\mathrm{\infty }}\frac{q^{(s)}(0)}{x^{s+1}},$$ $x\to +\mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $q(t)$: infinitely differentiable function

Keywords:

Laplace transform

Referenced by:

§2.3(i)

Permalink:

http://dlmf.nist.gov/2.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(i), §2.3 and Ch.2

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2.3.7 $$q(t)\sim \sum _{s=0}^{\mathrm{\infty }}{a}_s{t}^{(s+\lambda -\mu )/\mu },$$ $t\to 0+$,

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Defines:

$q(t)$: function (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: positive constant, $\lambda$: positive constant and $\mu$: positive constant

Referenced by:

§2.3(ii), §2.3(ii), §2.3(iv), §2.4(i), §2.4(ii)

Permalink:

http://dlmf.nist.gov/2.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(ii), §2.3 and Ch.2

… ►Other types of singular behavior in the integrand can be treated in an analogous manner. … ►

As $t\to a+$

2.3.14

$$p(t)\sim p(a)+\sum _{s=0}^{\mathrm{\infty }}{p}_s{(t-a)}^{s+\mu },$$

$$q(t)\sim \sum _{s=0}^{\mathrm{\infty }}{q}_s{(t-a)}^{s+\lambda -1},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: left endpoint, $p(t)$: real function, $p_s$: coefficients, $q(t)$: function, $q_s$: coefficients, $\lambda$: positive constant and $\mu$: positive constant

Referenced by:

item (b)

Permalink:

http://dlmf.nist.gov/2.3.E14

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §2.3(iii), §2.3 and Ch.2

and the expansion for $p(t)$ is differentiable. Again $\lambda$ and $\mu$ are positive constants. Also $p_0>0$ (consistent with (a)).

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2.3.16 $$\frac{q(t)}{p^{\prime }(t)}\sim \sum _{s=0}^{\mathrm{\infty }}{b}_s{v}^{(s+\lambda -\mu )/\mu },$$ $v\to 0+$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $b$: right endpoint, $p(t)$: real function, $q(t)$: function, $\lambda$: positive constant and $\mu$: positive constant

Permalink:

http://dlmf.nist.gov/2.3.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(iii), §2.3 and Ch.2

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7: 12.14 The Function $W(a,x)$

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12.14.20 $$s_1(a,x)\sim 1+\frac{d_2}{1!2x^2}-\frac{c_4}{2!2^2{x}^4}-\frac{d_6}{3!2^3{x}^6}+\frac{c_8}{4!2^4{x}^8}+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $x$: real variable, $a$: real or complex parameter, $c_{2r}$: coefficients, $d_{2r}$: coefficients and $s_n(a,x)$: coefficients

A&S Ref:

19.21.5

Permalink:

http://dlmf.nist.gov/12.14.E20

Encodings:

pMML, png, TeX

See also:

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

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12.14.23 $$s_1(a,x)+\mathrm{i}{s}_2(a,x)\sim \sum _{r=0}^{\mathrm{\infty }}{(-\mathrm{i})}^r\frac{{\left(\frac{1}{2}+\mathrm{i}a\right)}_{2r}}{2^{r}r!x^{2r}}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $x$: real variable, $a$: real or complex parameter and $s_n(a,x)$: coefficients

A&S Ref:

19.21.8 (modification of)

Permalink:

http://dlmf.nist.gov/12.14.E23

Encodings:

pMML, png, TeX

See also:

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

… ►Airy-type uniform asymptotic expansions can be used to include either one of the turning points $\pm 1$. … ►

12.14.29 $$l(\mu )\sim \frac{2^{\frac{1}{4}}}{{\mu }^{\frac{1}{2}}}\sum _{s=0}^{\mathrm{\infty }}\frac{l_s}{{\mu }^{4s}},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $l(\mu )$ and $l_s$: coefficient

Permalink:

http://dlmf.nist.gov/12.14.E29

Encodings:

pMML, png, TeX

See also:

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

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Airy-type Uniform Expansions

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8: 8.12 Uniform Asymptotic Expansions for Large Parameter

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8.12.7 $$S(a,\eta )\sim \frac{{\mathrm{e}}^{-\frac{1}{2}a{\eta }^2}}{\sqrt{2\pi a}}\sum _{k=0}^{\mathrm{\infty }}{c}_k(\eta )a^{-k},$$

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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, $a$: parameter, $k$: nonnegative integer, $\eta (\lambda )$ , $S(a,\eta )$: function and $c_k(\eta )$: coefficients

Referenced by:

§8.12

Permalink:

http://dlmf.nist.gov/8.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

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8.12.8 $$T(a,\eta )\sim \frac{{\mathrm{e}}^{\frac{1}{2}a{\eta }^2}}{\sqrt{2\pi a}}\sum _{k=0}^{\mathrm{\infty }}{c}_k(\eta ){(-a)}^{-k},$$

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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, $a$: parameter, $k$: nonnegative integer, $\eta (\lambda )$ , $T(a,\eta )$: function and $c_k(\eta )$: coefficients

Permalink:

http://dlmf.nist.gov/8.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

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8.12.15 $$Q(a,a)\sim \frac{1}{2}+\frac{1}{\sqrt{2\pi a}}\sum _{k=0}^{\mathrm{\infty }}{c}_k(0)a^{-k},$$ $|\mathrm{ph}a|\le \pi -\delta$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $Q(a,z)$: normalized incomplete gamma function, $\mathrm{ph}$: phase, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant and $c_k(\eta )$: coefficients

Permalink:

http://dlmf.nist.gov/8.12.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

… ►A different type of uniform expansion with coefficients that do not possess a removable singularity at $z=a$ is given by … ►

8.12.22 $$x(a,\frac{1}{2})\sim a-\frac{1}{3}+\frac{8}{405}{a}^{-1}+\frac{184}{25515}{a}^{-2}+\frac{2248}{\mathrm{34\hspace{0.33em}44525}}{a}^{-3}+\mathrm{\cdots },$$ $a\to \mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.12.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12, §8.12 and Ch.8

9: 12.11 Zeros

… ►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). … ►

12.11.4 $$u_{a,s}\sim 2^{\frac{1}{2}}\mu \left(p_0(\alpha )+\frac{p_1(\alpha )}{{\mu }^4}+\frac{p_2(\alpha )}{{\mu }^8}+\mathrm{\cdots }\right),$$

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Defines:

$u_{a,s}$: zeros (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $a$: real or complex parameter, $\alpha$ and $p_n(\zeta )$: coefficients

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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12.11.7 $$u_{a,s}^{\prime }\sim 2^{\frac{1}{2}}\mu \left(q_0(\beta )+\frac{q_1(\beta )}{{\mu }^4}+\frac{q_2(\beta )}{{\mu }^8}+\mathrm{\cdots }\right),$$

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Defines:

$u_{a,s}^{\prime }$: zeros (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $a$: real or complex parameter and $\beta$

Permalink:

http://dlmf.nist.gov/12.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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12.11.9 $$u_{a,1}\sim 2^{\frac{1}{2}}\mu \left(1-\mathrm{1.85575\hspace{0.33em}708}{\mu }^{-4/3}-\mathrm{0.34438\hspace{0.33em}34}{\mu }^{-8/3}-\mathrm{0.16871\hspace{0.33em}5}{\mu }^{-4}-0.11414{\mu }^{-16/3}-0.0808{\mu }^{-20/3}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: real or complex parameter and $u_{a,s}$: zeros

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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10: 2.9 Difference Equations

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2.9.8 $$w_j(n)\sim {\rho }_j^n{n}^{{\alpha }_j}\sum _{s=0}^{\mathrm{\infty }}\frac{a_{s,j}}{n^s},$$ $n\to \mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $n$: nonnegative integer, $a_{s,j}$: coefficients, ${\rho }_j$: roots and $w_j(n)$: solutions

Permalink:

http://dlmf.nist.gov/2.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §2.9(i), §2.9 and Ch.2

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2.9.9 $$w_j(n)\sim {\rho }^n\exp \left({(-1)}^j\kappa \sqrt{n}\right)n^{\alpha }\sum _{s=0}^{\mathrm{\infty }}{(-1)}^{js}\frac{c_s}{n^{s/2}},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\exp z$: exponential function, $\rho$: roots, $\kappa$ , $c$: constant and $w_j(n)$: solutions

Permalink:

http://dlmf.nist.gov/2.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §2.9(ii), §2.9 and Ch.2

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2.9.12 $$w_j(n)\sim {\rho }^n{n}^{{\alpha }_j}\sum _{s=0}^{\mathrm{\infty }}\frac{a_{s,j}}{n^s},$$ $n\to \mathrm{\infty }$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\rho$: roots and $w_j(n)$: solutions

Referenced by:

§2.9(ii)

Permalink:

http://dlmf.nist.gov/2.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.9(ii), §2.9 and Ch.2

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2.9.13 $$w_2(n)\sim {\rho }^n{n}^{{\alpha }_2}\sum _{\begin{array}{c}s=0\\ s\ne {\alpha }_2-{\alpha }_1\end{array}}^{\mathrm{\infty }}\frac{b_s}{n^s}+c{w}_1(n)\ln n,$$ $n\to \mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\ln z$: principal branch of logarithm function, $\rho$: roots, $b_s$: coefficients, $c$: constant and $w_j(n)$: solutions

Permalink:

http://dlmf.nist.gov/2.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §2.9(ii), §2.9 and Ch.2

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§2.9(iii) Other Approximations

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