expansion of arbitrary function

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1: 13.24 Series

… ►For expansions of arbitrary functions in series of $M_{\kappa ,\mu }\left(z\right)$ functions see Schäfke (1961b). …

2: 10.23 Sums

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§10.23(iii) Series Expansions of Arbitrary Functions

… ►For other types of expansions of arbitrary functions in series of Bessel functions, see Watson (1944, Chapters 17–19) and Erdélyi et al. (1953b, §§ 7.10.2–7.10.4). …

3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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1.18.16 $$\underset{m\to \mathrm{\infty }}{lim}{\int }_a^b{\left|f(x)-\sum _{n=0}^m{c}_n{\varphi }_n(x)\right|}^{2}dx=0.$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.18.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(ii), §1.18 and Ch.1

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4: 14.18 Sums

… ►For expansions of arbitrary functions in series of Legendre polynomials see §18.18(i), and for expansions of arbitrary functions in series of associated Legendre functions see Schäfke (1961b). …

5: 18.18 Sums

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§18.18(i) Series Expansions of Arbitrary Functions

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6: 18.24 Hahn Class: Asymptotic Approximations

… ►This expansion is in terms of the parabolic cylinder function and its derivative. … ►This expansion is in terms of confluent hypergeometric functions. … ►The first expansion holds uniformly for $\delta \le x\le 1+\delta$, and the second for $1-\delta \le x\le 1+{\delta }^{-1}$, $\delta$ being an arbitrary small positive constant. Both expansions are in terms of parabolic cylinder functions. … ►Taken together, these expansions are uniformly valid for and for $a$ in unbounded intervals—each of which contains $[0,(1-\delta )n]$, where $\delta$ again denotes an arbitrary small positive constant. …

7: 11.6 Asymptotic Expansions

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11.6.1 $${𝐊}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.29

Referenced by:

§11.6(i), §11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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11.6.2 $${𝐌}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.2.6

Referenced by:

§11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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11.6.3 $${\int }_0^z{𝐊}_0\left(t\right)dt-\frac{2}{\pi }(\ln \left(2z\right)+\gamma )\sim \frac{2}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{(2k)!(2k-1)!}{{(k!)}^2{(2z)}^{2k}},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\gamma$: Euler’s constant, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.32

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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11.6.6 $${𝐊}_{\nu }\left(\lambda \nu \right)\sim \frac{{(\frac{1}{2}\lambda \nu )}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{k!c_k(\lambda )}{{\nu }^k},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $c_k(\lambda )$: expansion function

A&S Ref:

12.1.34

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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11.6.7 $${𝐌}_{\nu }\left(\lambda \nu \right)\sim -\frac{{(\frac{1}{2}\lambda \nu )}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{k!c_k(\mathrm{i}\lambda )}{{\nu }^k},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $c_k(\lambda )$: expansion function

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

…

8: 8.11 Asymptotic Approximations and Expansions

… ►where $\delta$ denotes an arbitrary small positive constant. …

9: 11.11 Asymptotic Expansions of Anger–Weber Functions

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11.11.8 $${𝐀}_{\nu }\left(\lambda \nu \right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$,

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Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Referenced by:

(11.11.18), (11.11.19), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E8

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which originally was $\nu \to \mathrm{\infty }$, , has been replaced to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.10 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim -\frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(-\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

ⓘ

Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Proof sketch:

Apply Laplace’s method to the integral (11.10.4).

Referenced by:

(11.11.18), (11.11.19), §11.11(iii), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which was originally $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.11 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim {\left(\frac{2}{\pi \nu }\right)}^{1/2}{\mathrm{e}}^{-\nu \mu }\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_k{b}_k(\lambda )}{{\nu }^k},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $b_k(\lambda )$: expansion function

Sources:

Dingle (1973, p. 388); Olver (1997b, pp. 103 and 352)

Proof sketch:

Derivable by applying Laplace’s method to the integral (11.10.4).

Permalink:

http://dlmf.nist.gov/11.11.E11

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which was originally $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.18 $${𝐉}_{\nu }\left(\nu \right)\sim \frac{2^{1/3}}{3^{2/3}\mathrm{\Gamma }\left(\frac{2}{3}\right){\nu }^{1/3}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$,

ⓘ

Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Source:

Olver (1997b, pp. 103 and 352)

Proof sketch:

Dervivable using (11.10.15), (11.10.16), (11.11.8), (11.11.10), (11.11.16) and the expansions in Watson (1944, §8.42).

Referenced by:

§11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E18

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which originally was $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.19 $${𝐄}_{\nu }\left(\nu \right)\sim \frac{2^{1/3}}{3^{7/6}\mathrm{\Gamma }\left(\frac{2}{3}\right){\nu }^{1/3}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Source:

Olver (1997b, pp. 103 and 352)

Proof sketch:

Dervivable using (11.10.15), (11.10.16), (11.11.8), (11.11.10), (11.11.16) and the expansions in Watson (1944, §8.42).

Referenced by:

§11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E19

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which originally was $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

10: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

… ►where $n=1,2,3,\mathrm{\dots }$, and $A$, $B$ are arbitrary constants. … ►

Parabolic Cylinder Functions

… ►

§7.18(vi) Asymptotic Expansion

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