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41: 10.69 Uniform Asymptotic Expansions for Large Order

§10.69 Uniform Asymptotic Expansions for Large Order

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42: 14.1 Special Notation

§14.1 Special Notation

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$x$, $y$, $\tau$	real variables.
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$m$, $n$	unless stated otherwise, nonnegative integers, used for order and degree, respectively.
$\mu$, $\nu$	general order and degree, respectively.
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43: 22.10 Maclaurin Series

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22.10.1 $$\mathrm{sn}(z,k)=z-\left(1+k^2\right)\frac{z^3}{3!}+\left(1+14k^2+k^4\right)\frac{z^5}{5!}-\left(1+135k^2+135k^4+k^6\right)\frac{z^7}{7!}+O\left(z^9\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.1

Permalink:

http://dlmf.nist.gov/22.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

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22.10.2 $$\mathrm{cn}(z,k)=1-\frac{z^2}{2!}+\left(1+4k^2\right)\frac{z^4}{4!}-\left(1+44k^2+16k^4\right)\frac{z^6}{6!}+O\left(z^8\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.2

Permalink:

http://dlmf.nist.gov/22.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

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22.10.3 $$\mathrm{dn}(z,k)=1-k^2\frac{z^2}{2!}+k^2\left(4+k^2\right)\frac{z^4}{4!}-k^2\left(16+44k^2+k^4\right)\frac{z^6}{6!}+O\left(z^8\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.3

Permalink:

http://dlmf.nist.gov/22.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

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22.10.4 $$\mathrm{sn}(z,k)=\sin z-\frac{k^2}{4}(z-\sin z\cos z)\cos z+O\left(k^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.1

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.6 $$\mathrm{dn}(z,k)=1-\frac{k^2}{2}{\sin }^{2}z+O\left(k^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.3

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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44: 2.1 Definitions and Elementary Properties

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§2.1(i) Asymptotic and Order Symbols

… ►As $x\to c$ in $𝐗$ … ►

2.1.2 $f(x)=o\left(\varphi (x)\right)⟺f(x)/\varphi (x)\to 0.$

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Defines:

$o\left(x\right)$: order less than

Permalink:

http://dlmf.nist.gov/2.1.E2

Encodings:

pMML, png, TeX

See also:

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

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§2.1(ii) Integration and Differentiation

►Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. …

45: 10.64 Integral Representations

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46: Bibliography Q

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Higher-Order SUSY, Exactly Solvable Potentials, and Exceptional Orthogonal Polynomials. Modern Physics Letters A 26, pp. 1843–1852.

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External Links:

Document, Link

Referenced by:

§18.38(iii).

Encodings:

BibTeX

47: 13.2 Definitions and Basic Properties

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13.2.13 $$M(a,b,z)=1+O\left(z\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E13

Encodings:

pMML, png, TeX

See also:

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

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13.2.14 $$U(-n,b,z)={(-1)}^n{\left(b\right)}_n+O\left(z\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E14

Encodings:

pMML, png, TeX

See also:

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

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13.2.15 $$U(-n+b-1,b,z)={(-1)}^n{\left(2-b\right)}_n{z}^{1-b}+O\left(z^{2-b}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E15

Encodings:

pMML, png, TeX

See also:

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

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13.2.16 $$U(a,b,z)=\frac{\mathrm{\Gamma }\left(b-1\right)}{\mathrm{\Gamma }\left(a\right)}{z}^{1-b}+O\left(z^{2-\mathrm{\Re }b}\right),$$ $\mathrm{\Re }b\ge 2$, $b\ne 2$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

13.5.6 (with order estimate corrected)

Permalink:

http://dlmf.nist.gov/13.2.E16

Encodings:

pMML, png, TeX

See also:

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

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13.2.17 $$U(a,2,z)=\frac{1}{\mathrm{\Gamma }\left(a\right)}{z}^{-1}+O\left(\ln z\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

13.5.7

Permalink:

http://dlmf.nist.gov/13.2.E17

Encodings:

pMML, png, TeX

See also:

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

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48: 2.10 Sums and Sequences

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2.10.15 $$j^{\alpha }-{(j+1)}^{\alpha }=-\alpha j^{\alpha -1}+\alpha (\alpha -1)O\left(j^{\alpha -2}\right)$$

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Symbols:

$O\left(x\right)$: order not exceeding

Permalink:

http://dlmf.nist.gov/2.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

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2.10.17 $$S(\alpha ,\beta ,n)=O\left(n^{\alpha }\right)+O\left(1\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding and $S(\alpha ,\beta ,n)$: sum

Permalink:

http://dlmf.nist.gov/2.10.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

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2.10.29 $$f_n=g_n+o\left(r^{-n}\right),$$ $n\to \pm \mathrm{\infty }$.

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Symbols:

$o\left(x\right)$: order less than, $f_n$: coefficients, $r$: radious of annulus and $g_n$: coefficients

Referenced by:

§2.10(iv)

Permalink:

http://dlmf.nist.gov/2.10.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

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On the circle $|z|=r$, the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity $z_j$, say,

2.10.30 $$f(z)-g(z)=O\left({(z-z_j)}^{{\sigma }_j-1}\right),$$ $z\to z_j$,

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Symbols:

$O\left(x\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and ${\sigma }_j$: positive constant

Permalink:

http://dlmf.nist.gov/2.10.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

where ${\sigma }_j$ is a positive constant.

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2.10.32 $$f^{(m)}(z)-g^{(m)}(z)=O\left({(z-z_j)}^{{\sigma }_j-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and ${\sigma }_j$: positive constant

Permalink:

http://dlmf.nist.gov/2.10.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

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49: 24.14 Sums

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§24.14(i) Quadratic Recurrence Relations

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§24.14(ii) Higher-Order Recurrence Relations

… ►These identities can be regarded as higher-order recurrences. …

50: 30.3 Eigenvalues

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30.3.2 $${\lambda }_n^m\left({\gamma }^2\right)=n(n+1)-\frac{1}{2}{\gamma }^2+O\left(n^{-2}\right),$$ $n\to \mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.3(ii), §30.3 and Ch.30

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30.3.4

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.3(ii), §30.3 and Ch.30

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