DLMF: Untitled Document

§28.16 Asymptotic Expansions for Large $q$

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28.16.1

\lambda_{\nu}\left(h^{2}\right)\sim-2h^{2}+2sh-\dfrac{1}{8}(s^{2}+1)-\dfrac{1}% {2^{7}h}(s^{3}+3s)-\dfrac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\dfrac{1}{2^{17}h^% {3}}(33s^{5}+410s^{3}+405s)-\dfrac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}% +486)-\dfrac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.

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Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\sim$ : Poincaré asymptotic expansion, $h$ : parameter and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.16 and Ch.28

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Chapter 20 Theta Functions

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Ronald F. Boisvert, Editor at Large, NIST

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William P. Reinhardt, University of Washington, Chaps. 20, 22, 23

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Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23

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William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23

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Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23

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§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

… ►Formulas involving

\Pi\left(\phi,\alpha^{2},k\right)

that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using

R_{C}\left(x,y\right)

. … ►When

x

and

y

are positive,

R_{C}\left(x,y\right)

is an inverse circular function if

x<y

and an inverse hyperbolic function (or logarithm) if

x>y

: …For the special cases of

R_{C}\left(x,x\right)

and

R_{C}\left(0,y\right)

see (19.6.15). … ►

19.2.21

R_{C}\left(x,y\right)=\int_{0}^{1}(v^{2}x+(1-v^{2})y)^{-1/2}\,\mathrm{d}v,

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral
Referenced by:: §19.2(iv)
Permalink:: http://dlmf.nist.gov/19.2.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §19.2(iv), §19.2 and Ch.19

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B_{-2}^{5}=\tfrac{1}{120}(6-10t-15t^{2}+20t^{3}-5t^{4}),

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B_{3}^{7}=\tfrac{1}{720}(48+8t-192t^{2}-20t^{3}+85t^{4}+6t^{5}-7t^{6}),

… ►where

C

is a simple closed contour described in the positive rotational sense such that

C

and its interior lie in the domain of analyticity of

f

, and

x_{0}

is interior to

C

. Taking

C

to be a circle of radius

r

centered at

x_{0}

, we obtain … ►With the choice

r=k

(which is crucial when

k

is large because of numerical cancellation) the integrand equals

e^{k}

at the dominant points

\theta=0,2\pi

, and in combination with the factor

k^{-k}

in front of the integral sign this gives a rough approximation to

1/k!

. …

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•

Blanch and Clemm (1965) includes values of ${\operatorname{Mc}^{(2)}_{n}}\left(x,\sqrt{q}\right)$ , ${\operatorname{Mc}^{(2)}_{n}}'\left(x,\sqrt{q}\right)$ for $n=0(1)7$ , $x=0(.02)1$ ; $n=8(1)15$ , $x=0(.01)1$ . Also ${\operatorname{Ms}^{(2)}_{n}}\left(x,\sqrt{q}\right)$ , ${\operatorname{Ms}^{(2)}_{n}}'\left(x,\sqrt{q}\right)$ for $n=1(1)7$ , $x=0(.02)1$ ; $n=8(1)15$ , $x=0(.01)1$ . In all cases $q=0(.05)1$ . Precision is generally 7D. Approximate formulas and graphs are also included.

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Ince (1932) includes eigenvalues $a_{n}$ , $b_{n}$ , and Fourier coefficients for $n=0$ or $1(1)6$ , $q=0(1)10(2)20(4)40$ ; 7D. Also $\operatorname{ce}_{n}\left(x,q\right)$ , $\operatorname{se}_{n}\left(x,q\right)$ for $q=0(1)10$ , $x=1(1)90$ , corresponding to the eigenvalues in the tables; 5D. Notation: $a_{n}=\mathit{be}_{n}-2q$ , $b_{n}=\mathit{bo}_{n}-2q$ .

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Kirkpatrick (1960) contains tables of the modified functions $\operatorname{Ce}_{n}\left(x,q\right)$ , $\operatorname{Se}_{n+1}\left(x,q\right)$ for $n=0(1)5$ , $q=1(1)20$ , $x=0.1(.1)1$ ; 4D or 5D.

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National Bureau of Standards (1967) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n}\left(q\right)$ for $n=0(1)3$ with $q=0(.2)20(.5)37(1)100$ , and $n=4(1)15$ with $q=0(2)100$ ; Fourier coefficients for $\operatorname{ce}_{n}\left(x,q\right)$ and $\operatorname{se}_{n}\left(x,q\right)$ for $n=0(1)15$ , $n=1(1)15$ , respectively, and various values of $q$ in the interval $[0,100]$ ; joining factors $g_{\mathit{e},n}(\sqrt{q})$ , $f_{\mathit{e},n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\mbox{ to }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$ . Precision is generally 8D.

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Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n+1}\left(q\right)$ for $n=0(1)4$ , $q=0(1)50$ ; $n=0(1)20$ ( $a$ ’s) or 19 ( $b$ ’s), $q=1,3,5,10,15,25,50(50)200$ . Fourier coefficients for $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , $n=0(1)7$ . Mathieu functions $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , and their first $x$ -derivatives for $n=0(1)4$ , $x=0(5^{\circ})90^{\circ}$ . Modified Mathieu functions ${\operatorname{Mc}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ , ${\operatorname{Ms}^{(j)}_{n+1}}\left(x,\sqrt{10}\right)$ , and their first $x$ -derivatives for $n=0(1)4$ , $j=1,2$ , $x=0(.2)4$ . Precision is mostly 9S.

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§28.8 Asymptotic Expansions for Large $q$

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§28.8(ii) Sips’ Expansions

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§28.8(iii) Goldstein’s Expansions

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Barrett’s Expansions

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Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

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Bickley et al. (1952) tabulates $x^{-n}I_{n}\left(x\right)$ or $e^{-x}I_{n}\left(x\right)$ , $x^{n}K_{n}\left(x\right)$ or $e^{x}K_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ (.01 or .1) 10(.1) 20, 8S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)20$ , 10S.

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Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_{n}\left(z\right)$ and $K_{n}'\left(z\right)$ , for $n=2(1)20$ , 9S.

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Zhang and Jin (1996, p. 322) tabulates $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , $x=0(1)20$ , 7S.

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Zhang and Jin (1996, p. 323) tabulates the first $20$ real zeros of $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , 8D.

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Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

for%20large%20%7C%CE%B32%7C

1—10 of 410 matching pages

1: 28.16 Asymptotic Expansions for Large $q$

§28.16 Asymptotic Expansions for Large $q$

2: 20 Theta Functions

Chapter 20 Theta Functions

3: Staff

4: 19.2 Definitions

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

5: 23 Weierstrass Elliptic and Modular
Functions

6: 3.4 Differentiation

7: 28.35 Tables

8: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

§28.8(ii) Sips’ Expansions

§28.8(iii) Goldstein’s Expansions

Barrett’s Expansions

9: 10.75 Tables

10: 8.26 Tables

for%20large%20%7C%CE%B32%7C

1—10 of 410 matching pages

1: 28.16 Asymptotic Expansions for Large q

§28.16 Asymptotic Expansions for Large q

2: 20 Theta Functions

Chapter 20 Theta Functions

3: Staff

4: 19.2 Definitions

§19.2(iv) A Related Function: R C ⁡ ( x , y )

5: 23 Weierstrass Elliptic and Modular Functions

6: 3.4 Differentiation

7: 28.35 Tables

8: 28.8 Asymptotic Expansions for Large q

§28.8 Asymptotic Expansions for Large q

§28.8(ii) Sips’ Expansions

§28.8(iii) Goldstein’s Expansions

Barrett’s Expansions

9: 10.75 Tables

10: 8.26 Tables

1: 28.16 Asymptotic Expansions for Large $q$

§28.16 Asymptotic Expansions for Large $q$

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

5: 23 Weierstrass Elliptic and Modular
Functions

8: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$