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11: 13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20 Uniform Asymptotic Approximations for Large $\mu$

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§13.20(i) Large $\mu$, Fixed $\kappa$

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§13.20(v) Large $\mu$, Other Expansions

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12: 10.17 Asymptotic Expansions for Large Argument

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10.17.4 $$Y_{\nu }\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k}(\nu )}{z^{2k}}+\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k+1}(\nu )}{z^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$

A&S Ref:

9.2.6

Referenced by:

§10.17(iii), §10.17(iv), §10.18(iii), §10.24, §10.7(ii), §11.6(i)

Permalink:

http://dlmf.nist.gov/10.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(i), §10.17 and Ch.10

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13: 2.1 Definitions and Elementary Properties

… ►In (2.1.5) $ℝ$ can be replaced by any fixed ray in the sector , or by the whole of the sector $|\mathrm{ph}x|\le \frac{1}{2}\pi -\delta$. …But (2.1.5) does not hold as $x\to \mathrm{\infty }$ in (for example, set $x=1+\mathrm{i}t$ and let $t\to \pm \mathrm{\infty }$.) … ►For example, if $f(z)$ is analytic for all sufficiently large $|z|$ in a sector $𝐒$ and $f(z)=O\left(z^{\nu }\right)$ as $z\to \mathrm{\infty }$ in $𝐒$, $\nu$ being real, then $f^{\prime }(z)=O\left(z^{\nu -1}\right)$ as $z\to \mathrm{\infty }$ in any closed sector properly interior to $𝐒$ and with the same vertex (Ritt’s theorem). … ►If $\sum a_s{x}^{-s}$ converges for all sufficiently large $|x|$, then it is automatically the asymptotic expansion of its sum as $x\to \mathrm{\infty }$ in $ℂ$. … ►As an example, in the sector $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$ () each of the functions $0,{\mathrm{e}}^{-z}$, and ${\mathrm{e}}^{-\sqrt{z}}$ (principal value) has the null asymptotic expansion …

14: 12.2 Differential Equations

… ►Its importance is that when $a$ is negative and $|a|$ is large, $U(a,x)$ and $\overline{U}(a,x)$ asymptotically have the same envelope (modulus) and are $\frac{1}{2}\pi$ out of phase in the oscillatory interval . …

15: 13.7 Asymptotic Expansions for Large Argument

§13.7 Asymptotic Expansions for Large Argument

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§13.7(ii) Error Bounds

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§13.7(iii) Exponentially-Improved Expansion

… ►Then as $z\to \mathrm{\infty }$ with $\left|\left|z\right|-n\right|$ bounded and $a,b,m$ fixed … ►For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).

16: 8.21 Generalized Sine and Cosine Integrals

… ►Elsewhere in the sector $|\mathrm{ph}z|\le \pi$ the principal values are defined by analytic continuation from $\mathrm{ph}z=0$; compare §4.2(i). … ►When and , … ►When , … ►

§8.21(viii) Asymptotic Expansions

►When $z\to \mathrm{\infty }$ with $|\mathrm{ph}z|\le \pi -\delta$ (), …

17: 13.8 Asymptotic Approximations for Large Parameters

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§13.8(i) Large $|b|$, Fixed $a$ and $z$

… ►When the foregoing results are combined with Kummer’s transformation (13.2.39), an approximation is obtained for the case when $|b|$ is large, and $|b-a|$ and $|z|$ are bounded. … ►

§13.8(iii) Large $a$

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§13.8(iv) Large $a$ and $b$

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18: 2.8 Differential Equations with a Parameter

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2.8.29 $$W_{n,3}(u,\xi )={|\xi |}^{1/2}{J}_{\nu }\left(u{|\xi |}^{1/2}\right)\left(\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)-|\xi |J_{\nu +1}\left(u{|\xi |}^{1/2}\right)\left(\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+1}}+O\left(\frac{1}{u^{2n-2}}\right)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter

Referenced by:

§2.8(iv), §2.8(iv)

Permalink:

http://dlmf.nist.gov/2.8.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

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2.8.30 $$W_{n,4}(u,\xi )={|\xi |}^{1/2}{Y}_{\nu }\left(u{|\xi |}^{1/2}\right)\left(\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)-|\xi |Y_{\nu +1}\left(u{|\xi |}^{1/2}\right)\left(\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+1}}+O\left(\frac{1}{u^{2n-2}}\right)\right).$$

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter

Referenced by:

§2.8(iv), §2.8(iv)

Permalink:

http://dlmf.nist.gov/2.8.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

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2.8.35 $$W_{n,3}(u,\xi )={|\xi |}^{1/2}{J}_{\nu }\left(u{|\xi |}^{1/2}\right)\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}-|\xi |J_{\nu +1}\left(u{|\xi |}^{1/2}\right)\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+1}}+{|\xi |}^{1/2}\mathrm{env}{J}_{\nu }\left(u{|\xi |}^{1/2}\right)O\left(\frac{1}{u^{2n-1}}\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$ , $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter

Permalink:

http://dlmf.nist.gov/2.8.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

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2.8.36 $$W_{n,4}(u,\xi )={|\xi |}^{1/2}{Y}_{\nu }\left(u{|\xi |}^{1/2}\right)\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}-|\xi |Y_{\nu +1}\left(u{|\xi |}^{1/2}\right)\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+1}}+{|\xi |}^{1/2}\mathrm{env}{Y}_{\nu }\left(u{|\xi |}^{1/2}\right)O\left(\frac{1}{u^{2n-1}}\right),$$

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter

Permalink:

http://dlmf.nist.gov/2.8.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

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19: 13.9 Zeros

… ►where $n$ is a large positive integer, and the logarithm takes its principal value (§4.2(i)). … ►where $n$ is a large positive integer. … ►For fixed $a$ and $b$ in $ℂ$, $U(a,b,z)$ has a finite number of $z$-zeros in the sector . … ►In Wimp (1965) it is shown that if $a,b\in ℝ$ and $2a-b>-1$, then $U(a,b,z)$ has no zeros in the sector $|\mathrm{ph}z|\le \frac{1}{2}\pi$. … ►where $n$ is a large positive integer. …

20: 2.4 Contour Integrals

… ►Then …as $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$ (), with $z^{(s+\lambda )/\mu }$ assigned its principal value. … ►If this integral converges uniformly at each limit for all sufficiently large $t$, then by the Riemann–Lebesgue lemma (§1.8(i)) … ►

$z$ ranges along a ray or over an annular sector ${\theta }_1\le \theta \le {\theta }_2$, $|z|\ge Z$, where $\theta =\mathrm{ph}z$, , and $Z>0$. $I(z)$ converges at $b$ absolutely and uniformly with respect to $z$.

… ►For large $|z|$, $I(\alpha ,z)$ is approximated uniformly by the integral that corresponds to (2.4.19) when $f(\alpha ,w)$ is replaced by a constant. …