for large ℜz

SearchAdvancedHelp

(0.004 seconds)

11—20 of 132 matching pages

11: 13.9 Zeros

… ►For fixed $a,b\in ℂ$ the large $z$-zeros of $M(a,b,z)$ satisfy … ►For fixed $b$ and $z$ in $ℂ$ the large $a$-zeros of $M(a,b,z)$ are given by … ►For fixed $b$ and $z$ in $ℂ$ the large $a$-zeros of $U(a,b,z)$ are given by …

12: 4.38 Inverse Hyperbolic Functions: Further Properties

… ►

4.38.1 $$\mathrm{arcsinh}z=z-\frac{1}{2}\frac{z^3}{3}+\frac{1\cdot 3}{2\cdot 4}\frac{z^5}{5}-\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{z^7}{7}+\mathrm{\cdots },$$ .

ⓘ

Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function and $z$: complex variable

A&S Ref:

4.6.31

Referenced by:

§4.38(i)

Permalink:

http://dlmf.nist.gov/4.38.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(i), §4.38 and Ch.4

… ►

4.38.5 $$\mathrm{arctanh}z=z+\frac{z^3}{3}+\frac{z^5}{5}+\frac{z^7}{7}+\mathrm{\cdots },$$ $\left|z\right|\le 1$, $z\ne \pm 1$.

ⓘ

Symbols:

$\mathrm{arctanh}z$: inverse hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.6.33

Permalink:

http://dlmf.nist.gov/4.38.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(i), §4.38 and Ch.4

… ►

4.38.7 $$\mathrm{arctanh}z=\frac{z}{1-z^2}\left(1+\frac{2}{3}\frac{z^2}{z^2-1}+\frac{2\cdot 4}{3\cdot 5}{\left(\frac{z^2}{z^2-1}\right)}^2+\mathrm{\cdots }\right),$$ ,

ⓘ

Symbols:

$\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§4.38(i)

Permalink:

http://dlmf.nist.gov/4.38.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(i), §4.38 and Ch.4

►which requires $z$ $(=x+\mathrm{i}y)$ to lie between the two rectangular hyperbolas given by … ►

4.38.13 $$\frac{d}{dz}\mathrm{arcsech}z=-\frac{1}{z{(1-z^2)}^{1/2}}.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arcsech}z$: inverse hyperbolic secant function and $z$: complex variable

A&S Ref:

4.6.41 (has an error.)

Permalink:

http://dlmf.nist.gov/4.38.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(ii), §4.38 and Ch.4

…

13: 11.6 Asymptotic Expansions

… ►

§11.6(i) Large $|z|$, Fixed $\nu$

… ►

§11.6(ii) Large $|\nu |$, Fixed $z$

… ►

§11.6(iii) Large $|\nu |$, Fixed $z/\nu$

…

14: 11.11 Asymptotic Expansions of Anger–Weber Functions

… ►

§11.11(i) Large $|z|$, Fixed $\nu$

… ►

§11.11(ii) Large $|\nu |$, Fixed $z$

… ►

§11.11(iii) Large $\nu$, Fixed $z/\nu$

…

15: 13.29 Methods of Computation

… ►Although the Maclaurin series expansion (13.2.2) converges for all finite values of $z$, it is cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. For large $|z|$ the asymptotic expansions of §13.7 should be used instead. …

16: 6.18 Methods of Computation

… ►For large $x$ or $|z|$ these series suffer from slow convergence or cancellation (or both). … ►For large $x$ and $\left|z\right|$, expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available. … ► $A_0$, $B_0$, and $C_0$ can be computed by Miller’s algorithm (§3.6(iii)), starting with initial values $(A_N,B_N,C_N)=(1,0,0)$, say, where $N$ is an arbitrary large integer, and normalizing via $C_0=1/z$. …

17: 13.8 Asymptotic Approximations for Large Parameters

… ►

§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(ii) Large $b$ and $z$, Fixed $a$ and $b/z$

… ►For other asymptotic expansions for large $b$ and $z$ see López and Pagola (2010). … ►For generalizations in which $z$ is also allowed to be large see Temme and Veling (2022).

18: 10.74 Methods of Computation

… ►If $x$ or $|z|$ is large compared with ${|\nu |}^2$, then the asymptotic expansions of §§10.17(i)–10.17(iv) are available. … ►Moreover, because of their double asymptotic properties (§10.41(v)) these expansions can also be used for large $x$ or $|z|$, whether or not $\nu$ is large. … ►And since there are no error terms they could, in theory, be used for all values of $z$; however, there may be severe cancellation when $|z|$ is not large compared with $n^2$. …

19: 10.41 Asymptotic Expansions for Large Order

… ►

10.41.4 $$K_{\nu }\left(\nu z\right)\sim {\left(\frac{\pi }{2\nu }\right)}^{\frac{1}{2}}\frac{{\mathrm{e}}^{-\nu \eta }}{{(1+z^2)}^{\frac{1}{4}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k(p)}{{\nu }^k},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $U_k(p)$: polynomial coefficient

A&S Ref:

9.7.8

Permalink:

http://dlmf.nist.gov/10.41.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

… ►The series (10.41.3)–(10.41.6) can also be regarded as generalized asymptotic expansions for large $|z|$. … ►To establish (10.41.12) we substitute into (10.34.3), with $m=0$ and $z$ replaced by $\nu z$, by means of (10.41.13) observing that when $|z|$ is large the effect of replacing $z$ by $z{\mathrm{e}}^{\pm \pi \mathrm{i}}$ is to replace $\eta$, ${(1+z^2)}^{\frac{1}{4}}$, and $p$ by $-\eta$, $\pm \mathrm{i}{(1+z^2)}^{\frac{1}{4}}$, and $-p$, respectively. …

20: Bibliography W

… ►

An asymptotic expansion of $W_{k,m}(z)$ with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.

ⓘ

External Links:

FullText, Document, MathReview (J. A. Donaldson), ZentralBlatt

Referenced by:

§13.19.

Encodings:

BibTeX

…