turning points

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21: 12.14 The Function $W(a,x)$

… ►Airy-type uniform asymptotic expansions can be used to include either one of the turning points $\pm 1$. … ►In this case there are no real turning points, and the solutions of (12.2.3), with $z$ replaced by $x$, oscillate on the entire real $x$-axis. …

22: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►With the lower sign there are turning points at $t=\pm 1$, which need to be excluded from the regions of validity. … ►The turning points can be included if expansions in terms of Airy functions are used instead of elementary functions (§2.8(iii)). …

23: 9.13 Generalized Airy Functions

… ►are used in approximating solutions to differential equations with multiple turning points; see §2.8(v). …

24: 2.5 Mellin Transform Methods

… ►The domain of analyticity of $ℳf\left(z\right)$ is usually an infinite strip parallel to the imaginary axis. … ►If $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ can be continued analytically to meromorphic functions in a left half-plane, and if the contour $\mathrm{\Re }z=c$ can be translated to $\mathrm{\Re }z=d$ with , then ►

2.5.6

ⓘ

Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\mathrm{\Re }$: real part, $res$: residue, $f(x)$: locally integrable function, $c$: point, $I(x)$: convolution integral, $h(x)$: function, $d$: point and $E(x)$: function

Keywords:

Mellin transform

Referenced by:

§2.5(i), §2.5(i)

Permalink:

http://dlmf.nist.gov/2.5.E6

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(i), §2.5 and Ch.2

… ►

2.5.16 $$I(x)=\sum _{s=0}^{n-1}(c_s\ln x+d_s)x^{-2s-1}+O\left(x^{-2n-1+ϵ}\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\ln z$: principal branch of logarithm function, $ϵ$: parameter, $c$: point, $I(x)$: convolution integral and $d$: point

Permalink:

http://dlmf.nist.gov/2.5.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

… ►This, in turn, requires , , and either or . …