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11: 9.13 Generalized Airy Functions

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9.13.10

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Symbols:

$O\left(x\right)$: order not exceeding, $A_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (14)–(15))

Permalink:

http://dlmf.nist.gov/9.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

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9.13.12

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Symbols:

$O\left(x\right)$: order not exceeding, $B_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\sin z$: sine function, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (17)–(18))

Permalink:

http://dlmf.nist.gov/9.13.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

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12: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►We integrate by parts twice giving: … ►For $f(x)$ even in $x$ this yields the Fourier cosine transform pair (1.14.9) & (1.14.11), and for $f(x)$ odd the Fourier sine transform pair (1.14.10) & (1.14.12). … ►Note that the notations of (1.18.32) and (1.18.47) are used to distinguish the contributions from the discrete and continuous parts of the spectrum. … ►Surprisingly, if on any interval on the real line, even if positive elsewhere, as long as ${\int }_{X}q(x)dx\le 0$, see Simon (1976, Theorem 2.5), then there will be at least one eigenfunction with a negative eigenvalue, with corresponding $L^2\left(X\right)$ eigenfunction. … ►In unusual cases $N=\mathrm{\infty }$, even for all $\mathrm{\ell }$, such as in the case of the Schrödinger–Coulomb problem ($V=-r^{-1}$) discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at $\lambda =0$, implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66). …

13: 36.8 Convergent Series Expansions

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${\mathrm{\Psi }}_K\left(𝐱\right)={\displaystyle \frac{2}{K+2}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}\exp \left(\mathrm{i}{\displaystyle \frac{\pi (2n+1)}{2(K+2)}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{2n+1}{K+2}}\right)a_{2n}(𝐱),$ $K$ even,

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36.8.4 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)=2{\pi }^2{\left(\frac{2}{3}\right)}^{2/3}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-\mathrm{i}{(2/3)}^{2/3}z\right)}^n}{n!}\mathrm{\Re }\left(f_n(\frac{x+\mathrm{i}y}{{12}^{1/3}},\frac{x-\mathrm{i}y}{{12}^{1/3}})\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter and $f_n$: function

Referenced by:

§36.8

Permalink:

http://dlmf.nist.gov/36.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

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14: 2.11 Remainder Terms; Stokes Phenomenon

… ►Even when the series converges this is unwise: the tail needs to be majorized rigorously before the result can be guaranteed. For divergent expansions the situation is even more difficult. … ►By integration by parts (§2.3(i)) … ►If we permit the use of nonelementary functions as approximants, then even more powerful re-expansions become available. … ►That the change in their forms is discontinuous, even though the function being approximated is analytic, is an example of the Stokes phenomenon. …

15: 1.15 Summability Methods

… ►Then $\mathrm{\Phi }(z)$ is an analytic function in the upper half-plane and its real part is the Poisson integral $h(x,y)$; compare (1.9.34). The imaginary part …Moreover, ${lim}_{y\to 0+}\mathrm{\Im }\mathrm{\Phi }(x+\mathrm{i}y)$ is the Hilbert transform of $f(x)$ (§1.14(v)). … ►For $\mathrm{\Re }\alpha >0$ and $x\ge 0$, the Riemann-Liouville fractional integral of order $\alpha$ is defined by … ►In that case we must also replace $(x-t)$ in the integrand by $(t-x)$ and we can even set $a=\mathrm{\infty }$. …

16: 31.7 Relations to Other Functions

… ►The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …

17: 3.5 Quadrature

… ►where $n$ is even and … ►Here $f(\zeta )$ is assumed analytic in the half-plane $\mathrm{\Re }\zeta >c_0$ and bounded as $\zeta \to \mathrm{\infty }$ in $\left|\mathrm{ph}\zeta \right|\le \frac{1}{2}\pi$. …The complex Gauss nodes ${\zeta }_k$ have positive real part for all $s>0$. … ►The steepest descent path is given by $\mathrm{\Im }\left(t-2\sqrt{t}\right)=0$, or in polar coordinates $t=r{\mathrm{e}}^{\mathrm{i}\theta }$ we have $r={\sec }^2\left(\frac{1}{2}\theta \right)$. … ►Table 3.5.20 gives the results of applying the composite trapezoidal rule (3.5.2) with step size $h$; $n$ indicates the number of function values in the rule that are larger than ${10}^{-15}$ (we exploit the fact that the integrand is even). …

18: 2.5 Mellin Transform Methods

… ►From (2.5.12) and (2.5.13), it is seen that $a_s=b_s=0$ when $s$ is even. … ►with $b+\mathrm{\Re }{\beta }_0>1$, and …with $c+\mathrm{\Re }{\alpha }_0>-1$. …This, in turn, requires , , and either or . … ►for . …

19: 36.2 Catastrophes and Canonical Integrals

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36.2.15

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit and $K$: codimension

Referenced by:

§36.2(ii)

Permalink:

http://dlmf.nist.gov/36.2.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(ii), §36.2 and Ch.36

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$$\frac{{\partial }^{2q+1}}{{\partial x_1}^{2q+1}}{\mathrm{\Psi }}_K\left(\mathrm{𝟎}\right)=0,$$ $K$ even,

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$$\frac{{\partial }^{2q}}{{\partial x_1}^{2q}}{\mathrm{\Psi }}_K\left(\mathrm{𝟎}\right)=\frac{2}{K+2}\mathrm{\Gamma }\left(\frac{2q+1}{K+2}\right)\exp \left(\mathrm{i}\frac{\pi }{2}\left(\frac{2q+1}{K+2}+2q\right)\right),$$ $K$ even.

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36.2.20 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,0\right)=2{\pi }^2{(\frac{2}{3})}^{2/3}\mathrm{\Re }\left(\mathrm{Ai}\left(\frac{x+\mathrm{i}y}{{12}^{1/3}}\right)\mathrm{Bi}\left(\frac{x-\mathrm{i}y}{{12}^{1/3}}\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $y$: real parameter and $x$: real parameter

Referenced by:

§36.2(ii), §36.7(iii), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(ii), §36.2 and Ch.36

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20: 28.29 Definitions and Basic Properties

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28.29.6

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Symbols:

$\mathrm{\Re }$: real part and $\nu$: complex parameter

Referenced by:

§28.29(ii)

Permalink:

http://dlmf.nist.gov/28.29.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(ii), §28.29 and Ch.28

… ►In the symmetric case $Q(z)=Q(-z)$, $w_{\text{I}}(z,\lambda )$ is an even solution and $w_{\text{II}}(z,\lambda )$ is an odd solution; compare §28.2(ii). …