doubly-periodic forms

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31: 31.14 General Fuchsian Equation

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Normal Form

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32: 33.21 Asymptotic Approximations for Large $|r|$

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§33.21(i) Limiting Forms

►We indicate here how to obtain the limiting forms of $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, and $c(ϵ,\mathrm{\ell };r)$ as $r\to \pm \mathrm{\infty }$, with $ϵ$ and $\mathrm{\ell }$ fixed, in the following cases: …

33: Guide to Searching the DLMF

… ►To recognize the math symbols and structures, and to accommodate equivalence between various notations and various forms of expression, the search system maps the math part of your queries into a different form. … ►Note that the first form may match other functions $K$ than the Bessel $K$ function, so if you are sure you want Bessel $K$, you might as well enter one of the other 3 forms. …

34: 7.7 Integral Representations

… ►Integrals of the type $\int {\mathrm{e}}^{-z^2}R(z)dz$, where $R(z)$ is an arbitrary rational function, can be written in closed form in terms of the error functions and elementary functions. ►

7.7.1 $$\mathrm{erfc}z=\frac{2}{\pi }{\mathrm{e}}^{-z^2}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-z^2{t}^2}}{t^2+1}dt,$$ $|\mathrm{ph}z|\le \frac{1}{4}\pi$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

7.4.11 (in different form)

Referenced by:

§7.11, §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.2 $$w\left(z\right)=\frac{1}{\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}dt}{t-z}=\frac{2z}{\pi \mathrm{i}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}dt}{t^2-z^2},$$ $\mathrm{\Im }z>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

A&S Ref:

7.1.4 (in different form)

Referenced by:

§7.19(i), §7.22(i), §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.6 $${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-(a{t}^2+2bt+c)}dt=\frac{1}{2}\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{(b^2-ac)/a}\mathrm{erfc}\left(\sqrt{a}x+\frac{b}{\sqrt{a}}\right),$$ $\mathrm{\Re }a>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

A&S Ref:

7.4.32 (in different form)

Referenced by:

§7.14(i), §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.9 $${\int }_0^x\mathrm{erf}tdt=x\mathrm{erf}x+\frac{1}{\sqrt{\pi }}\left({\mathrm{e}}^{-x^2}-1\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable

A&S Ref:

7.4.35 (in different form)

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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35: 10.52 Limiting Forms

§10.52 Limiting Forms

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36: 20.13 Physical Applications

… ►For $\tau =\mathrm{i}t$, with $\alpha ,t,z$ real, (20.13.1) takes the form of a real-time $t$ diffusion equation … ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.

37: 36.15 Methods of Computation

… ►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. …

38: 9.17 Methods of Computation

… ►However, in the case of $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$ this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v). … ►For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). …

39: 11.13 Methods of Computation

… ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. …

40: 18.11 Relations to Other Functions

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§18.11(ii) Formulas of Mehler–Heine Type

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Jacobi

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Laguerre

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Hermite

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