doubly-periodic forms

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21: 27.22 Software

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Maple. isprime combines a strong pseudoprime test and a Lucas pseudoprime test. ifactor uses cfrac (§27.19) after exhausting trial division. Brent–Pollard rho, Square Forms Factorization, and ecm are available also; see §27.19.

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22: 18.32 OP’s with Respect to Freud Weights

… ►A Freud weight is a weight function of the form … ►Generalized Freud weights have the form … ►All of these forms appear in applications, see §18.39(iii) and Table 18.39.1, albeit sometimes with $x\in [0,\mathrm{\infty })$, where the term half-Freud weight is used; or on $x\in [-1,1]$ or $[0,1]$, where the term Rys weight is employed, see Rys et al. (1983). …

23: 28.15 Expansions for Small $q$

§28.15 Expansions for Small $q$

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28.15.1 $${\lambda }_{\nu }\left(q\right)={\nu }^2+\frac{1}{2({\nu }^2-1)}{q}^2+\frac{5{\nu }^2+7}{32{({\nu }^2-1)}^3({\nu }^2-4)}{q}^4+\frac{9{\nu }^4+58{\nu }^2+29}{64{({\nu }^2-1)}^5({\nu }^2-4)({\nu }^2-9)}{q}^6+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

A&S Ref:

20.3.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

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24: 26.3 Lattice Paths: Binomial Coefficients

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26.3.4 $$\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)x^m=\frac{1}{{(1-x)}^{n+1}},$$ .

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.1 (in slightly different form)

Permalink:

http://dlmf.nist.gov/26.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(ii), §26.3 and Ch.26

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§26.3(v) Limiting Form

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25: 6.7 Integral Representations

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6.7.1 $${\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-at}}{t+b}dt={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{\mathrm{i}at}}{t+\mathrm{i}b}dt={\mathrm{e}}^{ab}{E}_1\left(ab\right),$$ $a>0$, $b>0$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $\int$: integral

A&S Ref:

5.1.28 (modified form of) 5.1.29 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.3 $${\int }_x^{\mathrm{\infty }}\frac{{\mathrm{e}}^{\mathrm{i}t}}{a^2+t^2}dt=\frac{\mathrm{i}}{2a}\left({\mathrm{e}}^a{E}_1\left(a-\mathrm{i}x\right)-{\mathrm{e}}^{-a}{E}_1\left(-a-\mathrm{i}x\right)\right),$$ $a>0$, $x>0$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral and $x$: real variable

A&S Ref:

5.1.41 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.4 $${\int }_x^{\mathrm{\infty }}\frac{t{\mathrm{e}}^{\mathrm{i}t}}{a^2+t^2}dt=\frac{1}{2}\left({\mathrm{e}}^a{E}_1\left(a-\mathrm{i}x\right)+{\mathrm{e}}^{-a}{E}_1\left(-a-\mathrm{i}x\right)\right),$$ $a>0$, $x>0$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral and $x$: real variable

A&S Ref:

5.1.42 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.5 $${\int }_x^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{a^2+t^2}dt=-\frac{1}{2a\mathrm{i}}\left({\mathrm{e}}^{\mathrm{i}a}{E}_1\left(x+\mathrm{i}a\right)-{\mathrm{e}}^{-\mathrm{i}a}{E}_1\left(x-\mathrm{i}a\right)\right),$$ $a>0$, $x\in ℝ$,

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line and $x$: real variable

A&S Ref:

5.1.43 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

… ►Many integrals with exponentials and rational functions, for example, integrals of the type $\int {\mathrm{e}}^{z}R(z)dz$, where $R(z)$ is an arbitrary rational function, can be represented in finite form in terms of the function $E_1\left(z\right)$ and elementary functions; see Lebedev (1965, p. 42). …

26: 23.15 Definitions

… ►The set of all bilinear transformations of this form is denoted by SL$(2,\mathbb{Z})$ (Serre (1973, p. 77)). … ►If, as a function of $q$, $f(\tau )$ is analytic at $q=0$, then $f(\tau )$ is called a modular form. If, in addition, $f(\tau )\to 0$ as $q\to 0$, then $f(\tau )$ is called a cusp form. …

27: 28.8 Asymptotic Expansions for Large $q$

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28.8.6 $${\widehat{C}}_m\sim {\left(\frac{\pi h}{2{(m!)}^2}\right)}^{1/4}{\left(1+\frac{2m+1}{8h}+\frac{m^4+2m^3+263m^2+262m+108}{2048h^2}+\mathrm{\cdots }\right)}^{-1/2},$$

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Defines:

${\widehat{C}}_m$: coefficient (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer and $h$: parameter

A&S Ref:

20.9.19 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

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28.8.7 $${\widehat{S}}_m\sim {\left(\frac{\pi h}{2{(m!)}^2}\right)}^{1/4}{\left(1-\frac{2m+1}{8h}+\frac{m^4+2m^3-121m^2-122m-84}{2048h^2}+\mathrm{\cdots }\right)}^{-1/2}.$$

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Defines:

${\widehat{S}}_m$: coefficient (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer and $h$: parameter

A&S Ref:

20.9.20 (in slightly different form)

Referenced by:

§28.8(ii), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)

Permalink:

http://dlmf.nist.gov/28.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

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28.8.9 $$W_m^{\pm }(x)=\frac{{\mathrm{e}}^{\pm 2h\sin x}}{{(\cos x)}^{m+1}}\{\begin{array}{cc}{\left(\cos \left(\frac{1}{2}x+\frac{1}{4}\pi \right)\right)}^{2m+1},\hfill & \\ {\left(\sin \left(\frac{1}{2}x+\frac{1}{4}\pi \right)\right)}^{2m+1},\hfill & \end{array}$$

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Defines:

$W_m^{\pm }$: function (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.13 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

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28.8.11 $$P_m(x)\sim 1+\frac{s}{2^{3}h{\cos }^{2}x}+\frac{1}{h^2}\left(\frac{s^4+86s^2+105}{2^{11}{\cos }^{4}x}-\frac{s^4+22s^2+57}{2^{11}{\cos }^{2}x}\right)+\mathrm{\cdots },$$

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Defines:

$P_m(x)$ (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.14 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

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28.8.12 $$Q_m(x)\sim \frac{\sin x}{{\cos }^{2}x}\left(\frac{1}{2^{5}h}(s^2+3)+\frac{1}{2^9{h}^2}\left(s^3+3s+\frac{4s^3+44s}{{\cos }^{2}x}\right)\right)+\mathrm{\cdots }.$$

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Defines:

$Q_m(x)$ (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $\sin z$: sine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.14 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

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28: 15.11 Riemann’s Differential Equation

§15.11 Riemann’s Differential Equation

… ►The most general form is given by … ►A conformal mapping of the extended complex plane onto itself has the form … ►for arbitrary $\lambda$ and $\mu$.

29: 29.5 Special Cases and Limiting Forms

§29.5 Special Cases and Limiting Forms

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30: 23.18 Modular Transformations

… ►according as the elements of $𝒜$ in (23.15.3) have the respective forms …and $\lambda \left(\tau \right)$ is a cusp form of level zero for the corresponding subgroup of SL$(2,\mathbb{Z})$. … ► $J\left(\tau \right)$ is a modular form of level zero for SL$(2,\mathbb{Z})$. …