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11: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

… ►For $x\ge 0$ and $p>1$ let $x=\lambda p$ and define $A_0(\lambda )=1$, …so that $A_k(\lambda )$ is a polynomial in $\lambda$ of degree $k-1$ when $k\ge 1$. … ►

$A_1(\lambda )=1,$

►

$A_2(\lambda )=1-2\lambda ,$

►

$A_3(\lambda )=1-8\lambda +6{\lambda }^2.$

…

12: 23.15 Definitions

… ►Also $𝒜$ denotes a bilinear transformation on $\tau$, given by … ►A modular function $f(\tau )$ is a function of $\tau$ that is meromorphic in the half-plane $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{SL}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL$(2,\mathbb{Z})$, ►

23.15.5 $$f(𝒜\tau )=c_{𝒜}{(c\tau +d)}^{\mathrm{\ell }}f(\tau ),$$ $\mathrm{\Im }\tau >0$,

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Symbols:

$\mathrm{\Im }$: imaginary part, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer, $f(\tau )$: modular function, $c_{𝒜}$: constant and $\mathrm{\ell }$: level

Permalink:

http://dlmf.nist.gov/23.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(i), §23.15 and Ch.23

►where $c_{𝒜}$ is a constant depending only on $𝒜$, and $\mathrm{\ell }$ (the level) is an integer or half an odd integer. … ►

23.15.7 $$J\left(\tau \right)=\frac{{\left({\theta }_2^8(0,q)+{\theta }_3^8(0,q)+{\theta }_4^8(0,q)\right)}^3}{54{\left({\theta }_1^{\prime }(0,q)\right)}^8},$$

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Defines:

$J\left(\tau \right)$: Klein’s complete invariant

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

…

13: 26.10 Integer Partitions: Other Restrictions

… ►The set $\{n\ge 1|n\equiv \pm j(modk)\}$ is denoted by $A_{j,k}$. … ►Note that $p({𝒟}^{\prime }3,n)\le p(𝒟3,n)$, with strict inequality for $n\ge 9$. It is known that for $k>3$, $p(𝒟k,n)\ge p(\in A_{1,k+3},n)$, with strict inequality for $n$ sufficiently large, provided that $k=2^m-1,m=3,4,5$, or $k\ge 32$; see Yee (2004). … ►where $I_1\left(x\right)$ is the modified Bessel function (§10.25(ii)), and …The quantity $A_k(n)$ is real-valued. …

14: 36.7 Zeros

… ►Inside the cusp, that is, for , the zeros form pairs lying in curved rows. … ►Just outside the cusp, that is, for $x^2>8{|y|}^3/27$, there is a single row of zeros on each side. … ►

$y_n=-{\left({\displaystyle \frac{3\pi (8n+5)}{9+8{\xi }_n}}\right)}^{1/2},$

… ►

36.7.3 $$\frac{3\pi (8n+5)}{9+8{\xi }_n}{\xi }_n^{3/2}=\frac{27}{16}{\left(\frac{3}{2}\right)}^{1/2}\left(\ln \left(\frac{1}{{\xi }_n}\right)+3\ln \left(\frac{3}{2}\right)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $m$: integer

Referenced by:

§36.7(ii)

Permalink:

http://dlmf.nist.gov/36.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.7(ii), §36.7 and Ch.36

… ►The rings are almost circular (radii close to $(\Delta x)/9$ and varying by less than 1%), and almost flat (deviating from the planes $z_n$ by at most $(\Delta z)/36$). …

15: 1.1 Special Notation

… ► ►►►►►►

$x,y$	real variables.
…
${𝐀}^{-1}$	inverse of the square matrix $𝐀$
…
$det(𝐀)$	determinant of the square matrix $𝐀$
$\mathrm{tr}(𝐀)$	trace of the square matrix $𝐀$
$\mathrm{etr}\left(𝐀\right)$	exponential of $\mathrm{tr}(𝐀)$
…

►In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{\ast }$, $a$ being a complex number or a matrix; the Hermitian conjugate of $𝐀$ is usually being denoted ${𝐀}^{†}$.

16: 3.7 Ordinary Differential Equations

… ►Consideration will be limited to ordinary linear second-order differential equations … ►where $𝐀(\tau ,z)$ is the matrix … ►Let ${𝐀}_P$ be the $(2P)\times (2P+2)$ band matrix … ►If, for example, ${\beta }_0={\beta }_1=0$, then on moving the contributions of $w(z_0)$ and $w(z_P)$ to the right-hand side of (3.7.13) the resulting system of equations is not tridiagonal, but can readily be made tridiagonal by annihilating the elements of ${𝐀}_P$ that lie below the main diagonal and its two adjacent diagonals. … ►If $q(x)$ is $C^{\mathrm{\infty }}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used. …

17: Bibliography K

… ►

Special functions and the Bieberbach conjecture. Amer. Math. Monthly 95 (8), pp. 689–696.

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External Links:

ISSN 0002-9890, FullText, MathReview (S. L. Shukla), ZentralBlatt

Referenced by:

§16.23(iii).

Encodings:

BibTeX

… ►

Calculation of the complex zeros of the modified Bessel function of the second kind and its derivatives. Zh. Vychisl. Mat. i Mat. Fiz. 24 (8), pp. 1150–1163.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 24(4), pp. 115–123

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.42, §10.74(vi), 3rd item.

Encodings:

BibTeX

… ►

The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

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External Links:

Document, MathReview (M. Mikolas), ZentralBlatt

Referenced by:

(18.14.8), §18.14(i), §18.17(ii), §18.18(ii).

Encodings:

BibTeX

… ►

Bessel Functions and their Applications. Analytical Methods and Special Functions, Vol. 8, Taylor & Francis Ltd., London-New York.

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Notes:

Translated from the Russian by E. V. Pankratiev

External Links:

ISBN 0-415-28130-X, MathReview (L. Debnath), ZentralBlatt

Referenced by:

§10.73(i), §10.73(i), §10.73(i).

Encodings:

BibTeX

… ►

Some special cases of the generalized hypergeometric function ${}_{q+1}{}^{}F_q^{}$ . J. Comput. Appl. Math. 78 (1), pp. 79–95.

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External Links:

ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§16.7.

Encodings:

BibTeX

…

18: 3.3 Interpolation

… ►where $C$ is a simple closed contour in $D$ described in the positive rotational sense and enclosing the points $z,z_1,z_2,\mathrm{\dots },z_n$. … ►and $A_k^n$ are the Lagrangian interpolation coefficients defined by … ►Let $c_n$ be defined by … ►where ${\omega }_{n+1}(\zeta )$ is given by (3.3.3), and $C$ is a simple closed contour in $D$ described in the positive rotational sense and enclosing $z_0,z_1,\mathrm{\dots },z_n$. … ►Then by using $x_3$ in Newton’s interpolation formula, evaluating $\left[x_0,x_1,x_2,x_3\right]f=-\mathrm{0.26608\hspace{0.33em}28233}$ and recomputing $f^{\prime }(x)$, another application of Newton’s rule with starting value $x_3$ gives the approximation $x=\mathrm{2.33810\hspace{0.33em}7373}$, with 8 correct digits. …

19: 27.2 Functions

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer $n>1$ can be represented uniquely as a product of prime powers, …($\nu \left(1\right)$ is defined to be 0.) …It can be expressed as a sum over all primes $p\le x$: … ►is the sum of the $\alpha$th powers of the divisors of $n$, where the exponent $\alpha$ can be real or complex. Note that ${\sigma }_0\left(n\right)=d\left(n\right)$. …

20: 12.14 The Function $W(a,x)$

… ►the branch of $\mathrm{ph}$ being zero when $a=0$ and defined by continuity elsewhere. … ►Other expansions, involving $\cos \left(\frac{1}{4}{x}^2\right)$ and $\sin \left(\frac{1}{4}{x}^2\right)$, can be obtained from (12.4.3) to (12.4.6) by replacing $a$ by $-\mathrm{i}a$ and $z$ by $x{\mathrm{e}}^{\pi \mathrm{i}/4}$; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). … ►Here ${𝒜}_s(t)$ is as in §12.10(ii), $\sigma$ is defined by … ►uniformly for $t\in [-1+\delta ,1-\delta ]$, with $\eta$ given by (12.10.23) and ${\widetilde{𝒜}}_s(t)$ given by (12.10.24). … ►uniformly for $t\in [-1+\delta ,\mathrm{\infty })$, with $\zeta$, $\varphi (\zeta )$, $A_s(\zeta )$, and $B_s(\zeta )$ as in §12.10(vii). …