fundamental unit cell

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11: 4.23 Inverse Trigonometric Functions

… ►In (4.23.3) the integration path may not intersect $\pm \mathrm{i}$. …$\mathrm{Arctan}z$ and $\mathrm{Arccot}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. … ►Care needs to be taken on the cuts, for example, if then $1/(x+\mathrm{i}0)=(1/x)-\mathrm{i}0$. ►

§4.23(v) Fundamental Property

… ►where $z=x+\mathrm{i}y$ and $\pm z\notin (1,\mathrm{\infty })$ in (4.23.34) and (4.23.35), and in (4.23.36). …

12: 13.14 Definitions and Basic Properties

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13.14.10 $$M_{\kappa ,\mu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \mathrm{i}{\mathrm{e}}^{\pm \mu \pi \mathrm{i}}{M}_{-\kappa ,\mu }\left(z\right).$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(ii), §13.14 and Ch.13

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13.14.11 $$M_{\kappa ,\mu }\left(z{\mathrm{e}}^{2m\pi \mathrm{i}}\right)={(-1)}^m{\mathrm{e}}^{2m\mu \pi \mathrm{i}}{M}_{\kappa ,\mu }\left(z\right).$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer and $z$: complex variable

Referenced by:

§13.14(ii)

Permalink:

http://dlmf.nist.gov/13.14.E11

Encodings:

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See also:

Annotations for §13.14(ii), §13.14 and Ch.13

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§13.14(v) Numerically Satisfactory Solutions

►Fundamental pairs of solutions of (13.14.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are … ►A fundamental pair of solutions that is numerically satisfactory in the sector $|\mathrm{ph}z|\le \pi$ near the origin is …

13: 36.7 Zeros

… ►Near $z=z_n$, and for small $x$ and $y$, the modulus $|{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)|$ has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose $z$ and $x$ repeat distances are given by … ►

36.7.6 $$\exp \left(-2\pi \mathrm{i}\left(\frac{z-z_n}{\Delta z}+\frac{2x}{\Delta x}\right)\right)\left(2\exp \left(\frac{-6\pi \mathrm{i}x}{\Delta x}\right)\cos \left(\frac{2\sqrt{3}\pi y}{\Delta x}\right)+1\right)=\sqrt{3}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $m$: integer and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.7.E6

Encodings:

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See also:

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

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14: 1.4 Calculus of One Variable

… ►For historical reasons, $w(x)$ is also sometimes referred to as a density, as, for example, the mass per unit length at point $x$, see Shohat and Tamarkin (1970, p vii). … ►If, for example, $\alpha (x)=H\left(x-x_n\right)$, the Heaviside unit step-function (1.16.14), then the corresponding measure $d\alpha (x)$ is $\delta \left(x-x_n\right)dx$, where $\delta \left(x-x_n\right)$ is the Dirac $\delta$-function of §1.17, such that, for $f(x)$ a continuous function on $(a,b)$, ${\int }_a^{b}f(x)d\alpha (x)=f(x_n)$ for $x_n\in (a,b)$ and $0$ otherwise. … ►

Fundamental Theorem of Calculus

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15: 3.12 Mathematical Constants

… ►The fundamental constant …

16: 8.24 Physical Applications

… ►With more general values of $p$, $E_p\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).

17: 10.73 Physical Applications

… ►and on separation of variables we obtain solutions of the form ${\mathrm{e}}^{\pm \mathrm{i}n\varphi }{\mathrm{e}}^{\pm \kappa z}{J}_n\left(\kappa r\right)$, from which a solution satisfying prescribed boundary conditions may be constructed. … ►on assuming a time dependence of the form ${\mathrm{e}}^{\pm \mathrm{i}kt}$. …It is fundamental in the study of electromagnetic wave transmission. …

18: 27.14 Unrestricted Partitions

… ►A fundamental problem studies the number of ways $n$ can be written as a sum of positive integers $\le n$, that is, the number of solutions of … ►

27.14.10 $$A_k(n)=\sum _{\begin{array}{c}h=1\\ \left(h,k\right)=1\end{array}}^k\exp \left(\pi \mathrm{i}s(h,k)-2\pi \mathrm{i}n\frac{h}{k}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\left(m,n\right)$: greatest common divisor (gcd), $\mathrm{i}$: imaginary unit, $k$: positive integer, $n$: positive integer, $A_k(n)$: coefficients and $s(h,k)$: Dedekind sum

A&S Ref:

24.2.1 I.C

Permalink:

http://dlmf.nist.gov/27.14.E10

Encodings:

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See also:

Annotations for §27.14(iii), §27.14 and Ch.27

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27.14.14 $$\eta \left(\frac{a\tau +b}{c\tau +d}\right)=\epsilon {(-\mathrm{i}(c\tau +d))}^{\frac{1}{2}}\eta \left(\tau \right),$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $a$: integer, $b$: integer, $c$: integer and $d$: integer

Referenced by:

§17.2(i)

Permalink:

http://dlmf.nist.gov/27.14.E14

Encodings:

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See also:

Annotations for §27.14(iv), §27.14 and Ch.27

►where $\epsilon =\exp \left(\pi \mathrm{i}(((a+d)/(12c))-s(d,c))\right)$ and $s(d,c)$ is given by (27.14.11). … ►The 24th power of $\eta \left(\tau \right)$ in (27.14.12) with ${\mathrm{e}}^{2\pi \mathrm{i}\tau }=x$ is an infinite product that generates a power series in $x$ with integer coefficients called Ramanujan’s tau function $\tau \left(n\right)$: …

19: 18.38 Mathematical Applications

… ►Classical OP’s play a fundamental role in Gaussian quadrature. … ►also the case $\beta =0$ of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. … ►The $3j$ symbol (34.2.6), with an alternative expression as a terminating ${}_3{}^{}F_2^{}$ of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials. … ►The $6j$ symbol (34.4.3), with an alternative expression as a terminating balanced ${}_4{}^{}F_3^{}$ of unit argument, can be expressend in terms of Racah polynomials (18.26.3). … ►

18.38.10 $$f^{\prime }(x)+\left(\alpha +\frac{1}{2}\right)\frac{f(x)-f(-x)}{x}=\mathrm{i}\lambda f(x)$$

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Symbols:

$\mathrm{i}$: imaginary unit and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.38.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.38(iii), §18.38(iii), §18.38 and Ch.18

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20: 27.2 Functions

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§27.2(i) Definitions

►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, …