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1: 15.13 Zeros

… ►If $a$, $b$, $c$ are real, $a$, $b$, $c$, $c-a$, $c-b\ne 0,-1,-2,\mathrm{\dots }$, and, without loss of generality, $b\ge a$, $c\ge a+b$ (compare (15.8.1)), then ►

15.13.1

ⓘ

Symbols:

$\lfloor x\rfloor$: floor of $x$ , $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $N(a,b,c)$: number of zeros and $S$

Permalink:

http://dlmf.nist.gov/15.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.13 and Ch.15

… ►For further information on the location of real zeros see Zarzo et al. (1995) and Dominici et al. (2013). …

2: 16.21 Differential Equation

… ►

16.21.1 $$\left({(-1)}^{p-m-n}z(\mathit{\vartheta }-a_1+1)\mathrm{\cdots }(\mathit{\vartheta }-a_p+1)-(\mathit{\vartheta }-b_1)\mathrm{\cdots }(\mathit{\vartheta }-b_q)\right)w=0,$$

ⓘ

Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $w$: solution and $\mathit{\vartheta }$: differential operator

Referenced by:

§16.21, §16.21

Permalink:

http://dlmf.nist.gov/16.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.21 and Ch.16

… ►In consequence of (16.19.1) we may assume, without loss of generality, that $p\le q$. … ►

16.21.2 $$G_{p,q}^{1,p}(z{\mathrm{e}}^{(p-m-n-1)\pi \mathrm{i}};\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_j,b_1,\mathrm{\dots },b_{j-1},b_{j+1},\mathrm{\dots },b_q}),$$ $j=1,\mathrm{\dots },q$.

ⓘ

Symbols:

$G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.21 and Ch.16

…

3: 32.1 Special Notation

… ► ►►►►►

$m,n$	integers.
$x$	real variable.
$z$	complex variable.
$k$	real parameter.

…

4: 32.2 Differential Equations

… ►If $\gamma \delta \ne 0$ in ${\text{P}}_{\text{III}}$, then set $\gamma =1$ and $\delta =-1$, without loss of generality, by rescaling $w$ and $z$ if necessary. If $\gamma =0$ and $\alpha \delta \ne 0$ in ${\text{P}}_{\text{III}}$, then set $\alpha =1$ and $\delta =-1$, without loss of generality. Lastly, if $\delta =0$ and $\beta \gamma \ne 0$, then set $\beta =-1$ and $\gamma =1$, without loss of generality. ►If $\delta \ne 0$ in ${\text{P}}_{\text{V}}$, then set $\delta =-\frac{1}{2}$, without loss of generality. … ►

32.2.21 $$f_1(z)+f_3(z)=\sqrt{z},$$

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Symbols:

$z$: real and $f_j(z)$: solutions

Permalink:

http://dlmf.nist.gov/32.2.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(v), §32.2 and Ch.32

…

5: 19.14 Reduction of General Elliptic Integrals

… ►

19.14.5 $${\sin }^2\varphi =\frac{\gamma -\alpha }{U^2+\gamma },$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

… ►

19.14.7 $${\sin }^2\varphi =\frac{(\gamma -\alpha )x^2}{a_1{a}_2+\gamma x^2}.$$

ⓘ

Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

… ►

19.14.8 $${\sin }^2\varphi =\frac{\gamma -\alpha }{b_1{b}_2{y}^2+\gamma }.$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

… ►

19.14.9 $${\sin }^2\varphi =\frac{(\gamma -\alpha )(x^2-y^2)}{\gamma (x^2-y^2)-a_1(a_2+b_2{x}^2)}.$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

… ►It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. …

6: 25.5 Integral Representations

§25.5 Integral Representations

… ►

25.5.5 $$\zeta \left(s\right)=-s{\int }_0^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor -\frac{1}{2}}{x^{s+1}}dx,$$ .

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

Titchmarsh (1986b, (2.1.6), p. 15)

Permalink:

http://dlmf.nist.gov/25.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(i), §25.5 and Ch.25

… ►In (25.5.15)–(25.5.19), , $\psi \left(x\right)$ is the digamma function, and $\gamma$ is Euler’s constant (§5.2). (25.5.16) is also valid for , $s\ne 1$. … ►where the integration contour is a loop around the negative real axis; it starts at $-\mathrm{\infty }$, encircles the origin once in the positive direction without enclosing any of the points $z=\pm 2\pi \mathrm{i}$, $\pm 4\pi \mathrm{i}$, …, and returns to $-\mathrm{\infty }$. …

7: 4.2 Definitions

… ► $\ln z$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },0]$ and real-valued when $z$ ranges over the positive real numbers. … ►The real and imaginary parts of $\ln z$ are given by … ►The function $\exp$ is an entire function of $z$, with no real or complex zeros. … ►When $a$ is real … ►Again, without the closed definition the $\ge$ and $\le$ signs would have to be replaced by $>$ and , respectively.

8: Mathematical Introduction

… ► ►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$ℝ$	real line (excluding infinity).
…

… ►Special functions with one real variable are depicted graphically with conventional two-dimensional (2D) line graphs. … ►With two real variables, special functions are depicted as 3D surfaces, with vertical height corresponding to the value of the function, and coloring added to emphasize the 3D nature. … ►Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. … ►Another numerical convention is that decimals followed by dots are unrounded; without the dots they are rounded. …

9: 29.15 Fourier Series and Chebyshev Series

… ►

29.15.7 $$a_{\nu }^{2m}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$

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Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $H_p$: eigenvalues

Permalink:

http://dlmf.nist.gov/29.15.E7

Encodings:

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See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

… ►

29.15.12 $$a_{\nu }^{2m+1}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$

ⓘ

Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $H_p$: eigenvalues

Permalink:

http://dlmf.nist.gov/29.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

… ►

29.15.17 $$b_{\nu }^{2m+1}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$

ⓘ

Symbols:

$b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $H_p$: eigenvalues

Permalink:

http://dlmf.nist.gov/29.15.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

… ►

29.15.22 $$a_{\nu }^{2m}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$

ⓘ

Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $H_p$: eigenvalues

Permalink:

http://dlmf.nist.gov/29.15.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

… ►The set of coefficients of this polynomial (without normalization) can also be found directly as an eigenvector of an $(n+1)\times (n+1)$ tridiagonal matrix; see Arscott and Khabaza (1962). …

10: 28.29 Definitions and Basic Properties

… ► $Q(z)$ is either a continuous and real-valued function for $z\in ℝ$ or an analytic function of $z$ in a doubly-infinite open strip that contains the real axis. … ►Let $\nu$ be a real or complex constant satisfying (without loss of generality) … ►

§28.29(iii) Discriminant and Eigenvalues in the Real Case

► $Q(x)$ is assumed to be real-valued throughout this subsection. … ►To every equation (28.29.1), there belong two increasing infinite sequences of real eigenvalues: …