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21: 26.10 Integer Partitions: Other Restrictions

… ►The set $\{n\ge 1|n\equiv \pm j(modk)\}$ is denoted by $A_{j,k}$. … ►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $k$ for which $n-(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $m$ for which $n-\frac{1}{2}k{m}^2-m+\frac{1}{2}km\ge 0$. …

22: 3.5 Quadrature

… ►If in addition $f$ is periodic, $f\in C^k(ℝ)$, and the integral is taken over a period, then … ►

Table 3.5.1: Nodes and weights for the 5-point Gauss–Legendre formula.

► ►►

$\pm x_k$		$w_k$
…

►

►

Table 3.5.2: Nodes and weights for the 10-point Gauss–Legendre formula.

► ►►

$\pm x_k$	$w_k$
…

►

►

Table 3.5.3: Nodes and weights for the 20-point Gauss–Legendre formula.

► ►►

$\pm x_k$	$w_k$
…

►

… ►For integrals in higher dimensions, Monte Carlo methods are another—often the only—alternative. …

23: 10.43 Integrals

… ►

$$\int {\mathrm{e}}^{\pm z}{z}^{\nu }{𝒵}_{\nu }\left(z\right)dz=\frac{{\mathrm{e}}^{\pm z}{z}^{\nu +1}}{2\nu +1}\left({𝒵}_{\nu }\left(z\right)\mp {𝒵}_{\nu +1}\left(z\right)\right),$$ $\nu \ne -\frac{1}{2}$,

►

$$\int {\mathrm{e}}^{\pm z}{z}^{-\nu }{𝒵}_{\nu }\left(z\right)dz=\frac{{\mathrm{e}}^{\pm z}{z}^{-\nu +1}}{1-2\nu }\left({𝒵}_{\nu }\left(z\right)\mp {𝒵}_{\nu -1}\left(z\right)\right),$$ $\nu \ne \frac{1}{2}$.

►

§10.43(ii) Integrals over the Intervals $(0,x)$ and $(x,\mathrm{\infty })$

… ►

§10.43(iv) Integrals over the Interval ($0,\mathrm{\infty }$)

… ► …

24: 10.36 Other Differential Equations

… ►

10.36.2 $$z^2{w}^{\prime \prime }+z(1\pm 2z)w^{\prime }+(\pm z-{\nu }^2)w=0,$$ $w={\mathrm{e}}^{\mp z}{𝒵}_{\nu }\left(z\right)$.

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.41

Permalink:

http://dlmf.nist.gov/10.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

…

25: 4.4 Special Values and Limits

… ►

4.4.2 $$\ln \left(-1\pm \mathrm{i}0\right)=\pm \pi \mathrm{i},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\ln z$: principal branch of logarithm function

A&S Ref:

4.1.14 (modified)

Permalink:

http://dlmf.nist.gov/4.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(i), §4.4 and Ch.4

►

4.4.3 $$\ln \left(\pm \mathrm{i}\right)=\pm \frac{1}{2}\pi \mathrm{i}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\ln z$: principal branch of logarithm function

A&S Ref:

4.1.15

Permalink:

http://dlmf.nist.gov/4.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(i), §4.4 and Ch.4

… ►

4.4.6 $${\mathrm{e}}^{\pm \pi \mathrm{i}/2}=\pm \mathrm{i},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

A&S Ref:

4.2.27

Permalink:

http://dlmf.nist.gov/4.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

… ►

4.4.8 $${\mathrm{e}}^{\pm \pi \mathrm{i}/3}=\frac{1}{2}\pm \mathrm{i}\frac{\sqrt{3}}{2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/4.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

… ►

4.4.12 $${\mathrm{i}}^{\pm \mathrm{i}}={\mathrm{e}}^{\mp \pi /2}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/4.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

…

26: 33.2 Definitions and Basic Properties

… ►

§33.2(ii) Regular Solution $F_{\mathrm{\ell }}(\eta ,\rho )$

… ►

§33.2(iii) Irregular Solutions $G_{\mathrm{\ell }}(\eta ,\rho ),H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$

►The functions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are defined by … ►As in the case of $F_{\mathrm{\ell }}(\eta ,\rho )$, the solutions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ are analytic functions of $\rho$ when . Also, ${\mathrm{e}}^{\mp \mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)}{H}_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are analytic functions of $\eta$ when . …

27: 22.19 Physical Applications

… ►The periodicity and symmetry of the pendulum imply that the motion in each four intervals $\theta \in (0,\pm \alpha )$ and $\theta \in (\pm \alpha ,0)$ have the same “quarter periods” $K=K\left(\sin \left(\frac{1}{2}\alpha \right)\right)$. … ►

►►

Figure 22.19.1: Jacobi’s amplitude function $\mathrm{am}(x,k)$ for $0\le x\le 10\pi$ and $k=0.5,0.9999,1.0001,2$. …As $k\to 1-$, plateaus are seen as the motion approaches the separatrix where $\theta =n\pi$, $n=\pm 1,\pm 2,\mathrm{\dots }$, at which points the motion is time independent for $k=1$. … Magnify

… ►As $a\to \sqrt{1/\beta }$ from below the period diverges since $a=\pm \sqrt{1/\beta }$ are points of unstable equilibrium. … ►For an initial displacement with , bounded oscillations take place near one of the two points of stable equilibrium $x=\pm \sqrt{1/\beta }$. …For initial displacement with $|a|\ge \sqrt{2/\beta }$ the motion extends over the full range $-a\le x\le a$: …

28: 14.27 Zeros

… ► $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ (either side of the cut) has exactly one zero in the interval $(-\mathrm{\infty },-1)$ if either of the following sets of conditions holds: …For all other values of the parameters $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ has no zeros in the interval $(-\mathrm{\infty },-1)$. …

29: 36.5 Stokes Sets

… ►In the following subsections, only Stokes sets involving at least one real saddle are included unless stated otherwise. … ►The Stokes set consists of the rays $\mathrm{ph}x=\pm 2\pi /3$ in the complex $x$-plane. … ►

$x=B_{\pm }{|y|}^{4/3},$

►

$B_{\pm }={10}^{-1/3}\left(2x_{\pm }^{4/3}-\frac{1}{2}{x}_{\pm }^{-2/3}\right),$

►where $x_{\pm }$ are the two smallest positive roots of the equation …

30: 8.21 Generalized Sine and Cosine Integrals

… ►

8.21.1 $$\mathrm{ci}(a,z)\pm \mathrm{i}\mathrm{si}(a,z)={\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}a}\mathrm{\Gamma }(a,z{\mathrm{e}}^{\mp \frac{1}{2}\pi \mathrm{i}}),$$

ⓘ

Defines:

$\mathrm{ci}(a,z)$: generalized cosine integral and $\mathrm{si}(a,z)$: generalized sine integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.21(ii)

Permalink:

http://dlmf.nist.gov/8.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(i), §8.21 and Ch.8

►

8.21.2 $$\mathrm{Ci}(a,z)\pm \mathrm{i}\mathrm{Si}(a,z)={\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}a}\gamma (a,z{\mathrm{e}}^{\mp \frac{1}{2}\pi \mathrm{i}}).$$

ⓘ

Defines:

$\mathrm{Ci}(a,z)$: generalized cosine integral and $\mathrm{Si}(a,z)$: generalized sine integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.21(ii)

Permalink:

http://dlmf.nist.gov/8.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(i), §8.21 and Ch.8

… ►

8.21.3 $${\int }_0^{\mathrm{\infty }}{t}^{a-1}{\mathrm{e}}^{\pm \mathrm{i}t}dt={\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}a}\mathrm{\Gamma }\left(a\right),$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $a$: parameter

Referenced by:

§8.21(iv)

Permalink:

http://dlmf.nist.gov/8.21.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(ii), §8.21 and Ch.8

…