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11: 10.34 Analytic Continuation

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10.34.3 $$I_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=(\mathrm{i}/\pi )\left(\pm {\mathrm{e}}^{m\nu \pi \mathrm{i}}{K}_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp {\mathrm{e}}^{(m\mp 1)\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.34, §10.40(i), §10.40(iii), §10.41(iv)

Permalink:

http://dlmf.nist.gov/10.34.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.4 $$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=\csc \left(\nu \pi \right)\left(\pm \sin \left(m\nu \pi \right)K_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp \sin \left((m\mp 1)\nu \pi \right)K_{\nu }\left(z\right)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.34, §10.40(i)

Permalink:

http://dlmf.nist.gov/10.34.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.6 $$K_n\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=\pm {(-1)}^{n(m-1)}m{K}_n\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp {(-1)}^{nm}(m\mp 1)K_n\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $n$: integer and $z$: complex variable

Referenced by:

§10.40(i)

Permalink:

http://dlmf.nist.gov/10.34.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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12: 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 $n-\frac{1}{2}k{m}^2-m+\frac{1}{2}km\ge 0$. …

13: 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.

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$\pm x_k$		$w_k$
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Table 3.5.2: Nodes and weights for the 10-point Gauss–Legendre formula.

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$\pm x_k$	$w_k$
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Table 3.5.3: Nodes and weights for the 20-point Gauss–Legendre formula.

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$\pm x_k$	$w_k$
…

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Table 3.5.4: Nodes and weights for the 40-point Gauss–Legendre formula.

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$\pm x_k$	$w_k$
…

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14: 10.36 Other Differential Equations

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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

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15: 10.43 Integrals

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

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

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

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

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16: 4.4 Special Values and Limits

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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

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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

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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

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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

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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

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17: 36.5 Stokes Sets

… ►The Stokes set consists of the rays $\mathrm{ph}x=\pm 2\pi /3$ in the complex $x$-plane. … ►

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

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

►where $x_{\pm }$ are the two smallest positive roots of the equation … ►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …

18: 33.2 Definitions and Basic Properties

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

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§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 . …

19: 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” $K=K\left(\sin \left(\frac{1}{2}\alpha \right)\right)$. … ►This is an example of Duffing’s equation; see Ablowitz and Clarkson (1991, pp. 150–152) and Lawden (1989, pp. 117–119). … ►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 $-a\le x\le a$: …

20: 26.21 Tables

… ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. …