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11: 4.4 Special Values and Limits

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

ⓘ

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

►

4.4.9 $${\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}=-\frac{1}{2}\pm \mathrm{i}\frac{\sqrt{3}}{2},$$

ⓘ

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.E9

Encodings:

pMML, png, TeX

See also:

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

►

4.4.10 $${\mathrm{e}}^{\pm \pi \mathrm{i}/4}=\frac{1}{\sqrt{2}}\pm \mathrm{i}\frac{1}{\sqrt{2}},$$

ⓘ

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.E10

Encodings:

pMML, png, TeX

See also:

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

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

ⓘ

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

…

12: 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 $(-\mathrm{\infty },-1)$. ►For complex zeros of $P_{\nu }^{\mu }\left(z\right)$ see Hobson (1931, §§233, 234, and 238).

13: 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 . …

14: 26.21 Tables

… ►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s(n,k)$ and $S(n,k)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p(𝒟,n)$ for $n$ up to 500. ►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. …

15: 4.37 Inverse Hyperbolic Functions

… ►In (4.37.1) the integration path may not pass through either of the points $t=\pm \mathrm{i}$, and the function ${(1+t^2)}^{1/2}$ assumes its principal value when $t$ is real. In (4.37.2) the integration path may not pass through either of the points $\pm 1$, and the function ${(t^2-1)}^{1/2}$ assumes its principal value when $t\in (1,\mathrm{\infty })$. …In (4.37.3) the integration path may not intersect $\pm 1$. …$\mathrm{Arcsinh}z$ and $\mathrm{Arccsch}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. … ►For example, $\mathrm{arcsech}a=\mathrm{arccoth}\left({(1-a^2)}^{-1/2}\right)$.

16: 33.6 Power-Series Expansions in $\rho$

… ►where $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}=1$, $A_{\mathrm{\ell }+2}^{\mathrm{\ell }}=\eta /(\mathrm{\ell }+1)$, and ►

33.6.3 $$(k+\mathrm{\ell })(k-\mathrm{\ell }-1)A_k^{\mathrm{\ell }}=2\eta A_{k-1}^{\mathrm{\ell }}-A_{k-2}^{\mathrm{\ell }},$$ $k=\mathrm{\ell }+3,\mathrm{\ell }+4,\mathrm{\dots }$,

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Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.6

Permalink:

http://dlmf.nist.gov/33.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

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33.6.5 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )=\frac{{\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}}{(2\mathrm{\ell }+1)!\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k}{{\left(2\mathrm{\ell }+2\right)}_{k}k!}{(\mp 2\mathrm{i}\rho )}^{a+k}\left(\ln \left(\mp 2\mathrm{i}\rho \right)+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(2\mathrm{\ell }+2+k\right)\right)-\sum _{k=1}^{2\mathrm{\ell }+1}\frac{(2\mathrm{\ell }+1)!(k-1)!}{(2\mathrm{\ell }+1-k)!{\left(1-a\right)}_k}{(\mp 2\mathrm{i}\rho )}^{a-k}\right),$$

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Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a$: variable

A&S Ref:

14.1.14 (equivalent)

Referenced by:

§33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5)

Permalink:

http://dlmf.nist.gov/33.6.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.0.19):

On the right-hand side the factor in the denominator was incorrectly written as $\mathrm{\Gamma }\left(-\mathrm{\ell }+\mathrm{i}\eta \right)$. This has been corrected to $\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.6 and Ch.33

►where $a=1+\mathrm{\ell }\pm \mathrm{i}\eta$ and $\psi \left(x\right)={\mathrm{\Gamma }}^{\prime }\left(x\right)/\mathrm{\Gamma }\left(x\right)$ (§5.2(i)). … ►Corresponding expansions for $H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }(\eta ,\rho )$ can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).

17: 4.21 Identities

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4.21.1 $$\sin u\pm \cos u=\sqrt{2}\sin \left(u\pm \frac{1}{4}\pi \right)=\pm \sqrt{2}\cos \left(u\mp \frac{1}{4}\pi \right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $\sin z$: sine function

Referenced by:

§4.21(i), Erratum (V1.0.7) for Equation (4.21.1)

Permalink:

http://dlmf.nist.gov/4.21.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.0.7):

Originally the symbol $\pm$ was missing after the second equal sign.

Suggested 2012-09-27 by Dennis M. Heim

See also:

Annotations for §4.21(i), §4.21 and Ch.4

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4.21.4 $$\tan \left(u\pm v\right)=\frac{\tan u\pm \tan v}{1\mp \tan u\tan v},$$

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Symbols:

$\tan z$: tangent function

A&S Ref:

4.3.18

Referenced by:

(9.8.17)

Permalink:

http://dlmf.nist.gov/4.21.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(i), §4.21 and Ch.4

►

4.21.5 $$\cot \left(u\pm v\right)=\frac{\pm \cot u\cot v-1}{\cot u\pm \cot v}.$$

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Symbols:

$\cot z$: cotangent function

A&S Ref:

4.3.19

Permalink:

http://dlmf.nist.gov/4.21.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(i), §4.21 and Ch.4

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4.21.21 $$\sin \frac{z}{2}=\pm {\left(\frac{1-\cos z}{2}\right)}^{1/2},$$

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Symbols:

$\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.20

Referenced by:

§4.21(iii)

Permalink:

http://dlmf.nist.gov/4.21.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(iii), §4.21 and Ch.4

►

4.21.22 $$\cos \frac{z}{2}=\pm {\left(\frac{1+\cos z}{2}\right)}^{1/2},$$

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Symbols:

$\cos z$: cosine function and $z$: complex variable

A&S Ref:

4.3.21

Permalink:

http://dlmf.nist.gov/4.21.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(iii), §4.21 and Ch.4

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18: 32.7 Bäcklund Transformations

… ►and …with $\zeta =-2^{1/3}z$ and $\epsilon =\pm 1$, where $W(\zeta ;\frac{1}{2}\epsilon )$ satisfies ${\text{P}}_{\text{II}}$ with $z=\zeta$, $\alpha =\frac{1}{2}\epsilon$, and $w(z;0)$ satisfies ${\text{P}}_{\text{II}}$ with $\alpha =0$. ►The solutions $w_{\alpha }=w(z;\alpha )$, $w_{\alpha \pm 1}=w(z;\alpha \pm 1)$, satisfy the nonlinear recurrence relation … ►Let $w_0=w(z;{\alpha }_0,{\beta }_0)$ and $w_j^{\pm }=w(z;{\alpha }_j^{\pm },{\beta }_j^{\pm })$, $j=1,2,3,4$, be solutions of ${\text{P}}_{\text{IV}}$ with … ►with $\epsilon =\pm 1$. …

19: 19.34 Mutual Inductance of Coaxial Circles

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19.34.3 $$2abI({𝐞}_5)=a_{3}I(\mathrm{𝟎})-I({𝐞}_3)=a_{3}I(\mathrm{𝟎})-r_{+}^2{r}_{-}^{2}I(-{𝐞}_3)=2ab(I(\mathrm{𝟎})-r_{-}^{2}I({𝐞}_1-{𝐞}_3)),$$

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Symbols:

$I(𝐦)$: integral and $r_{\pm }$

Permalink:

http://dlmf.nist.gov/19.34.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

►where $a_1+b_{1}t=1+t$ and ►

19.34.4 $$r_{\pm }^2=a_3\pm 2ab=h^2+{(a\pm b)}^2$$

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Symbols:

$h$: distance and $r_{\pm }$

Permalink:

http://dlmf.nist.gov/19.34.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

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19.34.5 $$\frac{3c^2}{8\pi ab}M=3R_F(0,r_{+}^2,r_{-}^2)-2r_{-}^2{R}_D(0,r_{+}^2,r_{-}^2),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$

Permalink:

http://dlmf.nist.gov/19.34.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

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19.34.6 $$\frac{c^2}{2\pi }M=(r_{+}^2+r_{-}^2)R_F(0,r_{+}^2,r_{-}^2)-4R_G(0,r_{+}^2,r_{-}^2).$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$

Referenced by:

§19.34

Permalink:

http://dlmf.nist.gov/19.34.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

…

20: 4.38 Inverse Hyperbolic Functions: Further Properties

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4.38.15 $$\mathrm{Arcsinh}u\pm \mathrm{Arcsinh}v=\mathrm{Arcsinh}\left(u{(1+v^2)}^{1/2}\pm v{(1+u^2)}^{1/2}\right),$$

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Symbols:

$\mathrm{Arcsinh}z$: general inverse hyperbolic sine function

A&S Ref:

4.6.26

Permalink:

http://dlmf.nist.gov/4.38.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

►

4.38.16 $$\mathrm{Arccosh}u\pm \mathrm{Arccosh}v=\mathrm{Arccosh}\left(uv\pm {((u^2-1)(v^2-1))}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function

A&S Ref:

4.6.27

Permalink:

http://dlmf.nist.gov/4.38.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

►

4.38.17 $$\mathrm{Arctanh}u\pm \mathrm{Arctanh}v=\mathrm{Arctanh}\left(\frac{u\pm v}{1\pm uv}\right),$$

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Symbols:

$\mathrm{Arctanh}z$: general inverse hyperbolic tangent function

A&S Ref:

4.6.28

Permalink:

http://dlmf.nist.gov/4.38.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

►

4.38.18 $$\mathrm{Arcsinh}u\pm \mathrm{Arccosh}v=\mathrm{Arcsinh}\left(uv\pm {((1+u^2)(v^2-1))}^{1/2}\right)=\mathrm{Arccosh}\left(v{(1+u^2)}^{1/2}\pm u{(v^2-1)}^{1/2}\right),$$

ⓘ

Symbols:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function and $\mathrm{Arcsinh}z$: general inverse hyperbolic sine function

A&S Ref:

4.6.29

Permalink:

http://dlmf.nist.gov/4.38.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

►

4.38.19 $$\mathrm{Arctanh}u\pm \mathrm{Arccoth}v=\mathrm{Arctanh}\left(\frac{uv\pm 1}{v\pm u}\right)=\mathrm{Arccoth}\left(\frac{v\pm u}{uv\pm 1}\right).$$

ⓘ

Symbols:

$\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function and $\mathrm{Arctanh}z$: general inverse hyperbolic tangent function

A&S Ref:

4.6.30

Permalink:

http://dlmf.nist.gov/4.38.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

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