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31: 4.38 Inverse Hyperbolic Functions: Further Properties

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

ⓘ

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

ⓘ

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

ⓘ

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

…

32: 10.73 Physical Applications

… ►See Jackson (1999, Chapter 3, §§3.7, 3.8, 3.11, 3.13), Lamb (1932, Chapter V, §§100–102; Chapter VIII, §§186, 191–193; Chapter X, §§303, 304), Happel and Brenner (1973, Chapter 3, §3.3; Chapter 7, §7.3), Korenev (2002, Chapter 4, §43), and Gray et al. (1922, Chapter XI). …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 ${\mathrm{e}}^{\pm \mathrm{i}kt}$. …

33: 7.5 Interrelations

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7.5.6 $${\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}(\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right))=\frac{1}{2}(1\pm \mathrm{i})-(C\left(z\right)\pm \mathrm{i}S\left(z\right)).$$

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: 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

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.7 $$\zeta =\frac{1}{2}\sqrt{\pi }(1\mp \mathrm{i})z,$$

ⓘ

Defines:

$\zeta$: change of variable (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.8 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\mathrm{erf}\zeta .$$

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{erf}z$: error function, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5, §7.5, §7.6(i)

Permalink:

http://dlmf.nist.gov/7.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

►

7.5.9 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\left(1-{\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}w\left(\mathrm{i}\zeta \right)\right).$$

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.10 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=\frac{1}{2}(1\pm \mathrm{i}){\mathrm{e}}^{{\zeta }^2}\mathrm{erfc}\zeta .$$

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

A&S Ref:

7.3.22 (in different form)

Referenced by:

§7.12(ii), §7.13(iv), §7.5, §7.5

Permalink:

http://dlmf.nist.gov/7.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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34: 28.17 Stability as $x\to \pm \mathrm{\infty }$

§28.17 Stability as $x\to \pm \mathrm{\infty }$

►If all solutions of (28.2.1) are bounded when $x\to \pm \mathrm{\infty }$ along the real axis, then the corresponding pair of parameters $(a,q)$ is called stable. …

35: 12.2 Differential Equations

… ►Standard solutions are $U(a,\pm z)$, $V(a,\pm z)$, $\overline{U}(a,\pm x)$ (not complex conjugate), $U(-a,\pm \mathrm{i}z)$ for (12.2.2); $W(a,\pm x)$ for (12.2.3); $D_{\nu }\left(\pm z\right)$ for (12.2.4), where … ►The solutions $W(a,\pm x)$ are treated in §12.14. … ►

12.2.12 $$𝒲\left\{U(a,z),U(-a,\pm \mathrm{i}z)\right\}=\mp \mathrm{i}{\mathrm{e}}^{\pm \mathrm{i}\pi (\frac{1}{2}a+\frac{1}{4})}.$$

ⓘ

Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.4.1

Permalink:

http://dlmf.nist.gov/12.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §12.2(iii), §12.2 and Ch.12

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12.2.18 $$\sqrt{2\pi }U(a,z)=\mathrm{\Gamma }\left(\frac{1}{2}-a\right)\left({\mathrm{e}}^{\mp \mathrm{i}\pi (\frac{1}{2}a+\frac{1}{4})}U(-a,\pm \mathrm{i}z)+{\mathrm{e}}^{\pm \mathrm{i}\pi (\frac{1}{2}a+\frac{1}{4})}U(-a,\mp \mathrm{i}z)\right),$$

ⓘ

Symbols:

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

A&S Ref:

19.4.7 (modification of)

Referenced by:

§12.5(i), §12.9(i)

Permalink:

http://dlmf.nist.gov/12.2.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §12.2(v), §12.2 and Ch.12

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12.2.19 $$U(a,z)=\pm \mathrm{i}{\mathrm{e}}^{\pm \mathrm{i}\pi a}U(a,-z)+\frac{\sqrt{2\pi }}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\mathrm{e}}^{\pm \mathrm{i}\pi (\frac{1}{2}a-\frac{1}{4})}U(-a,\pm \mathrm{i}z).$$

ⓘ

Symbols:

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

Referenced by:

(12.14.4_5)

Permalink:

http://dlmf.nist.gov/12.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §12.2(v), §12.2 and Ch.12

…

36: 32.10 Special Function Solutions

… ►For example, if $\alpha =\frac{1}{2}\epsilon$, with $\epsilon =\pm 1$, then the Riccati equation is … ►Solutions for other values of $\alpha$ are derived from $w(z;\pm \frac{1}{2})$ by application of the Bäcklund transformations (32.7.1) and (32.7.2). … ►with $n\in \mathbb{Z}$, and ${\epsilon }_1=\pm 1$, ${\epsilon }_2=\pm 1$, independently. … ►with $n\in \mathbb{Z}$ and $\epsilon =\pm 1$. In the case when $n=0$ in (32.10.15), the Riccati equation is …

37: 10.13 Other Differential Equations

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10.13.4 $$w^{\prime \prime }+\frac{1\mp 2\nu }{z}{w}^{\prime }+{\lambda }^{2}w=0,$$ $w=z^{\pm \nu }{𝒞}_{\nu }\left(\lambda z\right)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.52

Referenced by:

Erratum (V1.0.8) for Equation (10.13.4)

Permalink:

http://dlmf.nist.gov/10.13.E4

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.8):

This equation has been generalized to allow an additional case. Originally only the case $w=z^{\nu }{𝒞}_{\nu }\left(\lambda z\right)$ was covered.

Suggested 2014-01-27 by Tom Koornwinder

See also:

Annotations for §10.13 and Ch.10

… ►See also Watson (1944, pp. 95–100).

38: 19.22 Quadratic Transformations

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19.22.5 $$2p_{\pm }=\sqrt{(p+x)(p+y)}\pm \sqrt{(p-x)(p-y)},$$

ⓘ

Defines:

$p_{\pm }$: modified parameter (locally)

Referenced by:

§19.22(iii), §19.22(iii)

Permalink:

http://dlmf.nist.gov/19.22.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.22(i), §19.22 and Ch.19

… ►

$4(p_{\pm }^2-a^2)={(\sqrt{p^2-x^2}\pm \sqrt{p^2-y^2})}^2.$

… ►

$2z_{\pm }=\sqrt{(z+x)(z+y)}\pm \sqrt{(z-x)(z-y)},$

… ►

$4(z_{\pm }^2-a^2)={(\sqrt{z^2-x^2}\pm \sqrt{z^2-y^2})}^2.$

… ►However, if $x$ and $y$ are complex conjugates and $z$ and $p$ are real, then the right-hand sides of all transformations in §§19.22(i) and 19.22(iii)—except (19.22.3) and (19.22.22)—are free of complex numbers and $p_{\pm }^2-p_{\mp }^2=\pm |p^2-x^2|\ne 0$. …

39: Errata

… ►

33.11.1 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k{\left(b\right)}_k}{k!{(\pm 2\mathrm{i}\rho )}^k}$$

Originally the factor in the denominator on the right-hand side was written incorrectly as ${(\mp 2\mathrm{i}\rho )}^k$. This has been corrected to ${(\pm 2\mathrm{i}\rho )}^k$.

Reported by Ian Thompson on 2018-05-17

… ►

8.12.18 $$\begin{array}{c}Q(a,z)\\ P(a,z)\end{array}\}\sim \frac{z^{a-\frac{1}{2}}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a\right)}\left(d(\pm \chi )\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\chi )}{z^{k/2}}\mp \sum _{k=1}^{\mathrm{\infty }}\frac{B_k(\chi )}{z^{k/2}}\right)$$

The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.

Reported 2017-01-28 by Richard Paris.

… ►

To be consistent with the notation used in (8.12.16), Equation (8.12.5) was changed to read

8.12.5 $$\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\mathrm{i}\sin (\pi a)}Q(-a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\pm \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)-\mathrm{i}T(a,\eta )$$

… ►

The correct corner coordinates for the 9-point square, given on the last line of this table, are $(\pm \sqrt{\frac{3}{5}}h,\pm \sqrt{\frac{3}{5}}h)$. Originally they were given incorrectly as $(\pm \sqrt{\frac{3}{5}}h,0)$, $(\pm \sqrt{\frac{3}{5}}h,0)$.

Diagram	$(x_j,y_j)$	$w_j$	$R$
⋮
	$(0,0)$	$\frac{16}{81}$	$O\left(h^6\right)$
	$(\pm \sqrt{\frac{3}{5}}h,0)$, $(0,\pm \sqrt{\frac{3}{5}}h)$	$\frac{10}{81}$
	$(\pm \sqrt{\frac{3}{5}}h,\pm \sqrt{\frac{3}{5}}h)$	$\frac{25}{324}$

Reported 2014-01-13 by Stanley Oleszczuk.

►

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

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

Reported 2012-09-27 by Dennis Heim.

…

40: 10.38 Derivatives with Respect to Order

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10.38.1 $$\frac{\partial I_{\pm \nu }\left(z\right)}{\partial \nu }=\pm I_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.42

Referenced by:

§10.38, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.38.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial I_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.38.2).

See also:

Annotations for §10.38 and Ch.10

… ►

10.38.6 $${\frac{\partial I_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=-\frac{1}{\sqrt{2\pi x}}\left(E_1\left(2x\right){\mathrm{e}}^x\pm \mathrm{Ei}\left(2x\right){\mathrm{e}}^{-x}\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.32 and 10.2.33 (both corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

►

10.38.7 $${\frac{\partial K_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=\pm \sqrt{\frac{\pi }{2x}}{E}_1\left(2x\right){\mathrm{e}}^x.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.34 (corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

…