variation of real or complex functions

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1: 1.4 Calculus of One Variable

… ►

Functions of Bounded Variation

… ► … ►Lastly, whether or not the real numbers

a

and

b

satisfy

a<b

, and whether or not they are finite, we define

\mathcal{V}_{a,b}\left(f\right)

by (1.4.34) whenever this integral exists. This definition also applies when

f(x)

is a complex function of the real variable

x

. …

2: 10.40 Asymptotic Expansions for Large Argument

… ►

10.40.6

I_{\nu}\left(z\right)K_{\nu}\left(z\right)\sim\frac{1}{2z}\left(1-\frac{1}{2}% \frac{\mu-1}{(2z)^{2}}+\frac{1\cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2z)^{4}% }-\dotsb\right),

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.7.5
Referenced by:: §10.40(i), §10.40(i)
Permalink:: http://dlmf.nist.gov/10.40.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §10.40(i), §10.40(i), §10.40 and Ch.10

►

10.40.7

I_{\nu}'\left(z\right)K_{\nu}'\left(z\right)\sim-\frac{1}{2z}\left(1+\frac{1}{% 2}\frac{\mu-3}{(2z)^{2}}-\frac{1}{2\cdot 4}\frac{(\mu-1)(\mu-45)}{(2z)^{4}}+% \dotsb\right),

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.7.6
Referenced by:: §10.40(i), §10.40(i)
Permalink:: http://dlmf.nist.gov/10.40.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §10.40(i), §10.40(i), §10.40 and Ch.10

… ►

§10.40(iii) Error Bounds for Complex Argument and Order

… ►where

\mathcal{V}

denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that

|\Re t|

changes monotonically. … ►

10.40.12

\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)\leq\begin{cases}|z|^{-\ell},&|% \operatorname{ph}z|\leq\tfrac{1}{2}\pi,\\ \chi(\ell)|z|^{-\ell},&\tfrac{1}{2}\pi\leq|\operatorname{ph}z|\leq\pi,\\ 2\chi(\ell)|\Re z|^{-\ell},&\pi\leq|\operatorname{ph}z|<\tfrac{3}{2}\pi,\end{cases}

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$ : total variation and $z$ : complex variable
Referenced by:: §10.40(i), §10.40(iii), Erratum (V1.0.11) for Equation (10.40.12)
Permalink:: http://dlmf.nist.gov/10.40.E12
Encodings:: pMML, png, TeX
Correction (effective with 1.0.11):: Originally the third constraint $\pi\leq|\operatorname{ph}z|<\frac{3}{2}\pi$ was written incorrectly as $\pi\leq|\operatorname{ph}z|\leq\frac{3}{2}\pi$ .
Suggested 2014-11-05 by Gergő Nemes
See also:: Annotations for §10.40(iii), §10.40 and Ch.10

…

3: 8.23 Statistical Applications

§8.23 Statistical Applications

…

4: 10.77 Software

… ►

§10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions

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§10.77(v) Bessel Functions–Real Order and Complex Argument (including Hankel Functions)

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§10.77(vii) Bessel Functions–Complex Order and Real Argument

… ►

§10.77(viii) Bessel Functions–Complex Order and Argument

…

5: 14.31 Other Applications

… ►

§14.31(iii) Miscellaneous

… ►Legendre functions

P_{\nu}\left(x\right)

of complex degree

\nu

appear in the application of complex angular momentum techniques to atomic and molecular scattering (Connor and Mackay (1979)). …

6: 30.6 Functions of Complex Argument

§30.6 Functions of Complex Argument

►The solutions … ►

30.6.3

\mathscr{W}\left\{\mathit{Ps}^{m}_{n}\left(z,\gamma^{2}\right),\mathit{Qs}^{m}% _{n}\left(z,\gamma^{2}\right)\right\}=\frac{(-1)^{m}(n+m)!}{(1-z^{2})(n-m)!}A_% {n}^{m}(\gamma^{2})A_{n}^{-m}(\gamma^{2}),

ⓘ

Symbols:: $\mathscr{W}$ : Wronskian, $!$ : factorial (as in $n!$ ), $\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $\mathit{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $z$ : complex variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $A_{n}^{m}(\gamma^{2})$ and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §30.6, §30.6 and Ch.30

… ►

30.6.4

\mathit{Ps}^{m}_{n}\left(x\pm\mathrm{i}0,\gamma^{2}\right)=(\mp\mathrm{i})^{m}% \mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),

ⓘ

Symbols:: $\mathrm{i}$ : imaginary unit, $\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §30.6, §30.6 and Ch.30

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7: 12.8 Recurrence Relations and Derivatives

… ►

12.8.1

zU\left(a,z\right)-U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+1,z\right)=0,

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.4
Referenced by:: §12.8(i)
Permalink:: http://dlmf.nist.gov/12.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

►

12.8.2

U'\left(a,z\right)+\tfrac{1}{2}zU\left(a,z\right)+(a+\tfrac{1}{2})U\left(a+1,z% \right)=0,

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.1
Permalink:: http://dlmf.nist.gov/12.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

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12.8.3

U'\left(a,z\right)-\tfrac{1}{2}zU\left(a,z\right)+U\left(a-1,z\right)=0,

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.2
Permalink:: http://dlmf.nist.gov/12.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

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12.8.4

2U'\left(a,z\right)+U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+1,z\right)=0.

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.3
Referenced by:: §12.8(i)
Permalink:: http://dlmf.nist.gov/12.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

… ►

12.8.5

zV\left(a,z\right)-V\left(a+1,z\right)+(a-\tfrac{1}{2})V\left(a-1,z\right)=0,

ⓘ

Symbols:: $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.8
Permalink:: http://dlmf.nist.gov/12.8.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

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8: 5.23 Approximations

… ►

§5.23(iii) Approximations in the Complex Plane

…

9: 22.17 Moduli Outside the Interval [0,1]

… ►

22.17.1

\operatorname{pq}\left(z,k\right)=\operatorname{pq}\left(z,-k\right),

ⓘ

Symbols:: $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$ : generic Jacobian elliptic function, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.17.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

… ►

22.17.2

\operatorname{sn}\left(z,1/k\right)=k\operatorname{sn}\left(z/k,k\right),

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.11.2
Permalink:: http://dlmf.nist.gov/22.17.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

►

22.17.3

\operatorname{cn}\left(z,1/k\right)=\operatorname{dn}\left(z/k,k\right),

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.11.3
Permalink:: http://dlmf.nist.gov/22.17.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

►

22.17.4

\operatorname{dn}\left(z,1/k\right)=\operatorname{cn}\left(z/k,k\right).

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.11.4
Permalink:: http://dlmf.nist.gov/22.17.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

… ►

§22.17(ii) Complex Moduli

…

10: 4.29 Graphics

… ►

§4.29(ii) Complex Arguments

… ►The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …