variation of real or complex functions

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1: 1.4 Calculus of One Variable

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Functions of Bounded Variation

… ► … ►Lastly, whether or not the real numbers $a$ and $b$ satisfy , and whether or not they are finite, we define ${𝒱}_{a,b}\left(f\right)$ 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

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10.40.6 $$I_{\nu }\left(z\right)K_{\nu }\left(z\right)\sim \frac{1}{2z}\left(1-\frac{1}{2}\frac{\mu -1}{{(2z)}^2}+\frac{1\cdot 3}{2\cdot 4}\frac{(\mu -1)(\mu -9)}{{(2z)}^4}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: 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

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10.40.7 $$I_{\nu }^{\prime }\left(z\right)K_{\nu }^{\prime }\left(z\right)\sim -\frac{1}{2z}\left(1+\frac{1}{2}\frac{\mu -3}{{(2z)}^2}-\frac{1}{2\cdot 4}\frac{(\mu -1)(\mu -45)}{{(2z)}^4}+\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: 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

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

… ►where $𝒱$ denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that $|\mathrm{\Re }t|$ changes monotonically. … ►

10.40.12

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, ${𝒱}_{a,b}\left(f\right)$: 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 was written incorrectly as $\pi \le |\mathrm{ph}z|\le \frac{3}{2}\pi$.

Suggested 2014-11-05 by Gergő Nemes

See also:

Annotations for §10.40(iii), §10.40 and Ch.10

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3: 8.23 Statistical Applications

§8.23 Statistical Applications

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4: 10.77 Software

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§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

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

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5: 14.31 Other Applications

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§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 $$𝒲\left\{{\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2),{\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)\right\}=\frac{{(-1)}^m(n+m)!}{(1-z^2)(n-m)!}{A}_n^m({\gamma }^2)A_n^{-m}({\gamma }^2),$$

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Symbols:

$𝒲$: Wronskian, $!$: factorial (as in $n!$), ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: 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

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30.6.4 $${\mathrm{𝑃𝑠}}_n^m(x\pm \mathrm{i}0,{\gamma }^2)={(\mp \mathrm{i})}^m{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$$

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Symbols:

$\mathrm{i}$: imaginary unit, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: 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

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12.8.1 $$zU(a,z)-U(a-1,z)+(a+\frac{1}{2})U(a+1,z)=0,$$

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Symbols:

$U(a,z)$: 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

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12.8.2 $$U^{\prime }(a,z)+\frac{1}{2}zU(a,z)+(a+\frac{1}{2})U(a+1,z)=0,$$

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Symbols:

$U(a,z)$: 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 $$U^{\prime }(a,z)-\frac{1}{2}zU(a,z)+U(a-1,z)=0,$$

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Symbols:

$U(a,z)$: 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 $$2U^{\prime }(a,z)+U(a-1,z)+(a+\frac{1}{2})U(a+1,z)=0.$$

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Symbols:

$U(a,z)$: 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

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12.8.5 $$zV(a,z)-V(a+1,z)+(a-\frac{1}{2})V(a-1,z)=0,$$

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Symbols:

$V(a,z)$: 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

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§5.23(iii) Approximations in the Complex Plane

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

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22.17.1 $$\mathrm{p}\mathrm{q}(z,k)=\mathrm{p}\mathrm{q}(z,-k),$$

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Symbols:

$\mathrm{p}\mathrm{q}(z,k)$: 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

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22.17.2 $$\mathrm{sn}(z,1/k)=k\mathrm{sn}(z/k,k),$$

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Symbols:

$\mathrm{sn}(z,k)$: 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

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22.17.3 $$\mathrm{cn}(z,1/k)=\mathrm{dn}(z/k,k),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: 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

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22.17.4 $$\mathrm{dn}(z,1/k)=\mathrm{cn}(z/k,k).$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: 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

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§22.17(ii) Complex Moduli

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10: 4.3 Graphics

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§4.3(ii) Complex Arguments: Conformal Maps

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Figure 4.3.3: $\ln \left(x+\mathrm{i}y\right)$ (principal value). … Magnify 3D Help

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Figure 4.3.4: ${\mathrm{e}}^{x+\mathrm{i}y}$. Magnify 3D Help