of imaginary argument

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11: 15.19 Methods of Computation

… ►For $z\in ℂ$ it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. This is because the linear transformations map the pair $\{{\mathrm{e}}^{\pi \mathrm{i}/3},{\mathrm{e}}^{-\pi \mathrm{i}/3}\}$ onto itself. …

12: 4.3 Graphics

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§4.3(i) Real Arguments

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§4.3(ii) Complex Arguments: Conformal Maps

►Figure 4.3.2 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the negative real axis, where $w={\mathrm{e}}^z$ and $z=\ln w$ (principal value). …Lines parallel to the real axis in the $z$-plane map onto rays in the $w$-plane, and lines parallel to the imaginary axis in the $z$-plane map onto circles centered at the origin in the $w$-plane. … ►

§4.3(iii) Complex Arguments: Surfaces

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13: 5.22 Tables

… ►This reference also includes $\psi \left(x+\mathrm{i}y\right)$ for the same arguments to 5D. …

14: Bibliography D

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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

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Notes:

Errata: In eq. (2.8) replace ${(x^2/4)}^2$ by ${(x^2/4)}^s$. In eq. (4.2) $\varphi (\zeta )$ should be replaced by $\psi (\zeta )$. In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$. In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.

External Links:

ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.24, §10.45, §2.8(iv).

Encodings:

BibTeX

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Conical functions of purely imaginary order and argument. Proc. Roy. Soc. Edinburgh Sect. A 143 (5), pp. 929–955.

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External Links:

ISSN 0308-2105, Document, Link, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§14.20(ix), §14.20(ix).

Encodings:

BibTeX

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15: Bibliography G

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Algorithm 831: Modified Bessel functions of imaginary order and positive argument. ACM Trans. Math. Software 30 (2), pp. 159–164.

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External Links:

ISSN 0098-3500, Document, GAMS (module 831; package TOMS), MathReview Entry, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments. ACM Trans. Math. Software 30 (2), pp. 145–158.

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External Links:

ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(viii).

Encodings:

BibTeX

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16: Software Index

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	Open Source													With Book						Commercial
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10.77(vi) Bessel Functions–Imaginary Order and Real Argument	✓	✓										✓	✓								✓	✓	✓
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17: 15.4 Special Cases

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18: 14.20 Conical (or Mehler) Functions

… ►For the case of purely imaginary order and argument see Dunster (2013). …

19: 10.61 Definitions and Basic Properties

… ►In general, Kelvin functions have a branch point at $x=0$ and functions with arguments $x{\mathrm{e}}^{\pm \pi \mathrm{i}}$ are complex. …

20: 33.11 Asymptotic Expansions for Large $\rho$

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33.11.4 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}}(f(\eta ,\rho )\pm \mathrm{i}g(\eta ,\rho )),$$

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Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $f(\eta ,\rho )$: function and $g(\eta ,\rho )$: function

Permalink:

http://dlmf.nist.gov/33.11.E4

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.22):

The arguments of some of the functions in these equations were included to improve clarity of the presentation.

See also:

Annotations for §33.11 and Ch.33

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