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21: 4.1 Special Notation

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$k,m,n$	integers.
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$\mathrm{e}$	base of natural logarithms.

►It is assumed the user is familiar with the definitions and properties of elementary functions of real arguments $x$. The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. ►The main functions treated in this chapter are the logarithm $\ln z$, $\mathrm{Ln}z$; the exponential $\exp z$, ${\mathrm{e}}^z$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\arcsin z$, $\mathrm{Arcsin}z$, etc. … ►Sometimes “arc” is replaced by the index “$-1$”, e. …

22: 19.11 Addition Theorems

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19.11.1 $$F(\theta ,k)+F(\varphi ,k)=F(\psi ,k),$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $\theta$: amplitude and $\psi$: angle

Referenced by:

§19.11(i)

Permalink:

http://dlmf.nist.gov/19.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(i), §19.11 and Ch.19

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19.11.2 $$E(\theta ,k)+E(\varphi ,k)=E(\psi ,k)+k^2\sin \theta \sin \varphi \sin \psi .$$

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Symbols:

$E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $\theta$: amplitude and $\psi$: angle

Permalink:

http://dlmf.nist.gov/19.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(i), §19.11 and Ch.19

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19.11.7 $$F(\varphi ,k)=K\left(k\right)-F(\theta ,k),$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument, $k$: real or complex modulus and $\theta$: amplitude

Permalink:

http://dlmf.nist.gov/19.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(ii), §19.11 and Ch.19

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19.11.8 $$E(\varphi ,k)=E\left(k\right)-E(\theta ,k)+k^2\sin \theta \sin \varphi ,$$

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Symbols:

$E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\theta$: amplitude

Permalink:

http://dlmf.nist.gov/19.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(ii), §19.11 and Ch.19

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19.11.9 $$\tan \theta =1/(k^{\prime }\tan \varphi ).$$

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Symbols:

$\tan z$: tangent function, $\varphi$: real or complex argument, $k^{\prime }$: complementary modulus and $\theta$: amplitude

Permalink:

http://dlmf.nist.gov/19.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(ii), §19.11 and Ch.19

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23: 4.29 Graphics

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§4.29(i) Real Arguments

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Figure 4.29.6: Principal values of $\mathrm{arccsch}x$ and $\mathrm{arcsech}x$. … Magnify

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§4.29(ii) Complex Arguments

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24: 16.27 Software

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§16.27(ii) Real Argument and Parameters

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§16.27(iii) Complex Argument and/or Parameters

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25: 23.24 Software

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§23.24(ii) Real Argument

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§23.24(iii) Complex Argument

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26: 13.7 Asymptotic Expansions for Large Argument

§13.7 Asymptotic Expansions for Large Argument

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§13.7(ii) Error Bounds

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§13.7(iii) Exponentially-Improved Expansion

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13.7.13

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $m$: integer, $n$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant and $R_n(a,b,z)$: remainder

Permalink:

http://dlmf.nist.gov/13.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(iii), §13.7 and Ch.13

… ►For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).

27: Bibliography T

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COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 36 (4), pp. 363–372.

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Notes:

Double-precision Fortran, minimum accuracy: 14D. See also Thompson (2004)

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item, §33.23(vi), 3rd item.

Encodings:

BibTeX

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Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§33.23(vi), I. J. Thompson and A. R. Barnett (1985).

Encodings:

BibTeX

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Algorithm 926: incomplete gamma functions with negative arguments. ACM Trans. Math. Software 39 (2), pp. Art. 14, 9.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

7th item, §8.28(ii).

Encodings:

BibTeX

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Evaluation of the exponential integral for large complex arguments. J. Research Nat. Bur. Standards 52, pp. 313–317.

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External Links:

MathReview (H. Bückner), ZentralBlatt

Referenced by:

§6.18(i).

Encodings:

BibTeX

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Simplified calculation of $\mathrm{Ei}(x)$ for positive arguments, and a short table of $\mathrm{Shi}(x)$ . Math. Comp. 22 (102), pp. 448–449.

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External Links:

FullText, Document, ZentralBlatt

Referenced by:

§6.18(i).

Encodings:

BibTeX

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28: 19.4 Derivatives and Differential Equations

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19.4.5 $$\frac{\partial F(\varphi ,k)}{\partial k}=\frac{E(\varphi ,k)-k_{}^{\prime }{}_{}{}^{2}F(\varphi ,k)}{kk_{}^{\prime }{}_{}{}^2}-\frac{k\sin \varphi \cos \varphi }{k_{}^{\prime }{}_{}{}^2\sqrt{1-k^2{\sin }^2\varphi }},$$

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.6 $$\frac{\partial E(\varphi ,k)}{\partial k}=\frac{E(\varphi ,k)-F(\varphi ,k)}{k},$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.30(i)

Permalink:

http://dlmf.nist.gov/19.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.7 $$\frac{\partial \mathrm{\Pi }(\varphi ,{\alpha }^2,k)}{\partial k}=\frac{k}{k_{}^{\prime }{}_{}{}^2(k^2-{\alpha }^2)}\left(E(\varphi ,k)-k_{}^{\prime }{}_{}{}^2\mathrm{\Pi }(\varphi ,{\alpha }^2,k)-\frac{k^2\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}\right).$$

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Symbols:

$\cos z$: cosine function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Permalink:

http://dlmf.nist.gov/19.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.9 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+k_{}^{\prime }{}_{}{}^2{D}_k+k)E(\varphi ,k)=\frac{k\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}.$$

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Symbols:

$\cos z$: cosine function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Permalink:

http://dlmf.nist.gov/19.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

►If $\varphi =\pi /2$, then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$. …

29: 10.77 Software

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§10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)

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§10.77(iii) Bessel Functions–Real Order and Argument

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§10.77(vi) Bessel Functions–Imaginary Order and Real Argument

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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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30: Bibliography N

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Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes. ACM Trans. Math. Software 18 (3), pp. 345–349.

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Notes:

Double-precision Fortran, minimum accuracy 9S, maximum accuracy 13S.

External Links:

ISSN 0098-3500, Document, GAMS (module 707; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes. J. Comput. Appl. Math. 39 (2), pp. 193–200.

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Notes:

Extended-precision Fortran evaluator, minimum accuracy 9S, maximum accuracy 13S.

External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§13.32(iii).

Encodings:

BibTeX

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On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

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External Links:

ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt

Referenced by:

§11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).

Encodings:

BibTeX

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Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl. Math. 150, pp. 141–177.

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External Links:

ISSN 0167-8019, Document, MathReview Entry, ZentralBlatt

Referenced by:

(9.7.16), (9.7.17), §9.7(iii), §9.7(iv).

Encodings:

BibTeX

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Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions. Stud. Appl. Math. 140 (4), pp. 508–541.

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External Links:

ISSN 0022-2526, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§11.11(i), §11.11(i).

Encodings:

BibTeX

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