complex arithmetic

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11: 19.8 Quadratic Transformations

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19.8.5 $$K\left(k\right)=\frac{\pi }{2M(1,k^{\prime })},$$ .

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(i), §19.8 and Ch.19

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19.8.6 $$E\left(k\right)=\frac{\pi }{2M(1,k^{\prime })}\left(a_0^2-\sum _{n=0}^{\mathrm{\infty }}{2}^{n-1}{c}_n^2\right)=K\left(k\right)\left(a_1^2-\sum _{n=2}^{\mathrm{\infty }}{2}^{n-1}{c}_n^2\right),$$ , $a_0=1$, $g_0=k^{\prime }$,

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $n$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, $a_n$: iterate and $g_n$: iterate

Permalink:

http://dlmf.nist.gov/19.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(i), §19.8 and Ch.19

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19.8.7 $$\mathrm{\Pi }({\alpha }^2,k)=\frac{\pi }{4M(1,k^{\prime })}\left(2+\frac{{\alpha }^2}{1-{\alpha }^2}\sum _{n=0}^{\mathrm{\infty }}{Q}_n\right),$$ , ,

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $n$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter and $Q_n$

Referenced by:

§19.8(i)

Permalink:

http://dlmf.nist.gov/19.8.E7

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See also:

Annotations for §19.8(i), §19.8 and Ch.19

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19.8.9 $$\mathrm{\Pi }({\alpha }^2,k)=\frac{\pi }{4M(1,k^{\prime })}\frac{k^2}{k^2-{\alpha }^2}\sum _{n=0}^{\mathrm{\infty }}{Q}_n,$$ , ,

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $n$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter and $Q_n$

Referenced by:

§19.8(i)

Permalink:

http://dlmf.nist.gov/19.8.E9

Encodings:

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See also:

Annotations for §19.8(i), §19.8 and Ch.19

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19.8.10 $$p_0^2=1-(k^2/{\alpha }^2).$$

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Symbols:

$k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Permalink:

http://dlmf.nist.gov/19.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(i), §19.8 and Ch.19

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12: 4.45 Methods of Computation

… ►For interval-arithmetic algorithms, see Markov (1981). … ►

§4.45(ii) Complex Variables

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4.45.15 $$\ln z=\ln |z|+\mathrm{i}\mathrm{ph}z,$$ $-\pi \le \mathrm{ph}z\le \pi$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.45.E15

Encodings:

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See also:

Annotations for §4.45(ii), §4.45 and Ch.4

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4.45.16 $${\mathrm{e}}^z={\mathrm{e}}^{\mathrm{\Re }z}(\cos \left(\mathrm{\Im }z\right)+\mathrm{i}\sin \left(\mathrm{\Im }z\right)).$$

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Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $\sin z$: sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.45.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.45(ii), §4.45 and Ch.4

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13: Bibliography

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Arb: A C Library for Arbitrary Precision Ball Arithmetic.

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Notes:

An extended-precision C code for special functions. The library uses arbitrary-precision ball arithmetic. Ball arithmetic is an efficient technique for performing numerical computations with automatic and rigorous tracking of error bounds. Arb is designed with computer algebra and computational number theory in mind, extending the principles behind FLINT to the domain of real and complex numbers.

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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14: Software Index

… ► ►►►►►►►

	Open Source													With Book						Commercial
…
4.48(ii) Interval Arithmetic	✓					✓						✓	✓								✓	✓	a		CoStLy
…
16.27(iii) Complex Arguments	✓											a	✓								✓	✓	✓
…
22.22(iii) Complex Argument	✓					✓						✓	✓	✓						✓	✓	✓	a	✓
…
23.24(iii) Complex Argument														✓						✓	✓	✓	a
…
33.26(iii) Complex arguments	✓											a	✓
…

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15: Bibliography I

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IEEE Standard for Floating-Point Arithmetic. The Institute of Electrical and Electronics Engineers, Inc..

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Notes:

IEEE Std 754-2008, Revision of IEEE Std 754-1985

External Links:

ISBN 978-0-7381-5753-5, Document

Referenced by:

§3.1(i).

Encodings:

BibTeX

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IEEE Standard for Interval Arithmetic: IEEE Std 1788-2015. The Institute of Electrical and Electronics Engineers, Inc..

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External Links:

ISBN 978-0-7381-9720-3, Document

Referenced by:

§3.1(ii), §3.1(ii), Erratum (V1.0.25) for Section 3.1.

Encodings:

BibTeX

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IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017. The Institute of Electrical and Electronics Engineers, Inc..

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External Links:

ISBN 978-1-5044-4602-0, Document

Referenced by:

§3.1(ii), §3.1(ii), Erratum (V1.0.25) for Section 3.1.

Encodings:

BibTeX

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IEEE International Standard for Information Technology—Microprocessor Systems—Floating-Point arithmetic: IEEE Std 754-2019. The Institute of Electrical and Electronics Engineers, Inc..

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Notes:

IEEE Std ISO/IEC/IEEE 60559, Revision of IEEE Std 754-1985

External Links:

ISBN 978-2-88912-566-1, Document

Referenced by:

Figure 3.1.1, Figure 3.1.1, §3.1(i), §3.1(i), Erratum (V1.0.25) for Section 3.1.

Encodings:

BibTeX

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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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External Links:

ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

BibTeX

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16: Bibliography C

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Level-Index Arithmetic: An Introductory Survey. In Numerical Analysis and Parallel Processing (Lancaster, 1987), P. R. Turner (Ed.), Lecture Notes in Math., Vol. 1397, pp. 95–168.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.1(iv).

Encodings:

BibTeX

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Polynomial approximations in the complex plane. J. Comput. Appl. Math. 18 (2), pp. 193–211.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.76(ii).

Encodings:

BibTeX

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An encyclopaedia of cubature formulas. J. Complexity 19 (3), pp. 445–453.

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Notes:

Numerical integration and its complexity (Oberwolfach, 2001)

External Links:

ISSN 0885-064X, Document, Web Page, MathReview, ZentralBlatt

Referenced by:

§3.5(x).

Encodings:

BibTeX

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The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30 (3-4), pp. 275–330.

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External Links:

ISSN 0013-8584, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.8(i).

Encodings:

BibTeX

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Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.

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External Links:

ISSN 0002-9920, MathReview (Willard Parker)

Referenced by:

§19.8(i).

Encodings:

BibTeX

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17: 25.15 Dirichlet $L$-functions

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25.15.1 $$L(s,\chi )=\sum _{n=1}^{\mathrm{\infty }}\frac{\chi (n)}{n^s},$$ $\mathrm{\Re }s>1$,

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Defines:

$L(s,\chi )$: Dirichlet $L$-function

Symbols:

$\mathrm{\Re }$: real part, $n$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, p. 249)

Referenced by:

(25.11.36), (25.11.36), §25.11(x), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

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25.15.2 $$L(s,\chi )=\prod _p{\left(1-\frac{\chi (p)}{p^s}\right)}^{-1},$$ $\mathrm{\Re }s>1$,

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\mathrm{\Re }$: real part, $p$: prime number, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

infinite product, product representation

Source:

Apostol (1976, p. 231)

Permalink:

http://dlmf.nist.gov/25.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

… ►where $\overline{\chi }$ is the complex conjugate of $\chi$, and … ►This result plays an important role in the proof of Dirichlet’s theorem on primes in arithmetic progressions (§27.11). Related results are: …

18: 13.29 Methods of Computation

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13.29.2 $$y(n)=z^{-n-\mu -\frac{1}{2}}{M}_{\kappa ,n+\mu }\left(z\right),$$

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Defines:

$y(n)$: solution (locally)

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.29.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

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13.29.5 $$(n+a)w(n)-\left(2(n+a+1)+z-b\right)w(n+1)+(n+a-b+2)w(n+2)=0,$$

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Symbols:

$n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.29.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

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13.29.6 $$w(n)={\left(a\right)}_{n}U(n+a,b,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.29.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

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13.29.7 $$z^{-a}=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a-b+1\right)}_s}{s!}w(s),$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.29.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

… ►The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.

19: 24.17 Mathematical Applications

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24.17.2 $$R_m(n)=\frac{1}{2(m-1)!}{\int }_a^n{f}^{(m)}(x){\tilde{E}}_{m-1}\left(h-x\right)dx.$$

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Symbols:

$dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${\tilde{E}}_n\left(x\right)$: periodic Euler functions, $m$: integer, $x$: real or complex, $h$: parameter, $a$: integer, $n$: integer, $f^{(m)}(x)$: integrable function and $R_m(n)$: remainder

Permalink:

http://dlmf.nist.gov/24.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(i), §24.17(i), §24.17 and Ch.24

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24.17.5

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Symbols:

$B_n$: Bernoulli numbers, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $n$: integer, $x$: real or complex and $M_n(x)$: function

Permalink:

http://dlmf.nist.gov/24.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

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24.17.7 $$M_n(x)=O\left({|x|}^{\gamma }\right),$$ $x\to \pm \mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $n$: integer, $x$: real or complex and $M_n(x)$: function

Permalink:

http://dlmf.nist.gov/24.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

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24.17.8 $$F(x)={\tilde{B}}_n\left(x\right)-2^{-n}{B}_n$$

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Symbols:

$B_n$: Bernoulli numbers, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.17.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

… ►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)). …

20: 18.39 Applications in the Physical Sciences

… ►Kuijlaars and Milson (2015, §1) refer to these, in this case complex zeros, as exceptional, as opposed to regular, zeros of the EOP’s, these latter belonging to the (real) orthogonality integration range. … ►

18.39.21 $$\mathscr{H}=-\frac{{\mathrm{\hslash }}^2}{2m}{\nabla }^2+V(𝐱),$$ $𝐱=(x,y,z)\in {ℝ}^3$,

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Symbols:

$\in$: element of, $ℝ$: real line, $y$: real variable, $z$: complex variable, $m$: nonnegative integer and $x$: real variable

Referenced by:

§18.39(ii)

Permalink:

http://dlmf.nist.gov/18.39.E21

Encodings:

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See also:

Annotations for §18.39(ii), §18.39 and Ch.18

… ►A major difficulty in such calculations, loss of precision, is addressed in Gautschi (2009) where use of variable precision arithmetic is discussed and employed. …