principle%20of%20the%20argument

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1: 35.1 Special Notation

$a, b$	complex variables.
…
$f(\mathbf{X})$	complex-valued function with $\mathbf{X}\in{\boldsymbol{\Omega}}$ .
…

►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively

\Gamma_{m}\left(a\right)

and

\mathrm{B}_{m}\left(a,b\right)

, and the special functions of matrix argument: Bessel (of the first kind)

A_{\nu}\left(\mathbf{T}\right)

and (of the second kind)

B_{\nu}\left(\mathbf{T}\right)

; confluent hypergeometric (of the first kind)

{{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)

\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)

and (of the second kind)

\Psi\left(a;b;\mathbf{T}\right)

; Gaussian hypergeometric

{{}_{2}F_{1}}\left(a_{1},a_{2};b;\mathbf{T}\right)

\displaystyle{{}_{2}F_{1}}\left({a_{1},a_{2}\atop b};\mathbf{T}\right)

; generalized hypergeometric

{{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)

\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)

. … ►Related notations for the Bessel functions are

\mathcal{J}_{\nu+\frac{1}{2}(m+1)}(\mathbf{T})=A_{\nu}\left(\mathbf{T}\right)/% A_{\nu}\left(\boldsymbol{{0}}\right)

(Faraut and Korányi (1994, pp. 320–329)),

K_{m}(0,\dots,0,\nu\mathpunct{|}\mathbf{S},\mathbf{T})=\left|\mathbf{T}\right|% ^{\nu}B_{\nu}\left(\mathbf{S}\mathbf{T}\right)

(Terras (1988, pp. 49–64)), and

\mathcal{K}_{\nu}(\mathbf{T})=\left|\mathbf{T}\right|^{\nu}B_{\nu}\left(% \mathbf{S}\mathbf{T}\right)

(Faraut and Korányi (1994, pp. 357–358)).

2: 35.5 Bessel Functions of Matrix Argument

§35.5 Bessel Functions of Matrix Argument

►

§35.5(i) Definitions

… ►

§35.5(ii) Properties

… ►

§35.5(iii) Asymptotic Approximations

►For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).

3: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

►

§35.8(i) Definition

… ►

Convergence Properties

… ►

Confluence

… ►

Invariance

…

4: 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

►

§35.6(i) Definitions

… ►

Laguerre Form

… ►

§35.6(ii) Properties

… ►

§35.6(iii) Relations to Bessel Functions of Matrix Argument

…

5: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

►

§35.7(i) Definition

… ►

Jacobi Form

… ►

Confluent Form

… ►

Integral Representation

…

7: Bibliography F

… ►

FDLIBM (free C library)

ⓘ

Notes:: The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.
External Links:: GAMS (package FDLIBM)
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

ⓘ

External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §6.15.
Encodings:: BibTeX

… ►

H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

ⓘ

Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

… ►

G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

ⓘ

External Links:: ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)
Referenced by:: §18.32.
Encodings:: BibTeX

… ►

B. Friedman (1990) Principles and Techniques of Applied Mathematics. Dover, New York.

ⓘ

Notes:: Reprint of First Edition, 1956, John-Wiley: New York
External Links:: ISBN 0-486-66444-9, MathReview Entry
Referenced by:: §1.16(ix), §1.17(ii), §1.17(ii), §1.17(i), §1.17(i), §1.17(iv), §1.18(ix), §1.18(iii), §1.18(vi), §1.18(i), §1.18(x).
Encodings:: BibTeX

…

8: 3.8 Nonlinear Equations

… ►Initial approximations to the zeros can often be found from asymptotic or other approximations to

f(z)

, or by application of the phase principle or Rouché’s theorem; see §1.10(iv). … ►

3.8.15

p(x)=(x-1)(x-2)\cdots(x-20)

ⓘ

Permalink:: http://dlmf.nist.gov/3.8.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

… ►Consider

x=20

and

j=19

. We have

p^{\mspace{1.0mu}\prime}(20)=19!

and

a_{19}=1+2+\dots+20=210

. … ►

3.8.16

\frac{\mathrm{d}x}{\mathrm{d}a_{19}}=-\frac{20^{19}}{19!}=(-4.30\dots)\times 1% 0^{7}.

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/3.8.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

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9: 8 Incomplete Gamma and Related
Functions

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10: 28 Mathieu Functions and Hill’s Equation

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principle%20of%20the%20argument

1—10 of 235 matching pages

1: 35.1 Special Notation

2: 35.5 Bessel Functions of Matrix Argument

§35.5 Bessel Functions of Matrix Argument

§35.5(i) Definitions

§35.5(ii) Properties

§35.5(iii) Asymptotic Approximations

3: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8(i) Definition

Convergence Properties

Confluence

Invariance

4: 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6(i) Definitions

Laguerre Form

§35.6(ii) Properties

§35.6(iii) Relations to Bessel Functions of Matrix Argument

5: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7(i) Definition

Jacobi Form

Confluent Form

Integral Representation

6: 20 Theta Functions

Chapter 20 Theta Functions

7: Bibliography F

8: 3.8 Nonlinear Equations

9: 8 Incomplete Gamma and Related
Functions

10: 28 Mathieu Functions and Hill’s Equation

principle%20of%20the%20argument

1—10 of 235 matching pages

1: 35.1 Special Notation

2: 35.5 Bessel Functions of Matrix Argument

§35.5 Bessel Functions of Matrix Argument

§35.5(i) Definitions

§35.5(ii) Properties

§35.5(iii) Asymptotic Approximations

3: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8(i) Definition

Convergence Properties

Confluence

Invariance

4: 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6(i) Definitions

Laguerre Form

§35.6(ii) Properties

§35.6(iii) Relations to Bessel Functions of Matrix Argument

5: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7(i) Definition

Jacobi Form

Confluent Form

Integral Representation

6: 20 Theta Functions

Chapter 20 Theta Functions

7: Bibliography F

8: 3.8 Nonlinear Equations

9: 8 Incomplete Gamma and Related Functions

10: 28 Mathieu Functions and Hill’s Equation

9: 8 Incomplete Gamma and Related
Functions