modulus and phase

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31—40 of 134 matching pages

31: 13.4 Integral Representations

… ►

13.4.12

{\mathbf{M}}\left(a,c,z\right)=\frac{\Gamma\left(b\right)}{2\pi\mathrm{i}}z^{1% -b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;% \ifrac{1}{t}\right)\,\mathrm{d}t,

b\neq 0,-1,-2,\dots

\left|\operatorname{ph}z\right|<\frac{1}{2}\pi

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $\operatorname{ph}$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.4.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §13.4(ii), §13.4 and Ch.13

…

32: 7.12 Asymptotic Expansions

… ►When

|\operatorname{ph}z|\leq\frac{1}{4}\pi

the remainder terms are bounded in magnitude by the first neglected terms, and have the same sign as these terms when

\operatorname{ph}z=0

. When

\frac{1}{4}\pi\leq|\operatorname{ph}z|<\frac{1}{2}\pi

the remainder terms are bounded in magnitude by

\csc\left(2|\operatorname{ph}z|\right)

times the first neglected terms. … ►When

|\operatorname{ph}z|\leq\frac{1}{8}\pi

R_{n}^{(\mathrm{f})}(z)

and

R_{n}^{(\mathrm{g})}(z)

are bounded in magnitude by the first neglected terms in (7.12.2) and (7.12.3), respectively, and have the same signs as these terms when

\operatorname{ph}z=0

. They are bounded by

|\csc\left(4\operatorname{ph}z\right)|

times the first neglected terms when

\frac{1}{8}\pi\leq|\operatorname{ph}z|<\frac{1}{4}\pi

. ►For other phase ranges use (7.4.7) and (7.4.8). …

33: 5.20 Physical Applications

… ►

Rutherford Scattering

►In nonrelativistic quantum mechanics, collisions between two charged particles are described with the aid of the Coulomb phase shift

\operatorname{ph}\Gamma\left(\ell+1+\mathrm{i}\eta\right)

; see (33.2.10) and Clark (1979). … ►

5.20.2

P(x_{1},\dots,x_{n})=C\exp\left(-W/(kT)\right),

ⓘ

Symbols:: $\exp\NVar{z}$ : exponential function, $n$ : nonnegative integer, $x$ : real variable, $W$ : potential energy, $T$ : temperature and $C$ : constant
Permalink:: http://dlmf.nist.gov/5.20.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §5.20, §5.20 and Ch.5

… ►Then the partition function (with

\beta=1/(kT)

) is given by ►

5.20.3

\psi_{n}(\beta)=\int_{{\mathbb{R}}^{n}}e^{-\beta W}\,\mathrm{d}x\\ =(2\pi)^{n/2}\beta^{-(n/2)-(\beta n(n-1)/4)}\*\left(\Gamma\left(1+\tfrac{1}{2}% \beta\right)\right)^{-n}\prod_{j=1}^{n}\Gamma\left(1+\tfrac{1}{2}j\beta\right).

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\mathbb{R}$ : real line, $j$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $W$ : potential energy, $\psi_{n}(\beta)$ : partition function and $\beta=1/(kT)$
Permalink:: http://dlmf.nist.gov/5.20.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §5.20, §5.20 and Ch.5

…

34: 15.8 Transformations of Variable

… ►

15.8.14

F\left({a,b\atop 2b};z\right)=\left(1-z\right)^{-\ifrac{a}{2}}F\left({\tfrac{1% }{2}a,b-\tfrac{1}{2}a\atop b+\tfrac{1}{2}};\frac{z^{2}}{4z-4}\right),

|\operatorname{ph}\left(1-z\right)|<\pi

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

… ►provided that

z

lies in the intersection of the open disks

\left|z-\frac{1}{4}\pm\frac{1}{4}\sqrt{3}\mathrm{i}\right|<\frac{1}{2}\sqrt{3}

, or equivalently,

\left|\operatorname{ph}\left(\ifrac{(1-z)}{(1+2z)}\right)\right|<\pi/3

. …

35: 15.9 Relations to Other Functions

… ►

15.9.17

\mathbf{F}\left({a,a+\tfrac{1}{2}\atop c};z\right)=2^{c-1}z^{\ifrac{(1-c)}{2}}% (1-z)^{-a+(\ifrac{(c-1)}{2})}\*P^{1-c}_{2a-c}\left(\frac{1}{\sqrt{1-z}}\right),

|\operatorname{ph}z|<\pi

and

|\operatorname{ph}\left(1-z\right)|<\pi

ⓘ

Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.9.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §15.9(iv), §15.9 and Ch.15

►

15.9.18

\mathbf{F}\left({a,b\atop a+b+\tfrac{1}{2}};z\right)=2^{a+b-(\ifrac{1}{2})}(-z% )^{(-a-b+(\ifrac{1}{2}))/2}\*P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left% (\sqrt{1-z}\right),

\left|\operatorname{ph}\left(-z\right)\right|<\pi

ⓘ

Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.9.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §15.9(iv), §15.9 and Ch.15

►

15.9.19

\mathbf{F}\left({a,b\atop a-b+1};z\right)=z^{\ifrac{(b-a)}{2}}(1-z)^{-b}\*P^{b% -a}_{-b}\left(\frac{1+z}{1-z}\right),

|\operatorname{ph}z|<\pi

and

|\operatorname{ph}\left(1-z\right)|<\pi

ⓘ

Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.9.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §15.9(iv), §15.9 and Ch.15

►

15.9.20

\mathbf{F}\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=\left(-z(1-z)\right)^{% \ifrac{(1-a-b)}{4}}\*P^{\ifrac{(1-a-b)}{2}}_{\ifrac{(a-b-1)}{2}}\left(1-2z% \right),

\left|\operatorname{ph}\left(-z\right)\right|<\pi

ⓘ

Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.9.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §15.9(iv), §15.9 and Ch.15

►

15.9.21

\mathbf{F}\left({a,1-a\atop c};z\right)=\left(\frac{-z}{1-z}\right)^{\ifrac{(1% -c)}{2}}\*P^{1-c}_{-a}\left(1-2z\right),

\left|\operatorname{ph}\left(-z\right)\right|<\pi

ⓘ

Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.9(iv)
Permalink:: http://dlmf.nist.gov/15.9.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §15.9(iv), §15.9 and Ch.15

…

36: 22.5 Special Values

… ►For example, at

z=K+iK^{\prime}

\operatorname{sn}\left(z,k\right)=1/k

\ifrac{\mathrm{d}\operatorname{sn}\left(z,k\right)}{\mathrm{d}z}=0

. (The modulus

k

is suppressed throughout the table.) … ►For example,

\operatorname{sn}\left(\frac{1}{2}K,k\right)=(1+k^{\prime})^{-1/2}

. … ►

§22.5(ii) Limiting Values of $k$

… ►For values of

K,K^{\prime}

when

k^{2}=\frac{1}{2}

(lemniscatic case) see §23.5(iii), and for

k^{2}=e^{\mathrm{i}\pi/3}

(equianharmonic case) see §23.5(v). …

37: 4.9 Continued Fractions

… ►

4.9.1

\ln\left(1+z\right)=\cfrac{z}{1+\cfrac{z}{2+\cfrac{z}{3+\cfrac{4z}{4+\cfrac{4z% }{5+\cfrac{9z}{6+\cfrac{9z}{7+}}}}}}}\cdots,

|\operatorname{ph}\left(1+z\right)|<\pi

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase and $z$ : complex variable
A&S Ref:: 4.1.39
Permalink:: http://dlmf.nist.gov/4.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §4.9(i), §4.9 and Ch.4

… ►

=1+\cfrac{z}{1-(z/2)+\cfrac{z^{2}/(4\cdot 3)}{1+\cfrac{z^{2}/(4\cdot 15)}{1+% \cfrac{z^{2}/(4\cdot 35)}{1+}}}}\cdots\cfrac{z^{2}/(4(4n^{2}-1))}{1+}\cdots

…

38: 4.45 Methods of Computation

… ►Suppose first

1/10\leq x\leq 10

. … ►We first compute

\xi=x/\pi

, followed by … ►From (4.24.15) with

u=v=((1+x^{2})^{1/2}-1)/x

, we have … ►For example,

\operatorname{arcsin}x=\operatorname{arctan}\left(x(1-x^{2})^{-1/2}\right)

. … ►See §1.9(i) for the precise relationship of

\operatorname{ph}z

to the arctangent function. …

39: 2.11 Remainder Terms; Stokes Phenomenon

… ►In

\mathbb{C}

both the modulus and phase of the asymptotic variable

z

need to be taken into account. …

40: 9.13 Generalized Airy Functions

… ►

9.13.9

A_{n}\left(z\right)=\sqrt{\ifrac{p}{\pi}}\sin\left(p\pi\right)z^{-n/4}e^{-% \zeta}\left(1+O\left(\zeta^{-1}\right)\right),

|\operatorname{ph}z|\leq 3p\pi-\delta

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $A_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $\delta$ : small positive constant, $n$ : parameter, $p$ : variable and $\zeta$ : variable
Source:: Swanson and Headley (1967, (13))
Permalink:: http://dlmf.nist.gov/9.13.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §9.13(i), §9.13 and Ch.9

►

9.13.10

A_{n}\left(-z\right)=\begin{cases}2\sqrt{p/\pi}\cos\left(\tfrac{1}{2}p\pi% \right)z^{-n/4}\left(\cos\left(\zeta-\tfrac{1}{4}\pi\right)+e^{|\Im\zeta|}O% \left(\zeta^{-1}\right)\right),&\text{$|\operatorname{ph}z|\leq 2p\pi-\delta$,% $n$ odd},\\ \sqrt{p/\pi}z^{-n/4}e^{\zeta}\left(1+O\left(\zeta^{-1}\right)\right),&\text{$|% \operatorname{ph}z|\leq p\pi-\delta$, $n$ even},\end{cases}

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $A_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $\operatorname{ph}$ : phase, $z$ : complex variable, $\delta$ : small positive constant, $n$ : parameter, $p$ : variable and $\zeta$ : variable
Source:: Swanson and Headley (1967, (14)–(15))
Permalink:: http://dlmf.nist.gov/9.13.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §9.13(i), §9.13 and Ch.9

►

9.13.11

B_{n}\left(z\right)={\pi}^{-1/2}z^{-n/4}e^{\zeta}\left(1+O\left(\zeta^{-1}% \right)\right),

|\operatorname{ph}z|\leq p\pi-\delta

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $B_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $z$ : complex variable, $\delta$ : small positive constant, $n$ : parameter, $p$ : variable and $\zeta$ : variable
Source:: Swanson and Headley (1967, (16))
Permalink:: http://dlmf.nist.gov/9.13.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §9.13(i), §9.13 and Ch.9

►

9.13.12

B_{n}\left(-z\right)=\begin{cases}-(\ifrac{2}{\sqrt{\pi}})\sin\left(\tfrac{1}{% 2}p\pi\right)z^{-n/4}\left(\sin\left(\zeta-\tfrac{1}{4}\pi\right)+e^{\left|\Im% \zeta\right|}O\left(\zeta^{-1}\right)\right),&\left|\operatorname{ph}z\right|% \leq 2p\pi-\delta,n\text{ odd},\\ (\ifrac{1}{\sqrt{\pi}})\sin\left(p\pi\right)z^{-n/4}e^{-\zeta}\left(1+O\left(% \zeta^{-1}\right)\right),&\left|\operatorname{ph}z\right|\leq 3p\pi-\delta,n% \text{ even}.\end{cases}

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $B_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $\delta$ : small positive constant, $n$ : parameter, $p$ : variable and $\zeta$ : variable
Source:: Swanson and Headley (1967, (17)–(18))
Permalink:: http://dlmf.nist.gov/9.13.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §9.13(i), §9.13 and Ch.9

… ►The function on the right-hand side is recessive in the sector

-(2j-1)\pi/m\leq\operatorname{ph}z\leq(2j+1)\pi/m

, and is therefore an essential member of any numerically satisfactory pair of solutions in this region. …