modulus%20and%20phase

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11—16 of 16 matching pages

11: 12.19 Tables

… ►

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Miller (1955) includes $W\left(a,x\right)$ , $W\left(a,-x\right)$ , and reduced derivatives for $a=-10(1)10$ , $x=0(.1)10$ , 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.

►

•

Fox (1960) includes modulus and phase functions for $W\left(a,x\right)$ and $W\left(a,-x\right)$ , and several auxiliary functions for $x^{-1}=0(.005)0.1$ , $a=-10(1)10$ , 8S.

… ►

•

Murzewski and Sowa (1972) includes $D_{-n}\left(x\right)$ $\left(=U\left(n-\tfrac{1}{2},x\right)\right)$ for $n=1(1)20$ , $x=0(.05)3$ , 7S.

…

12: 25.12 Polylogarithms

… ►When

z=e^{i\theta}

0\leq\theta\leq 2\pi

, (25.12.1) becomes … ►

See accompanying text — Figure 25.12.1: Dilogarithm function $\operatorname{Li}_{2}\left(x\right)$ , $-20\leq x<1.$ Magnify

►

… ►valid when

\Re s>0

and

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

, or

\Re s>1

and

z=1

. … ►Sometimes the factor

1/\Gamma\left(s+1\right)

is omitted. …

13: 10.3 Graphics

… ►

§10.3(i) Real Order and Variable

►For the modulus and phase functions

M_{\nu}\left(x\right)

\theta_{\nu}\left(x\right)

N_{\nu}\left(x\right)

, and

\phi_{\nu}\left(x\right)

see §10.18. … ►

… ►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. …

14: 3.8 Nonlinear Equations

… ►For multiple zeros the convergence is linear, but if the multiplicity

m

is known then quadratic convergence can be restored by multiplying the ratio

f(z_{n})/f^{\prime}(z_{n})

in (3.8.4) by

m

. … ►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

. …

15: 1.11 Zeros of Polynomials

… ►The sum and product of the roots are respectively

-b/a

and

c/a

. … ►Set

z=w-\tfrac{1}{3}a

to reduce

f(z)=z^{3}+az^{2}+bz+c

g(w)=w^{3}+pw+q

, with

p=(3b-a^{2})/3

q=(2a^{3}-9ab+27c)/27

. … ►Resolvent cubic is

z^{3}+12z^{2}+20z+9=0

with roots

\theta_{1}=-1

\theta_{2}=-\tfrac{1}{2}(11+\sqrt{85})

\theta_{3}=-\tfrac{1}{2}(11-\sqrt{85})

, and

\sqrt{-\theta_{1}}=1

\sqrt{-\theta_{2}}=\tfrac{1}{2}(\sqrt{17}+\sqrt{5})

\sqrt{-\theta_{3}}=\tfrac{1}{2}(\sqrt{17}-\sqrt{5})

. … ►are

1

{\mathrm{e}}^{2\pi\mathrm{i}/n}

{\mathrm{e}}^{4\pi\mathrm{i}/n},\dots,{\mathrm{e}}^{(2n-2)\pi\mathrm{i}/n}

, and of

z^{n}+1=0

they are

{\mathrm{e}}^{\pi\mathrm{i}/n},{\mathrm{e}}^{3\pi\mathrm{i}/n},\dots,{\mathrm{% e}}^{(2n-1)\pi\mathrm{i}/n}

. … ►where

R=(a^{2}+b^{2})^{1/2}

\alpha=\operatorname{ph}\left(a+\mathrm{i}b\right)

, with the principal value of phase (§1.9(i)), and

k=0,1,\dots,n-1

. …

16: 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. …Then numerical accuracy will disintegrate as the boundary rays

\operatorname{ph}z=\alpha

\operatorname{ph}z=\beta

are approached. … ►uniformly with respect to

\operatorname{ph}z

in each case. ►The relevant Stokes lines are

\operatorname{ph}z=\pm\pi

for

w_{1}(z)

, and

\operatorname{ph}z=0,2\pi

for

w_{2}(z)

. … ►For example, using double precision

d_{20}

is found to agree with (2.11.31) to 13D. …