金肯职业技术学院毕业证制作【言正微aptao168】0oB4ifw

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31—40 of 697 matching pages

31: 19.17 Graphics

… ►For

R_{F}

R_{G}

, and

R_{J}

, which are symmetric in

x, y, z

, we may further assume that

z

is the largest of

x, y, z

if the variables are real, then choose

z=1

, and consider only

0\leq x\leq 1

and

0\leq y\leq 1

. The cases

x=0

y=0

correspond to the complete integrals. … ►To view

R_{F}\left(0,y,1\right)

and

2R_{G}\left(0,y,1\right)

for complex

y

, put

y=1-k^{2}

, use (19.25.1), and see Figures 19.3.7–19.3.12. … ►To view

R_{F}\left(0,y,1\right)

and

2R_{G}\left(0,y,1\right)

for complex

y

, put

y=1-k^{2}

, use (19.25.1), and see Figures 19.3.7–19.3.12. … ►

See accompanying text — Figure 19.17.8: $R_{J}\left(0,y,1,p\right)$ , $0\leq y\leq 1$ , $-1\leq p\leq 2$ . Cauchy principal values are shown when $p<0$ . … Magnify 3D Help

32: 20.3 Graphics

… ►

►

…

33: 7.16 Generalized Error Functions

… ►Generalizations of the error function and Dawson’s integral are

\int_{0}^{x}e^{-t^{p}}\,\mathrm{d}t

and

\int_{0}^{x}e^{t^{p}}\,\mathrm{d}t

. …

34: 15.13 Zeros

… ►If

a

b

c

are real,

a

b

c

c-a

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

, and, without loss of generality,

b\geq a

c\geq a+b

(compare (15.8.1)), then ►

15.13.1

N(a,b,c)=\begin{cases}0,&a>0,\\ \left\lfloor-a\right\rfloor+\tfrac{1}{2}(1+S),&a<0,c-a>0,\\ \left\lfloor-a\right\rfloor+\tfrac{1}{2}(1+S)+\left\lfloor a-c+1\right\rfloor S% ,&a<0,c-a<0,\\ \end{cases}

ⓘ

Symbols:: $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $N(a,b,c)$ : number of zeros and $S$
Permalink:: http://dlmf.nist.gov/15.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.13 and Ch.15

… ►If

a

b

c

c-a

, or

c-b\in\{0,-1,-2,\dots\}

, then

F\left(a,b;c;z\right)

is not defined, or reduces to a polynomial, or reduces to

(1-z)^{c-a-b}

times a polynomial. …

35: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

… ►If

b

and

x

are fixed, with

b>0

and

0<x<1

, then as

a\to\infty

…for each

n=0,1,2,\dots

. … ►Then as

a\to\infty

, with

b

(

>0

) fixed, … ►with

\eta/(x-x_{0})>0

, and … ►uniformly for

b\in(0,\infty)

and

x\in(0,1)

. …

36: 28.7 Analytic Continuation of Eigenvalues

… ►To 4D the first branch points between

a_{0}\left(q\right)

and

a_{2}\left(q\right)

are at

q_{0}=\pm\mathrm{i}1.4688

with

a_{0}\left(q_{0}\right)=a_{2}\left(q_{0}\right)=2.0886

, and between

b_{2}\left(q\right)

and

b_{4}\left(q\right)

they are at

q_{1}=\pm\mathrm{i}6.9289

with

b_{2}\left(q_{1}\right)=b_{4}\left(q_{1}\right)=11.1904

. For real

q

with

|q|<|q_{0}|

a_{0}\left(\mathrm{i}q\right)

and

a_{2}\left(\mathrm{i}q\right)

are real-valued, whereas for real

q

with

|q|>|q_{0}|

a_{0}\left(\mathrm{i}q\right)

and

a_{2}\left(\mathrm{i}q\right)

are complex conjugates. … ►For a visualization of the first branch point of

a_{0}\left(\mathrm{i}\hat{q}\right)

and

a_{2}\left(\mathrm{i}\hat{q}\right)

see Figure 28.7.1. … ►All the

a_{2n}\left(q\right)

n=0,1,2,\dots

, can be regarded as belonging to a complete analytic function (in the large). …Analogous statements hold for

a_{2n+1}\left(q\right)

b_{2n+1}\left(q\right)

, and

b_{2n+2}\left(q\right)

, also for

n=0,1,2,\dots

. …

37: 13.27 Mathematical Applications

… ►

13.27.1

g=\begin{pmatrix}1&\alpha&\beta\\ 0&\gamma&\delta\\ 0&0&1\end{pmatrix},

ⓘ

Permalink:: http://dlmf.nist.gov/13.27.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.27 and Ch.13

►where

\alpha

\beta

\gamma

\delta

are real numbers, and

\gamma>0

. …

38: 14.27 Zeros

… ►

P^{\mu}_{\nu}\left(x\pm i0\right)

(either side of the cut) has exactly one zero in the interval

(-\infty,-1)

if either of the following sets of conditions holds: ►

(a)

$\mu<0$ , $\mu\notin\mathbb{Z}$ , $\nu\in\mathbb{Z}$ , and $\sin\left((\mu-\nu)\pi\right)$ and $\sin\left(\mu\pi\right)$ have opposite signs.

►

(b)

$\mu,\nu\in\mathbb{Z}$ , $\mu+\nu<0$ , and $\nu$ is odd.

►For all other values of the parameters

P^{\mu}_{\nu}\left(x\pm i0\right)

has no zeros in the interval

(-\infty,-1)

. …

39: 3.8 Nonlinear Equations

… ►If

f(z_{0})=0

and

f^{\prime}(z_{0})\neq 0

, then

z_{0}

is a simple zero of

f

. If

f(z_{0})=f^{\prime}(z_{0})=\cdots=f^{(m-1)}(z_{0})=0

and

f^{(m)}(z_{0})\neq 0

, then

z_{0}

is a zero of

f

of multiplicity $m$ ; compare §1.10(i). … ►

(a)

$f(x_{0})f^{\prime\prime}(x_{0})>0$ and $f^{\prime}(x)$ , $f^{\prime\prime}(x)$ do not change sign between $x_{0}$ and $\xi$ (monotonic convergence).

… ►From this graph we estimate an initial value

x_{0}=4.65

. …The choice of

x_{0}

here is critical. …

40: 1.10 Functions of a Complex Variable

… ►Then

z=z_{0}

is an isolated singularity of

f(z)

. … ►For example,

z=0

is a branch point of

\sqrt{z}

. … ►Suppose

f(z)

is analytic at

z=z_{0}

f^{\prime}(z_{0})\not=0

, and

f(z_{0})=w_{0}

. … ►where

\mu>0

f_{0}\not=0

, and the series converges in a neighborhood of

z_{0}

. …Let

w_{0}=f(z_{0})

. …

金肯职业技术学院毕业证制作【言正 微aptao168】0oB4ifw

31—40 of 697 matching pages

31: 19.17 Graphics

32: 20.3 Graphics

33: 7.16 Generalized Error Functions

34: 15.13 Zeros

35: 8.18 Asymptotic Expansions of I x ⁡ ( a , b )

36: 28.7 Analytic Continuation of Eigenvalues

37: 13.27 Mathematical Applications

38: 14.27 Zeros

39: 3.8 Nonlinear Equations

40: 1.10 Functions of a Complex Variable

金肯职业技术学院毕业证制作【言正微aptao168】0oB4ifw

35: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$