values at infinity

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21—30 of 132 matching pages

22: 3.7 Ordinary Differential Equations

… ►

§3.7(ii) Taylor-Series Method: Initial-Value Problems

… ►If the solution

w(z)

that we are seeking grows in magnitude at least as fast as all other solutions of (3.7.1) as we pass along

\mathscr{P}

from

a

b

, then

w(z)

and

w^{\prime}(z)

may be computed in a stable manner for

z=z_{0},z_{1},\dots,z_{P}

by successive application of (3.7.5) for

j=0,1,\dots,P-1

, beginning with initial values

w(a)

and

w^{\prime}(a)

. … ►Similarly, if

w(z)

is decaying at least as fast as all other solutions along

\mathscr{P}

, then we may reverse the labeling of the

z_{j}

along

\mathscr{P}

and begin with initial values

w(b)

and

w^{\prime}(b)

. ►

§3.7(iii) Taylor-Series Method: Boundary-Value Problems

… ►The latter is especially useful if the endpoint

b

\mathscr{P}

is at

\infty

, or if the differential equation is inhomogeneous. …

23: 4.15 Graphics

… ►

See accompanying text — Figure 4.15.4: $\operatorname{arctan}x$ and $\operatorname{arccot}x$ . Only principal values are shown. … Magnify

… ►Figure 4.15.7 illustrates the conformal mapping of the strip

-\tfrac{1}{2}\pi<\Re z<\tfrac{1}{2}\pi

onto the whole

w

-plane cut along the real axis from

-\infty

-1

and

1

\infty

, where

w=\sin z

and

z=\operatorname{arcsin}w

(principal value). …Lines parallel to the real axis in the

z

-plane map onto ellipses in the

w

-plane with foci at

w=\pm 1

, and lines parallel to the imaginary axis in the

z

-plane map onto rectangular hyperbolas confocal with the ellipses. In the labeling of corresponding points

r

is a real parameter that can lie anywhere in the interval

(0,\infty)

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

24: 1.10 Functions of a Complex Variable

… ►The singularities of

f(z)

at infinity are classified in the same way as the singularities of

f(1/z)

z=0

. … ►A function whose only singularities, other than the point at infinity, are poles is called a meromorphic function. … ►In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in

\mathbb{C}

with at most one exception. … ►Functions which have more than one value at a given point

z

are called multivalued (or many-valued) functions. … ►Then the value of

F(z)

at any other point is obtained by analytic continuation. …

25: 1.9 Calculus of a Complex Variable

… ►It is single-valued on

\mathbb{C}\setminus\{0\}

, except on the interval

(-\infty,0)

where it is discontinuous and two-valued. … ►

§1.9(iv) Conformal Mapping

►The extended complex plane,

\mathbb{C}\,\cup\,\{\infty\}

, consists of the points of the complex plane

\mathbb{C}

together with an ideal point

\infty

called the point at infinity. …A function

f(z)

is analytic at

\infty

g(z)=f(1/z)

is analytic at

z=0

, and we set

f^{\prime}(\infty)=g^{\prime}(0)

. …

26: 28.5 Second Solutions $\operatorname{fe}_{n}$ , $\operatorname{ge}_{n}$

… ► ►As a consequence of the factor

z

on the right-hand sides of (28.5.1), (28.5.2), all solutions of Mathieu’s equation that are linearly independent of the periodic solutions are unbounded as

z\to\pm\infty

\mathbb{R}

. …

27: 13.4 Integral Representations

… ►At the point where the contour crosses the interval

(1,\infty)

t^{-b}

and the

{{}_{2}{\mathbf{F}}_{1}}

function assume their principal values; compare §§15.1 and 15.2(i). …

28: 32.11 Asymptotic Approximations for Real Variables

… ►Next, for given initial conditions

w(0)=0

and

w^{\prime}(0)=k

, with

k

real,

w(x)

has at least one pole on the real axis. There are two special values of

k

k_{1}

and

k_{2}

, with the properties

-0.45142\;8<k_{1}<-0.45142\;7

1.85185\;3<k_{2}<1.85185\;5

, and such that: … ►Conversely, for any nonzero real

k

, there is a unique solution

w_{k}(x)

of (32.11.4) that is asymptotic to

k\operatorname{Ai}\left(x\right)

x\to+\infty

. … ►If

|k|>1

, then

w_{k}(x)

has a pole at a finite point

x=c_{0}

, dependent on

k

, and … ►Now suppose

x\to-\infty

. …

29: 22.11 Fourier and Hyperbolic Series

… ►

22.11.1

\operatorname{sn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n% +\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1-q^{2n+1}},

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $q$ : nome, $\sin\NVar{z}$ : sine function, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.1
Permalink:: http://dlmf.nist.gov/22.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.11 and Ch.22

►

22.11.2

\operatorname{cn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n% +\frac{1}{2}}\cos\left((2n+1)\zeta\right)}{1+q^{2n+1}},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$ : cosine function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.2
Permalink:: http://dlmf.nist.gov/22.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.11 and Ch.22

►

22.11.3

\operatorname{dn}\left(z,k\right)=\frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{% \infty}\frac{q^{n}\cos\left(2n\zeta\right)}{1+q^{2n}}.

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$ : cosine function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.3
Permalink:: http://dlmf.nist.gov/22.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.11 and Ch.22

… ►In (22.11.7)–(22.11.12) the left-hand sides are replaced by their limiting values at the poles of the Jacobian functions. … ►

22.11.14

k^{2}{\operatorname{sn}}^{2}\left(z,k\right)=\frac{{E^{\prime}}}{{K^{\prime}}}% -\left(\frac{\pi}{2{K^{\prime}}}\right)^{2}\sum_{n=-\infty}^{\infty}\left({% \operatorname{sech}}^{2}\left(\frac{\pi}{2{K^{\prime}}}(z-2nK)\right)\right),

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, ${E^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the second kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.11.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §22.11 and Ch.22

…

30: 2.10 Sums and Sequences

… ►In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at

s=2m-1

, where

m

is any positive integer satisfying

m\geq\frac{1}{2}(\alpha+1)

. ►For extensions of the Euler–Maclaurin formula to functions

f(x)

with singularities at

x=a

x=n

(or both) see Sidi (2004, 2012b, 2012a). … ►For an extension to integrals with Cauchy principal values see Elliott (1998). … ►The singularities of

f(z)

on the unit circle are branch points at

z=e^{\pm i\alpha}

. To match the limiting behavior of

f(z)

at these points we set …