DLMF: Untitled Document

… ►Standard solutions are

U\left(a,\pm z\right)

,

V\left(a,\pm z\right)

,

\overline{U}\left(a,\pm x\right)

(not complex conjugate),

U\left(-a,\pm iz\right)

for (12.2.2);

W\left(a,\pm x\right)

for (12.2.3);

D_{\nu}\left(\pm z\right)

for (12.2.4), where … ►

§12.2(vi) Solution $\overline{U}\left(a,x\right)$ ; Modulus and Phase Functions

►When

z

(=x)

is real the solution

\overline{U}\left(a,x\right)

is defined by …unless

a=\tfrac{1}{2},\tfrac{3}{2},\dots

, in which case

\overline{U}\left(a,x\right)

is undefined. …Properties of

\overline{U}\left(a,x\right)

follow immediately from those of

V\left(a,x\right)

via (12.2.21). …

… ►Here

u=u_{k}(x;\nu)

is the solution of … ►

32.3.3

u\sim kU\left(-\nu-\tfrac{1}{2},x\right),

x\to+\infty

.

ⓘ

Symbols:: $\sim$ : asymptotic equality, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $x$ : real, $k$ : real, $\nu$ : parameter and $u_{k}(x;\nu)$ : solution
Permalink:: http://dlmf.nist.gov/32.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §32.3(iii), §32.3 and Ch.32

… ►

32.3.4

w(x)=2\sqrt{2}u_{k}^{2}(\sqrt{2}x,\nu),

ⓘ

Symbols:: $x$ : real, $k$ : real, $\nu$ : parameter and $u_{k}(x;\nu)$ : solution
Permalink:: http://dlmf.nist.gov/32.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §32.3(iii), §32.3 and Ch.32

… ►If we set

\ifrac{{\mathrm{d}}^{2}u}{{\mathrm{d}x}^{2}}=0

in (32.3.2) and solve for

u

, then … ►

See accompanying text — Figure 32.3.7: $u_{k}(x;-\tfrac{1}{2})$ for $-12\leq x\leq 4$ with $k=0.33554\;691$ , $0.33554\;692$ . …The parabolas $u^{2}+\tfrac{1}{2}x=0$ , $u^{2}+\tfrac{1}{6}x=0$ are shown in black and green, respectively. Magnify

…

… ►

4.21.8

\cos u+\cos v=2\cos\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right),

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function
A&S Ref:: 4.3.36
Permalink:: http://dlmf.nist.gov/4.21.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §4.21(i), §4.21 and Ch.4

… ►

4.21.15

2\sin u\sin v=\cos\left(u-v\right)-\cos\left(u+v\right),

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function
A&S Ref:: 4.3.31
Permalink:: http://dlmf.nist.gov/4.21.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §4.21(ii), §4.21 and Ch.4

►

4.21.16

2\cos u\cos v=\cos\left(u-v\right)+\cos\left(u+v\right),

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function
A&S Ref:: 4.3.32
Permalink:: http://dlmf.nist.gov/4.21.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §4.21(ii), §4.21 and Ch.4

►

4.21.17

2\sin u\cos v=\sin\left(u-v\right)+\sin\left(u+v\right).

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function
A&S Ref:: 4.3.33
Permalink:: http://dlmf.nist.gov/4.21.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §4.21(ii), §4.21 and Ch.4

►

4.21.18

{\sin}^{2}u-{\sin}^{2}v=\sin\left(u+v\right)\sin\left(u-v\right),

ⓘ

Symbols:: $\sin\NVar{z}$ : sine function
A&S Ref:: 4.3.40
Permalink:: http://dlmf.nist.gov/4.21.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §4.21(ii), §4.21 and Ch.4

…

… ►

12.8.1

zU\left(a,z\right)-U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+1,z\right)=0,

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.4
Referenced by:: §12.8(i)
Permalink:: http://dlmf.nist.gov/12.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

►

12.8.2

U'\left(a,z\right)+\tfrac{1}{2}zU\left(a,z\right)+(a+\tfrac{1}{2})U\left(a+1,z% \right)=0,

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.1
Permalink:: http://dlmf.nist.gov/12.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

►

12.8.3

U'\left(a,z\right)-\tfrac{1}{2}zU\left(a,z\right)+U\left(a-1,z\right)=0,

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.2
Permalink:: http://dlmf.nist.gov/12.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

►

12.8.4

2U'\left(a,z\right)+U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+1,z\right)=0.

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.3
Referenced by:: §12.8(i)
Permalink:: http://dlmf.nist.gov/12.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

►(12.8.1)–(12.8.4) are also satisfied by

\overline{U}\left(a,z\right)

. …

…

… ►

3.9.9

t_{n,2k}=\frac{H_{k+1}(s_{n})}{H_{k}(\Delta^{2}s_{n})},

n=0,1,2,\dots

,

ⓘ

Symbols:: $s_{n}$ : sequence, $t_{n}$ : sequence and $H_{m}(u)$ : Hankel determinant
Referenced by:: §3.9(iv)
Permalink:: http://dlmf.nist.gov/3.9.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §3.9(iv), §3.9 and Ch.3

… ►

3.9.10

H_{m}(u_{n})=\begin{vmatrix}u_{n}&u_{n+1}&\cdots&u_{n+m-1}\\ u_{n+1}&u_{n+2}&\cdots&u_{n+m}\\ \vdots&\vdots&\ddots&\vdots\\ u_{n+m-1}&u_{n+m}&\cdots&u_{n+2m-2}\end{vmatrix}.

ⓘ

Defines:: $H_{m}(u)$ : Hankel determinant (locally)
Symbols:: $\det$ : determinant
Permalink:: http://dlmf.nist.gov/3.9.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §3.9(iv), §3.9 and Ch.3

… ►For examples and other transformations for convergent sequences and series, see Wimp (1981, pp. 156–199), Brezinski and Redivo Zaglia (1991, pp. 55–72), and Sidi (2003, Chapters 6, 12–13, 15–16, 19–24, and pp. 483–492). …

… ►

7.19.4

H\left(a,u\right)=\frac{a}{\pi}\int_{-\infty}^{\infty}\frac{e^{-t^{2}}\,% \mathrm{d}t}{(u-t)^{2}+a^{2}}=\frac{1}{a\sqrt{\pi}}\mathsf{U}\left(\frac{u}{a}% ,\frac{1}{4a^{2}}\right).

ⓘ

Defines:: $H\left(\NVar{a},\NVar{u}\right)$ : line-broadening function
Symbols:: $\mathsf{U}\left(\NVar{x},\NVar{t}\right)$ : Voigt 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 and $\int$ : integral
Permalink:: http://dlmf.nist.gov/7.19.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §7.19(i), §7.19 and Ch.7

►

H\left(a,u\right)

is sometimes called the line broadening function; see, for example, Finn and Mugglestone (1965). … ►

… ►

\mathsf{U}\left(-x,t\right)=\mathsf{U}\left(x,t\right),

… ►

0<\mathsf{U}\left(x,t\right)\leq 1,

…

… ►Then

u_{n}(z)=(a_{n}(z))^{2}

satisfies the nonlinear recurrence relation ►

32.15.3

(u_{n+1}+u_{n}+u_{n-1})u_{n}=n-2zu_{n},

ⓘ

Symbols:: $n$ : integer, $z$ : real and $u_{n}(z)$ : solution
Permalink:: http://dlmf.nist.gov/32.15.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §32.15 and Ch.32

…

… ►

4.35.7

\cosh u+\cosh v=2\cosh\left(\frac{u+v}{2}\right)\cosh\left(\frac{u-v}{2}\right),

ⓘ

Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function
A&S Ref:: 4.5.43
Permalink:: http://dlmf.nist.gov/4.35.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §4.35(i), §4.35 and Ch.4

… ►

4.35.14

2\sinh u\sinh v=\cosh\left(u+v\right)-\cosh\left(u-v\right),

ⓘ

Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function and $\sinh\NVar{z}$ : hyperbolic sine function
A&S Ref:: 4.5.38
Permalink:: http://dlmf.nist.gov/4.35.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §4.35(ii), §4.35 and Ch.4

►

4.35.15

2\cosh u\cosh v=\cosh\left(u+v\right)+\cosh\left(u-v\right),

ⓘ

Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function
A&S Ref:: 4.5.39
Permalink:: http://dlmf.nist.gov/4.35.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §4.35(ii), §4.35 and Ch.4

►

4.35.16

2\sinh u\cosh v=\sinh\left(u+v\right)+\sinh\left(u-v\right).

ⓘ

Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function and $\sinh\NVar{z}$ : hyperbolic sine function
A&S Ref:: 4.5.40
Permalink:: http://dlmf.nist.gov/4.35.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §4.35(ii), §4.35 and Ch.4

►

4.35.17

{\sinh}^{2}u-{\sinh}^{2}v=\sinh\left(u+v\right)\sinh\left(u-v\right),

ⓘ

Symbols:: $\sinh\NVar{z}$ : hyperbolic sine function
A&S Ref:: 4.5.47
Permalink:: http://dlmf.nist.gov/4.35.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §4.35(ii), §4.35 and Ch.4

…

… ►Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions

U\left(a,b,x\right)

and

M\left(a,b,x\right)

(§13.2(i)) whose regions of validity include intervals with endpoints

x=\infty

and

x=0

, respectively. As special cases of these results a Chebyshev-series expansion for

U\left(a,x\right)

valid when

\lambda\leq x<\infty

follows from (12.7.14), and Chebyshev-series expansions for

U\left(a,x\right)

and

V\left(a,x\right)

valid when

0\leq x\leq\lambda

follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13). …

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11—20 of 227 matching pages

11: 12.2 Differential Equations

§12.2(vi) Solution $\overline{U}\left(a,x\right)$ ; Modulus and Phase Functions

12: 32.3 Graphics

13: 4.21 Identities

14: 12.8 Recurrence Relations and Derivatives

15: 12.21 Software

16: 3.9 Acceleration of Convergence

17: 7.19 Voigt Functions

18: 32.15 Orthogonal Polynomials

19: 4.35 Identities

20: 12.20 Approximations

.ued在线最靠谱的博彩平台『世界杯佣金分红55%，咨询专员：@ky975』.ic6.k2q1w9-y0aqm00ek

11—20 of 227 matching pages

11: 12.2 Differential Equations

§12.2(vi) Solution U ¯ ⁡ ( a , x ) ; Modulus and Phase Functions

12: 32.3 Graphics

13: 4.21 Identities

14: 12.8 Recurrence Relations and Derivatives

15: 12.21 Software

16: 3.9 Acceleration of Convergence

17: 7.19 Voigt Functions

18: 32.15 Orthogonal Polynomials

19: 4.35 Identities

20: 12.20 Approximations

§12.2(vi) Solution $\overline{U}\left(a,x\right)$ ; Modulus and Phase Functions