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21: 36.12 Uniform Approximation of Integrals

… ►Also,

f

is real analytic, and

\ifrac{{\partial}^{K+2}f}{{\partial u}^{K+2}}>0

for all

\mathbf{y}

such that all

K+1

critical points coincide. …The critical points

u_{j}(\mathbf{y})

1\leq j\leq K+1

, are defined by … ►Define a mapping

u(t;\mathbf{y})

by relating

f(u;\mathbf{y})

to the normal form (36.2.1) of

\Phi_{K}\left(t;\mathbf{x}\right)

in the following way: …Correspondence between the

u_{j}(\mathbf{y})

and the

t_{j}(\mathbf{x})

is established by the order of critical points along the real axis when

\mathbf{y}

and

\mathbf{x}

are such that these critical points are all real, and by continuation when some or all of the critical points are complex. … ►For

K=1

, with a single parameter

y

, let the two critical points of

f(u;y)

be denoted by

u_{\pm}(y)

, with

u_{+}>u_{-}

for those values of

y

for which these critical points are real. …

22: 29.18 Mathematical Applications

… ►The wave equation ►

29.18.1

\nabla^{2}u+\omega^{2}u=0,

ⓘ

Symbols:: $u$ : solution and $\omega$ : parameter
Referenced by:: §29.18(ii)
Permalink:: http://dlmf.nist.gov/29.18.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §29.18(i), §29.18 and Ch.29

… ►

29.18.4

u(r,\beta,\gamma)=u_{1}(r)u_{2}(\beta)u_{3}(\gamma),

ⓘ

Symbols:: $u$ : solution, $r$ : sphero-conal coordinate, $\beta$ : sphero-conal coordinate, $\gamma$ : sphero-conal coordinate and $u_{j}$ : solutions
Permalink:: http://dlmf.nist.gov/29.18.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §29.18(i), §29.18 and Ch.29

►where

u_{1}

u_{2}

u_{3}

satisfy the differential equations … ►where

u_{1}

u_{2}

u_{3}

each satisfy the Lamé wave equation (29.11.1). …

23: 12.1 Special Notation

… ►The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions:

U\left(a,z\right)

V\left(a,z\right)

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

, and

W\left(a,z\right)

. …An older notation, due to Whittaker (1902), for

U\left(a,z\right)

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

. The notations are related by

U\left(a,z\right)=D_{-a-\frac{1}{2}}\left(z\right)

. …

24: 12.13 Sums

… ►

12.13.1

U\left(a,x+y\right)=e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac% {(-y)^{m}}{m!}U\left(a-m,x\right),

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $x$ : real variable, $y$ : real variable and $a$ : real or complex parameter
Referenced by:: §12.13(i)
Permalink:: http://dlmf.nist.gov/12.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §12.13(i), §12.13 and Ch.12

►

12.13.2

U\left(a,x+y\right)=e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \genfrac{(}{)}{0.0pt}{}{-a-\tfrac{1}{2}}{m}y^{m}U\left(a+m,x\right),

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $x$ : real variable, $y$ : real variable and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §12.13(i), §12.13 and Ch.12

… ►

12.13.5

U\left(a,x\cos t+y\sin t\right)\\ =e^{\frac{1}{4}(x\sin t-y\cos t)^{2}}\*\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt% }{}{-a-\tfrac{1}{2}}{m}(\tan t)^{m}U\left(m+a,x\right)U\left(-m-\tfrac{1}{2},y% \right),

\Re a\leq-\tfrac{1}{2},0\leq t\leq\tfrac{1}{4}\pi

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $x$ : real variable, $y$ : real variable and $a$ : real or complex parameter
Referenced by:: §12.13(i)
Permalink:: http://dlmf.nist.gov/12.13.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §12.13(i), §12.13 and Ch.12

►

12.13.6

n!U\left(n+\tfrac{1}{2},z\right)=i^{n}e^{-\frac{1}{2}z^{2}}\operatorname{erfc}% (z/\sqrt{2})U\left(-n-\tfrac{1}{2},iz\right)+\sum_{m=1}^{\left\lfloor\frac{1}{% 2}n+\frac{1}{2}\right\rfloor}U\left(2m-n-\tfrac{1}{2},z\right),

n=0,1,2,\dots.

ⓘ

Symbols:: $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\mathrm{i}$ : imaginary unit, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §12.13(i)
Permalink:: http://dlmf.nist.gov/12.13.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §12.13(i), §12.13 and Ch.12

…

25: 36.6 Scaling Relations

… ►

\Psi^{(\mathrm{U})}(\mathbf{x};k)=k^{\beta^{(\mathrm{U})}}\Psi^{(\mathrm{U})}% \left(\mathbf{y}^{(\mathrm{U})}(k)\right),

… ►

Indices for $k$ -Scaling of Magnitude of $\Psi_{K}$ or $\Psi^{(\mathrm{U})}$ (Singularity Index)

… ►

\text{umbilics: }\beta^{(\mathrm{U})}=\frac{1}{3}.

… ►

\text{umbilics: }\gamma_{x}^{(\mathrm{U})}=\tfrac{2}{3},

►

\gamma_{y}^{(\mathrm{U})}=\tfrac{2}{3},

…

26: 13.5 Continued Fractions

… ►

13.5.1

\frac{M\left(a,b,z\right)}{M\left(a+1,b+1,z\right)}=1+\cfrac{u_{1}z}{1+\cfrac{% u_{2}z}{1+\cdots}},

ⓘ

Symbols:: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $z$ : complex variable and $u_{n}$ : continued fraction coefficients
Referenced by:: §13.31(ii)
Permalink:: http://dlmf.nist.gov/13.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.5 and Ch.13

… ►

u_{2n+1}=\frac{a-b-n}{(b+2n)(b+2n+1)}

►

u_{2n}=\frac{a+n}{(b+2n-1)(b+2n)}

… ►

13.5.3

\frac{U\left(a,b,z\right)}{U\left(a,b-1,z\right)}=1+\cfrac{v_{1}/z}{1+\cfrac{v% _{2}/z}{1+\cdots}},

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $z$ : complex variable and $v_{n}$ : continued fraction coefficients
Permalink:: http://dlmf.nist.gov/13.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §13.5 and Ch.13

…

27: 31.17 Physical Applications

… ►The problem of adding three quantum spins

\mathbf{s}

\mathbf{t}

, and

\mathbf{u}

can be solved by the method of separation of variables, and the solution is given in terms of a product of two Heun functions. We use vector notation

[\mathbf{s},\mathbf{t},\mathbf{u}]

(respective scalar

(s,t,u)

) for any one of the three spin operators (respective spin values). ►Consider the following spectral problem on the sphere

S_{2}

\mathbf{x}^{2}=x_{s}^{2}+x_{t}^{2}+x_{u}^{2}=R^{2}

. …for the common eigenfunction

\Psi(\mathbf{x})=\Psi(x_{s},x_{t},x_{u})

, where

a

is the coupling parameter of interacting spins. … ►

\beta=j-s-t-u,

…

28: 13.3 Recurrence Relations and Derivatives

… ►

13.3.7

U\left(a-1,b,z\right)+(b-2a-z)U\left(a,b,z\right)+a(a-b+1)U\left(a+1,b,z\right% )=0,

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and $z$ : complex variable
A&S Ref:: 13.4.15
Permalink:: http://dlmf.nist.gov/13.3.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §13.3(i), §13.3 and Ch.13

►

13.3.8

(b-a-1)U\left(a,b-1,z\right)+(1-b-z)U\left(a,b,z\right)+zU\left(a,b+1,z\right)% =0,

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and $z$ : complex variable
A&S Ref:: 13.4.16
Permalink:: http://dlmf.nist.gov/13.3.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §13.3(i), §13.3 and Ch.13

►

13.3.9

U\left(a,b,z\right)-aU\left(a+1,b,z\right)-U\left(a,b-1,z\right)=0,

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and $z$ : complex variable
A&S Ref:: 13.4.17
Permalink:: http://dlmf.nist.gov/13.3.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §13.3(i), §13.3 and Ch.13

►

13.3.10

(b-a)U\left(a,b,z\right)+U\left(a-1,b,z\right)-zU\left(a,b+1,z\right)=0,

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and $z$ : complex variable
A&S Ref:: 13.4.18
Permalink:: http://dlmf.nist.gov/13.3.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §13.3(i), §13.3 and Ch.13

►

13.3.11

(a+z)U\left(a,b,z\right)-zU\left(a,b+1,z\right)+a(b-a-1)U\left(a+1,b,z\right)=0,

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and $z$ : complex variable
A&S Ref:: 13.4.19
Permalink:: http://dlmf.nist.gov/13.3.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §13.3(i), §13.3 and Ch.13

…

29: 2.8 Differential Equations with a Parameter

… ►in which

u

is a real or complex parameter, and asymptotic solutions are needed for large

|u|

that are uniform with respect to

z

in a point set

\mathbf{D}

\mathbb{R}

\mathbb{C}

. For example,

u

can be the order of a Bessel function or degree of an orthogonal polynomial. … ►The parameter

u

is assumed to be real and positive. … ►The transformed differential equation is … ►Then as

u\to\infty

…

30: 19.29 Reduction of General Elliptic Integrals

… ►where …where … ►where the arguments of the

R_{D}

function are, in order,

(a-b)(u-c)

(b-c)(a-u)

(a-b)(b-c)

. … ►where …If

x=\infty

, then

U

is found by taking the limit. …