房产继承公证书【WeChat微aptao168】83w

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21: 28.29 Definitions and Basic Properties

… ►The basic solutions

w_{\mbox{\tiny I}}(z,\lambda)

w_{\mbox{\tiny II}}(z,\lambda)

are defined in the same way as in §28.2(ii) (compare (28.2.5), (28.2.6)). … ►Then (28.29.1) has a nontrivial solution

w(z)

with the pseudoperiodic property … ►Let

w(z)

be a solution linearly independent of

P(z)

. … ►Furthermore, for each solution

w(z)

of (28.29.1) …A nontrivial solution

w(z)

is either a Floquet solution with respect to

\nu

, or

w(z+\pi)-e^{\mathrm{i}\nu\pi}w(z)

is a Floquet solution with respect to

-\nu

. …

22: 4.34 Derivatives and Differential Equations

… ►With

a\neq 0

, the general solutions of the differential equations ►

4.34.7

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-a^{2}w=0,

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $a$ : real or complex constant, $z$ : complex variable and $w$ : solution
Permalink:: http://dlmf.nist.gov/4.34.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §4.34 and Ch.4

►

4.34.8

\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}=1,

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $a$ : real or complex constant, $z$ : complex variable and $w$ : solution
Permalink:: http://dlmf.nist.gov/4.34.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §4.34 and Ch.4

►

4.34.9

\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}=-1,

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $a$ : real or complex constant, $z$ : complex variable and $w$ : solution
Permalink:: http://dlmf.nist.gov/4.34.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §4.34 and Ch.4

►

4.34.10

\frac{\mathrm{d}w}{\mathrm{d}z}+a^{2}w^{2}=1,

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $a$ : real or complex constant, $z$ : complex variable and $w$ : solution
Permalink:: http://dlmf.nist.gov/4.34.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §4.34 and Ch.4

…

23: 4.5 Inequalities

… ►For more inequalities involving the exponential function see Mitrinović (1964, pp. 73–77), Mitrinović (1970, pp. 266–271), and Bullen (1998, pp. 81–83).

24: 15.10 Hypergeometric Differential Equation

… ►

15.10.17

w_{3}(z)=\frac{\Gamma\left(1-c\right)\Gamma\left(a+b-c+1\right)}{\Gamma\left(a% -c+1\right)\Gamma\left(b-c+1\right)}w_{1}(z)+\frac{\Gamma\left(c-1\right)% \Gamma\left(a+b-c+1\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}w_{2}(z),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $w_{1}(z)$ : Kummer’s solution $w_{1}$ , $w_{2}(z)$ : Kummer’s solution $w_{2}$ and $w_{3}(z)$ : Kummer’s solution $w_{3}$
Permalink:: http://dlmf.nist.gov/15.10.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(ii), §15.10 and Ch.15

►

15.10.18

w_{4}(z)=\frac{\Gamma\left(1-c\right)\Gamma\left(c-a-b+1\right)}{\Gamma\left(1% -a\right)\Gamma\left(1-b\right)}w_{1}(z)+\frac{\Gamma\left(c-1\right)\Gamma% \left(c-a-b+1\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}w_{2}(z),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $w_{1}(z)$ : Kummer’s solution $w_{1}$ , $w_{2}(z)$ : Kummer’s solution $w_{2}$ and $w_{4}(z)$ : Kummer’s solution $w_{4}$
Permalink:: http://dlmf.nist.gov/15.10.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(ii), §15.10 and Ch.15

►

15.10.19

w_{5}(z)=\frac{\Gamma\left(1-c\right)\Gamma\left(a-b+1\right)}{\Gamma\left(a-c% +1\right)\Gamma\left(1-b\right)}w_{1}(z)+{\mathrm{e}}^{(c-1)\pi\mathrm{i}}% \frac{\Gamma\left(c-1\right)\Gamma\left(a-b+1\right)}{\Gamma\left(a\right)% \Gamma\left(c-b\right)}w_{2}(z),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $w_{1}(z)$ : Kummer’s solution $w_{1}$ , $w_{2}(z)$ : Kummer’s solution $w_{2}$ and $w_{5}(z)$ : Kummer’s solution $w_{5}$
Permalink:: http://dlmf.nist.gov/15.10.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(ii), §15.10 and Ch.15

… ►

15.10.21

w_{1}(z)=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a% \right)\Gamma\left(c-b\right)}w_{3}(z)+\frac{\Gamma\left(c\right)\Gamma\left(a% +b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}w_{4}(z),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $w_{1}(z)$ : Kummer’s solution $w_{1}$ , $w_{3}(z)$ : Kummer’s solution $w_{3}$ and $w_{4}(z)$ : Kummer’s solution $w_{4}$
Referenced by:: §15.8(i)
Permalink:: http://dlmf.nist.gov/15.10.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(ii), §15.10 and Ch.15

… ►

15.10.25

w_{1}(z)=\frac{\Gamma\left(c\right)\Gamma\left(b-a\right)}{\Gamma\left(b\right% )\Gamma\left(c-a\right)}w_{5}(z)+\frac{\Gamma\left(c\right)\Gamma\left(a-b% \right)}{\Gamma\left(a\right)\Gamma\left(c-b\right)}w_{6}(z),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $w_{1}(z)$ : Kummer’s solution $w_{1}$ , $w_{5}(z)$ : Kummer’s solution $w_{5}$ and $w_{6}(z)$ : Kummer’s solution $w_{6}$
Referenced by:: §15.8(i)
Permalink:: http://dlmf.nist.gov/15.10.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(ii), §15.10 and Ch.15

…

25: 32.2 Differential Equations

… ►be a nonlinear second-order differential equation in which

F

is a rational function of

w

and

\ifrac{\mathrm{d}w}{\mathrm{d}z}

, and is locally analytic in

z

, that is, analytic except for isolated singularities in

\mathbb{C}

. … ►They are distinct modulo Möbius (bilinear) transformations … ►In

\mbox{P}_{\mbox{\scriptsize V}}

, if

w(z)=(\coth u(\zeta))^{2}

with

\zeta=\ln z

, then … ►Then

w(z)=f_{1}(z)

satisfies

\mbox{P}_{\mbox{\scriptsize IV}}

with … ►Then

w(z)=1-(\sqrt{z}/f_{1}(z))

satisfies

\mbox{P}_{\mbox{\scriptsize V}}

with …

NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.2.0 of 2024-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

…

27: 28.2 Definitions and Basic Properties

… ►

§28.2(ii) Basic Solutions $w_{\mbox{\rm\tiny I}}$ , $w_{\mbox{\rm\tiny II}}$

… ►Furthermore, a solution

w

with given initial constant values of

w

and

w^{\prime}

at a point

z_{0}

is an entire function of the three variables

z

a

, and

q

. … ►(28.2.1) possesses a fundamental pair of solutions

w_{\mbox{\tiny I}}(z;a,q),w_{\mbox{\tiny II}}(z;a,q)

called basic solutions with …

w_{\mbox{\tiny I}}(z;a,q)

is even and

w_{\mbox{\tiny II}}(z;a,q)

is odd. … ►Even parity means

w(-z)=w(z)

, and odd parity means

w(-z)=-w(z)

. …

28: 13.14 Definitions and Basic Properties

… ►This equation is obtained from Kummer’s equation (13.2.1) via the substitutions

W=e^{-\frac{1}{2}z}z^{\frac{1}{2}+\mu}w

\kappa=\tfrac{1}{2}b-a

, and

\mu=\tfrac{1}{2}b-\tfrac{1}{2}

. … ►In general

M_{\kappa,\mu}\left(z\right)

and

W_{\kappa,\mu}\left(z\right)

are many-valued functions of

z

with branch points at

z=0

and

z=\infty

. … ►Also, unless specified otherwise

M_{\kappa,\mu}\left(z\right)

and

W_{\kappa,\mu}\left(z\right)

are assumed to have their principal values. … ►For

W_{\kappa,\mu}\left(z\right)

with

\Re\mu<0

use (13.14.31). … ►When

2\mu

is an integer we may use the results of §13.2(v) with the substitutions

b=2\mu+1

a=\mu-\kappa+\tfrac{1}{2}

, and

W=e^{-\frac{1}{2}z}z^{\frac{1}{2}+\mu}w

, where

W

is the solution of (13.14.1) corresponding to the solution

w

of (13.2.1). …

29: 13.15 Recurrence Relations and Derivatives

… ►

13.15.8

W_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)-\sqrt{z}W_{\kappa,\mu}% \left(z\right)+(\kappa-\mu-\tfrac{1}{2})W_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}% \left(z\right)=0,

ⓘ

Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §13.15(i), §13.15 and Ch.13

►

13.15.9

W_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-\sqrt{z}W_{\kappa,\mu}% \left(z\right)+(\kappa+\mu-\tfrac{1}{2})W_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}% \left(z\right)=0,

ⓘ

Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
A&S Ref:: 13.4.30 (in different form)
Permalink:: http://dlmf.nist.gov/13.15.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §13.15(i), §13.15 and Ch.13

►

13.15.10

2\mu W_{\kappa,\mu}\left(z\right)-\sqrt{z}W_{\kappa+\frac{1}{2},\mu+\frac{1}{2% }}\left(z\right)+\sqrt{z}W_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)=0,

ⓘ

Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
Referenced by:: §13.15(i)
Permalink:: http://dlmf.nist.gov/13.15.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §13.15(i), §13.15 and Ch.13

►

13.15.11

W_{\kappa+1,\mu}\left(z\right)+(2\kappa-z)W_{\kappa,\mu}\left(z\right)+(\kappa% -\mu-\tfrac{1}{2})(\kappa+\mu-\tfrac{1}{2})W_{\kappa-1,\mu}\left(z\right)=0,

ⓘ

Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
A&S Ref:: 13.4.31
Permalink:: http://dlmf.nist.gov/13.15.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §13.15(i), §13.15 and Ch.13

►

13.15.12

(\kappa-\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z% \right)+2\mu W_{\kappa,\mu}\left(z\right)-(\kappa+\mu-\tfrac{1}{2})\sqrt{z}W_{% \kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)=0,

ⓘ

Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §13.15(i), §13.15 and Ch.13

…

30: 2 Asymptotic Approximations

… …

房产继承公证书【WeChat微aptao168】83w

21—30 of 284 matching pages

21: 28.29 Definitions and Basic Properties

22: 4.34 Derivatives and Differential Equations

23: 4.5 Inequalities

24: 15.10 Hypergeometric Differential Equation

25: 32.2 Differential Equations

26: How to Cite

27: 28.2 Definitions and Basic Properties

§28.2(ii) Basic Solutions w I , w II

28: 13.14 Definitions and Basic Properties

29: 13.15 Recurrence Relations and Derivatives

30: 2 Asymptotic Approximations

§28.2(ii) Basic Solutions $w_{\mbox{\rm\tiny I}}$ , $w_{\mbox{\rm\tiny II}}$