办假的科罗拉多学院文凭毕业证【somewhat微aptao168】bigrf

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1: Software Index

	Open Source						With Book		Commercial
…
19.39(iv) $R_{C}\left(x,y\right)$ , $R_{F}\left(x,y,z\right)$ , $R_{D}\left(x,y,z\right)$ , $R_{J}\left(x,y,z,p\right)$	✓	✓	✓	✓	a	✓		✓	✓	✓	Derive
…

… ►

Open Source Collections and Systems.

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

…

2: 19.21 Connection Formulas

… ►

19.21.1

R_{F}\left(0,z+1,z\right)R_{D}\left(0,z+1,1\right)+R_{D}\left(0,z+1,z\right)R_% {F}\left(0,z+1,1\right)=3\pi/(2z),

z\in\mathbb{C}\setminus(-\infty,0]

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{C}$ : complex plane, $\in$ : element of, $(\NVar{a},\NVar{b}]$ : half-closed interval and $\setminus$ : set subtraction
Referenced by:: §19.21(i), §19.21(i), §19.7(i)
Permalink:: http://dlmf.nist.gov/19.21.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(i), §19.21 and Ch.19

… ►The complete cases of

R_{F}

and

R_{G}

have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … ►

19.21.4

R_{F}\left(0,z-1,z\right)=R_{F}\left(0,1-z,1\right)\mp\mathrm{i}R_{F}\left(0,z% ,1\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/19.21.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(i), §19.21 and Ch.19

… ►The complete case of

R_{J}

can be expressed in terms of

R_{F}

and

R_{D}

: … ►Because

R_{G}

is completely symmetric,

x, y, z

can be permuted on the right-hand side of (19.21.10) so that

(x-z)(y-z)\leq 0

if the variables are real, thereby avoiding cancellations when

R_{G}

is calculated from

R_{F}

and

R_{D}

(see §19.36(i)). …

3: 19.32 Conformal Map onto a Rectangle

… ►

19.32.1

z(p)=R_{F}\left(p-x_{1},p-x_{2},p-x_{3}\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $z(p)$ : function
Permalink:: http://dlmf.nist.gov/19.32.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.32 and Ch.19

… ►

z(x_{1})=R_{F}\left(0,x_{1}-x_{2},x_{1}-x_{3}\right)\quad\text{($>0$)},

… ►

z(x_{3})=R_{F}\left(x_{3}-x_{1},x_{3}-x_{2},0\right)=-iR_{F}\left(0,x_{1}-x_{3% },x_{2}-x_{3}\right).

…

4: 20.9 Relations to Other Functions

… ►

20.9.3

R_{F}\left(\frac{{\theta_{2}}^{2}\left(z,q\right)}{{\theta_{2}}^{2}\left(0,q% \right)},\frac{{\theta_{3}}^{2}\left(z,q\right)}{{\theta_{3}}^{2}\left(0,q% \right)},\frac{{\theta_{4}}^{2}\left(z,q\right)}{{\theta_{4}}^{2}\left(0,q% \right)}\right)=\frac{\theta_{1}'\left(0,q\right)}{\theta_{1}\left(z,q\right)}z,

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Referenced by:: §19.25(v), §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §20.9(i), §20.9 and Ch.20

►

20.9.4

R_{F}\left(0,{\theta_{3}}^{4}\left(0,q\right),{\theta_{4}}^{4}\left(0,q\right)% \right)=\tfrac{1}{2}\pi,

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and $q$ : nome
Referenced by:: §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §20.9(i), §20.9 and Ch.20

►

20.9.5

\exp\left(-\frac{\pi R_{F}\left(0,k^{2},1\right)}{R_{F}\left(0,{k^{\prime}}^{2% },1\right)}\right)=q.

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $q$ : nome and $k$ : modulus
Referenced by:: §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §20.9(i), §20.9 and Ch.20

…

5: 19.24 Inequalities

… ►For

x>0

y>0

, and

x\neq y

, the complete cases of

R_{F}

and

R_{G}

satisfy ►

R_{F}\left(x,y,0\right)R_{G}\left(x,y,0\right)>\tfrac{1}{8}\pi^{2},

… ►

19.24.10

\frac{3}{\sqrt{x}+\sqrt{y}+\sqrt{z}}\leq R_{F}\left(x,y,z\right)\leq\frac{1}{(% xyz)^{1/6}},

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.9(ii)
Permalink:: http://dlmf.nist.gov/19.24.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §19.24(ii), §19.24(ii), §19.24 and Ch.19

… ►Other inequalities for

R_{F}\left(x,y,z\right)

are given in Carlson (1970). … ►

R_{F}\left(x,y,z\right)R_{G}\left(x,y,z\right)>1,

…

6: 19.28 Integrals of Elliptic Integrals

… ►

19.28.1

\int_{0}^{1}t^{\sigma-1}R_{F}\left(0,t,1\right)\,\mathrm{d}t=\tfrac{1}{2}\left% (\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2},

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sigma$ : parameter
Referenced by:: §19.28, §19.28
Permalink:: http://dlmf.nist.gov/19.28.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.28 and Ch.19

… ►

19.28.5

\int_{z}^{\infty}R_{D}\left(x,y,t\right)\,\mathrm{d}t=6R_{F}\left(x,y,z\right),

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral
Referenced by:: §19.28, §19.28
Permalink:: http://dlmf.nist.gov/19.28.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.28 and Ch.19

… ►

19.28.7

\int_{0}^{\infty}R_{J}\left(x,y,z,r^{2}\right)\,\mathrm{d}r=\tfrac{3}{2}\pi R_% {F}\left(xy,xz,yz\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral
Referenced by:: §19.28
Permalink:: http://dlmf.nist.gov/19.28.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.28 and Ch.19

… ►

19.28.9

\int_{0}^{\pi/2}R_{F}\left({\sin}^{2}\theta{\cos}^{2}\left(x+y\right),{\sin}^{% 2}\theta{\cos}^{2}\left(x-y\right),1\right)\,\mathrm{d}\theta=R_{F}\left(0,{% \cos}^{2}x,1\right)R_{F}\left(0,{\cos}^{2}y,1\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sin\NVar{z}$ : sine function
Referenced by:: §19.28, §19.28
Permalink:: http://dlmf.nist.gov/19.28.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.28 and Ch.19

►

19.28.10

\int_{0}^{\infty}R_{F}\left((ac+bd)^{2},(ad+bc)^{2},4abcd{\cosh}^{2}z\right)\,% \mathrm{d}z=\tfrac{1}{2}R_{F}\left(0,a^{2},b^{2}\right)R_{F}\left(0,c^{2},d^{2% }\right),

a,b,c,d>0

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function and $\int$ : integral
Referenced by:: §19.28
Permalink:: http://dlmf.nist.gov/19.28.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §19.28 and Ch.19

…

7: 19.18 Derivatives and Differential Equations

… ►

19.18.1

\frac{\partial R_{F}\left(x,y,z\right)}{\partial z}=-\tfrac{1}{6}R_{D}\left(x,% y,z\right),

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : partial differential of $x$
Referenced by:: §19.18(i), §19.28
Permalink:: http://dlmf.nist.gov/19.18.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.18(i), §19.18 and Ch.19

►

19.18.2

\frac{\mathrm{d}}{\mathrm{d}x}R_{G}\left(x+a,x+b,x+c\right)=\tfrac{1}{2}R_{F}% \left(x+a,x+b,x+c\right).

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind and $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$
Referenced by:: §19.18(i)
Permalink:: http://dlmf.nist.gov/19.18.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.18(i), §19.18 and Ch.19

… ►

19.18.6

\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{% \partial z}\right)R_{F}\left(x,y,z\right)=\frac{-1}{2\sqrt{x}\sqrt{y}\sqrt{z}},

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : partial differential of $x$
Referenced by:: §19.18(ii), §19.32, Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.18.E6
Encodings:: pMML, png, TeX
Correction (effective with 1.1.3):: The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.
Suggested 2021-06-07 by Luc Maisonobe
See also:: Annotations for §19.18(ii), §19.18 and Ch.19

… ►

19.18.9

\left(x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}+z\frac{% \partial}{\partial z}\right)R_{F}\left(x,y,z\right)=-\tfrac{1}{2}R_{F}\left(x,% y,z\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : partial differential of $x$
Permalink:: http://dlmf.nist.gov/19.18.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.18(ii), §19.18 and Ch.19

… ►The next four differential equations apply to the complete case of

R_{F}

and

R_{G}

in the form

R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)

(see (19.16.20) and (19.16.23)). …

8: 19.22 Quadratic Transformations

… ►

19.22.1

R_{F}\left(0,x^{2},y^{2}\right)=R_{F}\left(0,xy,a^{2}\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.22(i), §19.28
Permalink:: http://dlmf.nist.gov/19.22.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(i), §19.22 and Ch.19

►

19.22.2

2R_{G}\left(0,x^{2},y^{2}\right)=4R_{G}\left(0,xy,a^{2}\right)-xyR_{F}\left(0,% xy,a^{2}\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.22(i)
Permalink:: http://dlmf.nist.gov/19.22.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(i), §19.22 and Ch.19

►

19.22.3

2y^{2}R_{D}\left(0,x^{2},y^{2}\right)=\tfrac{1}{4}(y^{2}-x^{2})R_{D}\left(0,xy% ,a^{2}\right)+3R_{F}\left(0,xy,a^{2}\right).

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.22(i), §19.22(i), §19.22(iii)
Permalink:: http://dlmf.nist.gov/19.22.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(i), §19.22 and Ch.19

… ►

19.22.8

\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}\right)=\frac{1}{M\left(a_{0},g_{% 0}\right)},

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.22.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(ii), §19.22 and Ch.19

… ►

19.22.18

R_{F}\left(x^{2},y^{2},z^{2}\right)=R_{F}\left(a^{2},z_{-}^{2},z_{+}^{2}\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.22(i), §19.22(iii), §19.22(iii), §19.29(iii), §19.36(ii), §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.22.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(iii), §19.22 and Ch.19

…

9: 19.36 Methods of Computation

… ►For

R_{F}

the polynomial of degree 7, for example, is … ►All cases of

R_{F}

R_{C}

R_{J}

, and

R_{D}

are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)). …Because of cancellations in (19.26.21) it is advisable to compute

R_{G}

from

R_{F}

and

R_{D}

by (19.21.10) or else to use §19.36(ii). … ►We compute

R_{F}\left(1,2,4\right)

by setting

\theta=1

t_{0}=c_{0}=1

, and

a_{0}=\sqrt{3}

. … ►Similarly, §19.26(ii) eases the computation of functions such as

R_{F}\left(x,y,z\right)

when

x

(

>0

) is small compared with

\min(y,z)

. …

10: 19.26 Addition Theorems

… ►

19.26.1

R_{F}\left(x+\lambda,y+\lambda,z+\lambda\right)+R_{F}\left(x+\mu,y+\mu,z+\mu% \right)=R_{F}\left(x,y,z\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\lambda$ : positive, $x$ : positive, $y$ : positive, $z$ : positive and $\mu>0$ : parameter
Referenced by:: §19.26(i)
Permalink:: http://dlmf.nist.gov/19.26.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(i), §19.26 and Ch.19

… ►

19.26.8

2R_{G}\left(x+\lambda,y+\lambda,z+\lambda\right)+2R_{G}\left(x+\mu,y+\mu,z+\mu% \right)=2R_{G}\left(x,y,z\right)+\lambda R_{F}\left(x+\lambda,y+\lambda,z+% \lambda\right)+\mu R_{F}\left(x+\mu,y+\mu,z+\mu\right)+\sqrt{\lambda+\mu+x+y+z}.

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\lambda$ : positive, $x$ : positive, $y$ : positive, $z$ : positive and $\mu>0$ : parameter
Permalink:: http://dlmf.nist.gov/19.26.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(i), §19.26 and Ch.19

… ►

19.26.16

R_{F}\left(\lambda,y+\lambda,z+\lambda\right)={R_{F}\left(0,y,z\right)-R_{F}% \left(\mu,y+\mu,z+\mu\right),}

\lambda\mu=yz

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\lambda$ : positive, $y$ : positive, $z$ : positive and $\mu>0$ : parameter
Permalink:: http://dlmf.nist.gov/19.26.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(ii), §19.26 and Ch.19

… ►

19.26.18

R_{F}\left(x,y,z\right)=2R_{F}\left(x+\lambda,y+\lambda,z+\lambda\right)=R_{F}% \left(\frac{x+\lambda}{4},\frac{y+\lambda}{4},\frac{z+\lambda}{4}\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\lambda$ : positive, $x$ : positive, $y$ : positive and $z$ : positive
Referenced by:: §19.26(iii), §19.36(i), §19.36(i)
Permalink:: http://dlmf.nist.gov/19.26.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(iii), §19.26 and Ch.19

… ►

19.26.21

2R_{G}\left(x,y,z\right)=4R_{G}\left(x+\lambda,y+\lambda,z+\lambda\right)-% \lambda R_{F}\left(x,y,z\right)-\sqrt{x}-\sqrt{y}-\sqrt{z}.

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\lambda$ : positive, $x$ : positive, $y$ : positive and $z$ : positive
Referenced by:: §19.36(i)
Permalink:: http://dlmf.nist.gov/19.26.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(iii), §19.26 and Ch.19

…