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21: 7.2 Definitions

… ► $\mathrm{erf}z$, $\mathrm{erfc}z$, and $w\left(z\right)$ are entire functions of $z$, as is $F\left(z\right)$ in the next subsection. … ►

7.2.8 $$S\left(z\right)={\int }_0^z\sin \left(\frac{1}{2}\pi t^2\right)dt,$$

ⓘ

Defines:

$S\left(z\right)$: Fresnel integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.2

Referenced by:

§7.5, §7.6(i)

Permalink:

http://dlmf.nist.gov/7.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(iii), §7.2 and Ch.7

► $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection. … ►

7.2.10 $$\mathrm{f}\left(z\right)=\left(\frac{1}{2}-S\left(z\right)\right)\cos \left(\frac{1}{2}\pi z^2\right)-\left(\frac{1}{2}-C\left(z\right)\right)\sin \left(\frac{1}{2}\pi z^2\right),$$

ⓘ

Defines:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals

Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.5

Referenced by:

§7.4, §7.5

Permalink:

http://dlmf.nist.gov/7.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(iv), §7.2 and Ch.7

►

7.2.11 $$\mathrm{g}\left(z\right)=\left(\frac{1}{2}-C\left(z\right)\right)\cos \left(\frac{1}{2}\pi z^2\right)+\left(\frac{1}{2}-S\left(z\right)\right)\sin \left(\frac{1}{2}\pi z^2\right).$$

ⓘ

Defines:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals

Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.6

Referenced by:

§7.4, §7.5

Permalink:

http://dlmf.nist.gov/7.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(iv), §7.2 and Ch.7

…

22: 19.14 Reduction of General Elliptic Integrals

… ►

19.14.5 $${\sin }^2\varphi =\frac{\gamma -\alpha }{U^2+\gamma },$$

ⓘ

Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

… ►

19.14.7 $${\sin }^2\varphi =\frac{(\gamma -\alpha )x^2}{a_1{a}_2+\gamma x^2}.$$

ⓘ

Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

… ►

19.14.8 $${\sin }^2\varphi =\frac{\gamma -\alpha }{b_1{b}_2{y}^2+\gamma }.$$

ⓘ

Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

… ►

19.14.9 $${\sin }^2\varphi =\frac{(\gamma -\alpha )(x^2-y^2)}{\gamma (x^2-y^2)-a_1(a_2+b_2{x}^2)}.$$

ⓘ

Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

… ►

19.14.10 $${\sin }^2\varphi =\frac{(\gamma -\alpha )(y^2-x^2)}{\gamma (y^2-x^2)-a_1(a_2+b_2{y}^2)}.$$

ⓘ

Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.15, §19.29(iii)

Permalink:

http://dlmf.nist.gov/19.14.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

…

23: 19.33 Triaxial Ellipsoids

… ►

19.33.2 $$\frac{S}{2\pi }=c^2+\frac{ab}{\sin \varphi }\left(E(\varphi ,k){\sin }^2\varphi +F(\varphi ,k){\cos }^2\varphi \right),$$ $a\ge b\ge c$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $S$: area

Permalink:

http://dlmf.nist.gov/19.33.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(i), §19.33 and Ch.19

… ►

19.33.5 $$V(\lambda )=Q{R}_F(a^2+\lambda ,b^2+\lambda ,c^2+\lambda ),$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $Q$: charge and $V(\lambda )$: potential

Permalink:

http://dlmf.nist.gov/19.33.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(ii), §19.33 and Ch.19

►and the electric capacity $C=Q/V(0)$ is given by ►

19.33.6 $$1/C=R_F(a^2,b^2,c^2).$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind and $C$: electric capacity

Permalink:

http://dlmf.nist.gov/19.33.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(ii), §19.33 and Ch.19

… ►

19.33.11 $$U=\frac{1}{2}{(\alpha \beta \gamma )}^2{R}_F({\alpha }^2,{\beta }^2,{\gamma }^2){\int }_0^{\mathrm{\infty }}{(g(r))}^{2}dr,$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $dx$: differential of $x$ , $\int$: integral, $\alpha$: positive constant, $\beta$: positive constant, $\gamma$: positive constant, $U$: energy and $g(r)$: function

Permalink:

http://dlmf.nist.gov/19.33.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(iv), §19.33 and Ch.19

…

24: 19.28 Integrals of Elliptic Integrals

… ►

19.28.1 $${\int }_0^1{t}^{\sigma -1}{R}_F(0,t,1)dt=\frac{1}{2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2,$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\mathrm{B}(a,b)$: beta function, $dx$: 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 $${\int }_z^{\mathrm{\infty }}{R}_D(x,y,t)dt=6R_F(x,y,z),$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $dx$: 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

… ►

19.28.8 $${\int }_0^{\mathrm{\infty }}{R}_J(tx,y,z,tp)dt=\frac{6}{\sqrt{p}}{R}_C(p,x)R_F(0,y,z).$$

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.28

Permalink:

http://dlmf.nist.gov/19.28.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

►

19.28.9 $${\int }_0^{\pi /2}{R}_F({\sin }^2\theta {\cos }^2\left(x+y\right),{\sin }^2\theta {\cos }^2\left(x-y\right),1)d\theta =R_F(0,{\cos }^{2}x,1)R_F(0,{\cos }^{2}y,1),$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $\sin z$: 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 $${\int }_0^{\mathrm{\infty }}{R}_F({(ac+bd)}^2,{(ad+bc)}^2,4abcd{\cosh }^{2}z)dz=\frac{1}{2}{R}_F(0,a^2,b^2)R_F(0,c^2,d^2),$$ $a,b,c,d>0$.

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $dx$: differential of $x$ , $\cosh z$: 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

…

25: 14.3 Definitions and Hypergeometric Representations

… ►

14.3.2

ⓘ

Defines:

: Ferrers function of the second kind

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.3(i)

Permalink:

http://dlmf.nist.gov/14.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14

… ►

14.3.3

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and : real variable

Permalink:

http://dlmf.nist.gov/14.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14

… ►

14.3.11

ⓘ

Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mu$: general order, $\nu$: general degree, : real variable and $w_2$: function $w_1$,

A&S Ref:

8.1.4 (modified)

Referenced by:

§10.59, §14.23, §14.3(iii), §14.5(i)

Permalink:

http://dlmf.nist.gov/14.3.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

… ►

14.3.20

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and : real variable

A&S Ref:

8.1.6 (modified)

Referenced by:

§14.21(iii), §14.3(iii)

Permalink:

http://dlmf.nist.gov/14.3.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

… ►In terms of the Gegenbauer function $C_{\alpha }^{(\beta )}\left(x\right)$ and the Jacobi function ${\varphi }_{\lambda }^{(\alpha ,\beta )}\left(t\right)$ (§§15.9(iii), 15.9(ii)): …

26: 19.3 Graphics

… ►

►

►

Figure 19.3.3: $F(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. If ${\sin }^2\varphi =1$ ($\ge k^2$), then the function reduces to $K\left(k\right)$, becoming infinite when $k^2=1$. … Magnify 3D Help

►

►

►

Figure 19.3.4: $E(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. … Magnify 3D Help

… ►

►

►

Figure 19.3.6: $\mathrm{\Pi }(\varphi ,2,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 3$, . Cauchy principal values are shown when ${\sin }^2\varphi >\frac{1}{2}$. … Magnify 3D Help

…

27: 7.5 Interrelations

… ►

7.5.1 $$F\left(z\right)=\frac{1}{2}\mathrm{i}\sqrt{\pi }\left({\mathrm{e}}^{-z^2}-w\left(z\right)\right)=-\frac{1}{2}\mathrm{i}\sqrt{\pi }{\mathrm{e}}^{-z^2}\mathrm{erf}\left(\mathrm{i}z\right).$$

ⓘ

Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

►

7.5.2 $$C\left(z\right)+\mathrm{i}S\left(z\right)=\frac{1}{2}(1+\mathrm{i})-ℱ\left(z\right).$$

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, $ℱ\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

►

7.5.3 $$C\left(z\right)=\frac{1}{2}+\mathrm{f}\left(z\right)\sin \left(\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right)\cos \left(\frac{1}{2}\pi z^2\right),$$

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.9

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

►

7.5.4 $$S\left(z\right)=\frac{1}{2}-\mathrm{f}\left(z\right)\cos \left(\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right)\sin \left(\frac{1}{2}\pi z^2\right).$$

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.10

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

… ►

7.5.13 $$G\left(x\right)=\sqrt{\pi }F\left(x\right)-\frac{1}{2}{\mathrm{e}}^{-x^2}\mathrm{Ei}\left(x^2\right),$$ $x>0$.

ⓘ

Symbols:

$F\left(z\right)$: Dawson’s integral, $G\left(z\right)$: Goodwin–Staton integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral and $x$: real variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

…

28: 15.8 Transformations of Variable

… ►

15.8.2 $$\frac{\sin \left(\pi (b-a)\right)}{\pi }𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)=\frac{{(-z)}^{-a}}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)}𝐅(\genfrac{}{}{0pt}{}{a,a-c+1}{a-b+1};\frac{1}{z})-\frac{{(-z)}^{-b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-b\right)}𝐅(\genfrac{}{}{0pt}{}{b,b-c+1}{b-a+1};\frac{1}{z}),$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\sin z$: sine function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.12(i), §15.16, §15.8(i), §15.8(i), §15.8(ii), §15.8(ii), §15.8(ii)

Permalink:

http://dlmf.nist.gov/15.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(i), §15.8 and Ch.15

►

15.8.3 $$\frac{\sin \left(\pi (b-a)\right)}{\pi }𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)=\frac{{(1-z)}^{-a}}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)}𝐅(\genfrac{}{}{0pt}{}{a,c-b}{a-b+1};\frac{1}{1-z})-\frac{{(1-z)}^{-b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-b\right)}𝐅(\genfrac{}{}{0pt}{}{b,c-a}{b-a+1};\frac{1}{1-z}),$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\sin z$: sine function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.8(i), §15.8(ii)

Permalink:

http://dlmf.nist.gov/15.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(i), §15.8 and Ch.15

►

15.8.4 $$\frac{\sin \left(\pi (c-a-b)\right)}{\pi }𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)=\frac{1}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}𝐅(\genfrac{}{}{0pt}{}{a,b}{a+b-c+1};1-z)-\frac{{(1-z)}^{c-a-b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}𝐅(\genfrac{}{}{0pt}{}{c-a,c-b}{c-a-b+1};1-z),$$ , .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\sin z$: sine function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.16, §15.19(iv), §15.2(i), §15.4(ii), §15.8(i), §15.8(i), §15.8(ii)

Permalink:

http://dlmf.nist.gov/15.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(i), §15.8 and Ch.15

►

15.8.5 $$\frac{\sin \left(\pi (c-a-b)\right)}{\pi }𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)=\frac{z^{-a}}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}𝐅(\genfrac{}{}{0pt}{}{a,a-c+1}{a+b-c+1};1-\frac{1}{z})-\frac{{(1-z)}^{c-a-b}{z}^{a-c}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}𝐅(\genfrac{}{}{0pt}{}{c-a,1-a}{c-a-b+1};1-\frac{1}{z}),$$ , .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\sin z$: sine function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.8(i), §15.8(i), §15.8(ii), §15.8(ii), §15.8(ii)

Permalink:

http://dlmf.nist.gov/15.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(i), §15.8 and Ch.15

… ►Alternatively, if $b-a$ is a negative integer, then we interchange $a$ and $b$ in $𝐅(a,b;c;z)$. …

29: 31.2 Differential Equations

… ►

31.2.5 $$z={\sin }^2\theta ,$$

ⓘ

Symbols:

$\sin z$: sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/31.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iii), §31.2 and Ch.31

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31.2.6 $$\frac{d^{2}w}{{d\theta }^2}+\left((2\gamma -1)\cot \theta -(2\delta -1)\tan \theta -\frac{ϵ\sin \left(2\theta \right)}{a-{\sin }^2\theta }\right)\frac{dw}{d\theta }+4\frac{\alpha \beta {\sin }^2\theta -q}{a-{\sin }^2\theta }w=0.$$

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Symbols:

$\cot z$: cotangent function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function, $\tan z$: tangent function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iii), §31.2 and Ch.31

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$F$-Homotopic Transformations

… ►There are $4!=24$ homographies $\tilde{z}(z)=(Az+B)/(Cz+D)$ that take $0,1,a,\mathrm{\infty }$ to some permutation of $0,1,a^{\prime },\mathrm{\infty }$, where $a^{\prime }$ may differ from $a$. … ►There are $8\cdot 24=192$ automorphisms of equation (31.2.1) by compositions of $F$-homotopic and homographic transformations. …

30: 19.8 Quadratic Transformations

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$$E(\varphi ,k)=\frac{1}{2}(1+k^{\prime })E({\varphi }_1,k_1)-k^{\prime }F(\varphi ,k)+\frac{1}{2}(1-k^{\prime })\sin {\varphi }_1.$$

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$$E(\varphi ,k)=(1+k)E({\varphi }_2,k_2)+(1-k)F({\varphi }_2,k_2)-k\sin \varphi .$$

… ►We consider only the descending Gauss transformation because its (ascending) inverse moves $F(\varphi ,k)$ closer to the singularity at $k=\sin \varphi =1$. … ►

$\sin {\psi }_1={\displaystyle \frac{(1+k^{\prime })\sin \varphi }{1+\mathrm{\Delta }}},$

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$\mathrm{\Delta }=\sqrt{1-k^2{\sin }^2\varphi }.$

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