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11: 15.13 Zeros

… ►Let $N(a,b,c)$ denote the number of zeros of $F(a,b;c;z)$ in the sector . If $a$, $b$, $c$ are real, $a$, $b$, $c$, $c-a$, $c-b\ne 0,-1,-2,\mathrm{\dots }$, and, without loss of generality, $b\ge a$, $c\ge a+b$ (compare (15.8.1)), then … ►where $S=\mathrm{sign}\left(\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)\right)$. … ►If $a$, $b$, $c$, $c-a$, or $c-b\in \{0,-1,-2,\mathrm{\dots }\}$, then $F(a,b;c;z)$ is not defined, or reduces to a polynomial, or reduces to ${(1-z)}^{c-a-b}$ times a polynomial. … ►A small table of zeros is given in Conde and Kalla (1981) and Segura (2008).

12: 4.42 Solution of Triangles

… ►

4.42.1 $$\sin A=\frac{a}{c}=\frac{1}{\csc A},$$

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Symbols:

$\csc z$: cosecant function, $\sin z$: sine function, $A$: angle, $a$: height and $c$: hypotenuse

A&S Ref:

4.3.147

Permalink:

http://dlmf.nist.gov/4.42.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(i), §4.42 and Ch.4

… ►

4.42.3 $$\tan A=\frac{a}{b}=\frac{1}{\cot A}.$$

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Symbols:

$\cot z$: cotangent function, $\tan z$: tangent function, $A$: angle, $a$: height and $b$: base

A&S Ref:

4.3.147

Permalink:

http://dlmf.nist.gov/4.42.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(i), §4.42 and Ch.4

… ►

4.42.7 $$\text{area}=\frac{1}{2}bc\sin A={\left(s(s-a)(s-b)(s-c)\right)}^{1/2},$$

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Symbols:

$\sin z$: sine function, $a$: length, $b$: base, $c$: length, $s$: semi-perimeter and $A$: angle

A&S Ref:

4.3.148

Permalink:

http://dlmf.nist.gov/4.42.E7

Encodings:

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See also:

Annotations for §4.42(ii), §4.42 and Ch.4

►where $s=\frac{1}{2}(a+b+c)$ (the semiperimeter). … ►

4.42.8 $$\cos a=\cos b\cos c+\sin b\sin c\cos A,$$

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Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $a$: arc length, $b$: arc length and $c$: arc length

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

…

13: 8.10 Inequalities

… ►The inequalities in (8.10.1) and (8.10.2) are reversed when $a\ge 1$. … ►where … ►Also, define …where …Equalities in (8.10.11) apply only when $a=1$. …

14: 8.17 Incomplete Beta Functions

… ►where, as in §5.12, $\mathrm{B}(a,b)$ denotes the beta function: … ►Addendum: For a companion equation see (8.17.24). … ►For a historical profile of ${\mathrm{B}}_x(a,b)$ see Dutka (1981). … ►With $a>0$, $b>0$, and , … ►The expansion (8.17.22) converges rapidly for . …

15: 8.1 Special Notation

… ► ►►►

$x$	real variable.
…
$a,p$	real or complex parameters.
…

… ►The functions treated in this chapter are the incomplete gamma functions $\gamma (a,z)$, $\mathrm{\Gamma }(a,z)$, ${\gamma }^{\ast }(a,z)$, $P(a,z)$, and $Q(a,z)$; the incomplete beta functions ${\mathrm{B}}_x(a,b)$ and $I_x(a,b)$; the generalized exponential integral $E_p\left(z\right)$; the generalized sine and cosine integrals $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$. ►Alternative notations include: Prym’s functions $P_z(a)=\gamma (a,z)$, $Q_z(a)=\mathrm{\Gamma }(a,z)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma (a+1,z)$, $[a,z]!=\mathrm{\Gamma }(a+1,z)$, Dingle (1973); $B(a,b,x)={\mathrm{B}}_x(a,b)$, $I(a,b,x)=I_x(a,b)$, Magnus et al. (1966); $\mathrm{Si}(a,x)\to \mathrm{Si}(1-a,x)$, $\mathrm{Ci}(a,x)\to \mathrm{Ci}(1-a,x)$, Luke (1975).

16: 8.5 Confluent Hypergeometric Representations

… ►

8.5.1 $$\gamma (a,z)=a^{-1}{z}^a{\mathrm{e}}^{-z}M(1,1+a,z)=a^{-1}{z}^{a}M(a,1+a,-z),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.12

Referenced by:

§8.11(ii), §8.14

Permalink:

http://dlmf.nist.gov/8.5.E1

Encodings:

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See also:

Annotations for §8.5 and Ch.8

►

8.5.2 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-z}𝐌(1,1+a,z)=𝐌(a,1+a,-z).$$

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Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.7

Permalink:

http://dlmf.nist.gov/8.5.E2

Encodings:

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See also:

Annotations for §8.5 and Ch.8

►

8.5.3 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-z}U(1-a,1-a,z)=z^a{\mathrm{e}}^{-z}U(1,1+a,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.14, §8.19(vi)

Permalink:

http://dlmf.nist.gov/8.5.E3

Encodings:

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See also:

Annotations for §8.5 and Ch.8

►

8.5.4 $$\gamma (a,z)=a^{-1}{z}^{\frac{1}{2}a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{M}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.5

Permalink:

http://dlmf.nist.gov/8.5.E4

Encodings:

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See also:

Annotations for §8.5 and Ch.8

►

8.5.5 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}a-\frac{1}{2}}{W}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

17: 13.30 Tables

… ►

Žurina and Osipova (1964) tabulates $M(a,b,x)$ and $U(a,b,x)$ for $b=2$, $a=-0.98(.02)1.10$, $x=0(.01)4$, 7D or 7S.

►

Slater (1960) tabulates $M(a,b,x)$ for $a=-1(.1)1$, $b=0.1(.1)1$, and $x=0.1(.1)10$, 7–9S; $M(a,b,1)$ for $a=-11(.2)2$ and $b=-4(.2)1$, 7D; the smallest positive $x$-zero of $M(a,b,x)$ for $a=-4(.1)-0.1$ and $b=0.1(.1)2.5$, 7D.

►

Abramowitz and Stegun (1964, Chapter 13) tabulates $M(a,b,x)$ for $a=-1(.1)1$, $b=0.1(.1)1$, and $x=0.1(.1)1(1)10$, 8S. Also the smallest positive $x$-zero of $M(a,b,x)$ for $a=-1(.1)-0.1$ and $b=0.1(.1)1$, 7D.

►

Zhang and Jin (1996, pp. 411–423) tabulates $M(a,b,x)$ and $U(a,b,x)$ for $a=-5(.5)5$, $b=0.5(.5)5$, and $x=0.1,1,5,10,20,30$, 8S (for $M(a,b,x)$) and 7S (for $U(a,b,x)$).

…

18: 16.9 Zeros

… ►Assume that $p=q$ and none of the $a_j$ is a nonpositive integer. Then ${}_p{}^{}F_p^{}(𝐚;𝐛;z)$ has at most finitely many zeros if and only if the $a_j$ can be re-indexed for $j=1,\mathrm{\dots },p$ in such a way that $a_j-b_j$ is a nonnegative integer. ►Next, assume that $p=q$ and that the $a_j$ and the quotients ${\left(𝐚\right)}_j/{\left(𝐛\right)}_j$ are all real. Then ${}_p{}^{}F_p^{}(𝐚;𝐛;z)$ has at most finitely many real zeros. …

19: 4.43 Cubic Equations

… ►

4.43.2 $$z^3+pz+q=0$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $p$: real constant and $q$: real constant

Referenced by:

(4.43.1), §4.43

Permalink:

http://dlmf.nist.gov/4.43.E2

Encodings:

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Errata (effective with 1.0.10):

Cases (a), (b), and (c) of of this equation have been corrected. Formerly, with constants $C$ and $D$ given in Eq. (4.43.1), they read

1. $A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=C$, when and $C\le 1$.

2. $A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=C$, when and $C>1$.

3. $B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=D$, when $p>0$.

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for §4.43 and Ch.4

… ►

$A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=4q/A^3$, when $4p^3+27q^2\le 0$.

►

$A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=-4q/A^3$, when , , and $4p^3+27q^2>0$.

►

$B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=-4q/B^3$, when $p>0$.

►Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. …

20: 15.5 Derivatives and Contiguous Functions

… ►

15.5.5 $${\left(z\frac{d}{dz}z\right)}^n\left(z^{c-a-1}{(1-z)}^{a+b-c}F(a,b;c;z)\right)={\left(c-a\right)}_n{z}^{c-a+n-1}{(1-z)}^{a-n+b-c}F(a-n,b;c;z).$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§18.17(iv)

Permalink:

http://dlmf.nist.gov/15.5.E5

Encodings:

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See also:

Annotations for §15.5(i), §15.5 and Ch.15

… ►The six functions $F(a\pm 1,b;c;z)$, $F(a,b\pm 1;c;z)$, $F(a,b;c\pm 1;z)$ are said to be contiguous to $F(a,b;c;z)$. … ►

15.5.12 $$(b-a)F(a,b;c;z)+aF(a+1,b;c;z)-bF(a,b+1;c;z)=0,$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(ii), §15.5 and Ch.15

… ►By repeated applications of (15.5.11)–(15.5.18) any function $F(a+k,b+\mathrm{\ell };c+m;z)$, in which $k,\mathrm{\ell },m$ are integers, can be expressed as a linear combination of $F(a,b;c;z)$ and any one of its contiguous functions, with coefficients that are rational functions of $a,b,c$, and $z$. … ►

15.5.20 $$z(1-z)\left(dF(a,b;c;z)/dz\right)=(c-a)F(a-1,b;c;z)+(a-c+bz)F(a,b;c;z)=(c-b)F(a,b-1;c;z)+(b-c+az)F(a,b;c;z),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.5.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(ii), §15.5 and Ch.15

…