real

SearchAdvancedHelp

(0.001 seconds)

21—30 of 641 matching pages

21: 5.12 Beta Function

… ►In (5.12.1)–(5.12.4) it is assumed $\mathrm{\Re }a>0$ and $\mathrm{\Re }b>0$. … ►

5.12.5 $${\int }_0^{\pi /2}{(\cos t)}^{a-1}\cos \left(bt\right)dt=\frac{\pi }{2^a}\frac{1}{a\mathrm{B}(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1))},$$ $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$\mathrm{B}(a,b)$: beta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§10.22(ii), §5.12, §7.6(ii)

Permalink:

http://dlmf.nist.gov/5.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.12, §5.12 and Ch.5

… ►

5.12.7 $${\int }_0^{\mathrm{\infty }}\frac{\cosh \left(2bt\right)}{{(\cosh t)}^{2a}}dt=4^{a-1}\mathrm{B}(a+b,a-b),$$ $\mathrm{\Re }a>|\mathrm{\Re }b|$.

ⓘ

Symbols:

$\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\mathrm{\Re }$: real part, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.12

Permalink:

http://dlmf.nist.gov/5.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.12, §5.12 and Ch.5

►

5.12.8 $$\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{dt}{{(w+\mathrm{i}t)}^a{(z-\mathrm{i}t)}^b}=\frac{{(w+z)}^{1-a-b}}{(a+b-1)\mathrm{B}(a,b)},$$ $\mathrm{\Re }\left(a+b\right)>1$, $\mathrm{\Re }w>0$, $\mathrm{\Re }z>0$.

ⓘ

Symbols:

$\mathrm{B}(a,b)$: beta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $w$: real or complex variable, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.12, §5.12

Permalink:

http://dlmf.nist.gov/5.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.12, §5.12 and Ch.5

… ►when $\mathrm{\Re }b>0$, $a$ is not an integer and the contour cuts the real axis between $-1$ and the origin. …

22: 20.16 Software

… ►

§20.16(ii) Real Argument and Parameter

…

23: 22.22 Software

… ►

§22.22(ii) Real Argument

…

24: 31.1 Special Notation

… ► ►►►

$x$, $y$	real variables.
…
$q,\alpha ,\beta ,\gamma ,\delta ,ϵ,\nu$	complex parameters.

…

25: 33.17 Recurrence Relations and Derivatives

… ►

33.17.1 $$(\mathrm{\ell }+1)rf(ϵ,\mathrm{\ell }-1;r)-(2\mathrm{\ell }+1)\left(\mathrm{\ell }(\mathrm{\ell }+1)-r\right)f(ϵ,\mathrm{\ell };r)+\mathrm{\ell }\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r)=0,$$

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

►

33.17.2 $$(\mathrm{\ell }+1)\left(1+{\mathrm{\ell }}^2ϵ\right)rh(ϵ,\mathrm{\ell }-1;r)-(2\mathrm{\ell }+1)\left(\mathrm{\ell }(\mathrm{\ell }+1)-r\right)h(ϵ,\mathrm{\ell };r)+\mathrm{\ell }rh(ϵ,\mathrm{\ell }+1;r)=0,$$

ⓘ

Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

►

33.17.3 $$(\mathrm{\ell }+1)r{f}^{\prime }(ϵ,\mathrm{\ell };r)=\left({(\mathrm{\ell }+1)}^2-r\right)f(ϵ,\mathrm{\ell };r)-\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r),$$

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

►

33.17.4 $$(\mathrm{\ell }+1)r{h}^{\prime }(ϵ,\mathrm{\ell };r)=\left({(\mathrm{\ell }+1)}^2-r\right)h(ϵ,\mathrm{\ell };r)-rh(ϵ,\mathrm{\ell }+1;r).$$

ⓘ

Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

26: 13.10 Integrals

… ►

13.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}𝐌(a,b,t)dt=z^{-b}{\left(1-\frac{1}{z}\right)}^{-a},$$ $\mathrm{\Re }b>0$, $\mathrm{\Re }z>1$,

ⓘ

Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

►

13.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{b-1}𝐌(a,c,t)dt=\frac{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a-b\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)},$$ $\mathrm{\Re }\left(c-a\right)>\mathrm{\Re }b>0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Permalink:

http://dlmf.nist.gov/13.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

… ►

13.10.10 $${\int }_0^{\mathrm{\infty }}{t}^{\lambda -1}𝐌(a,b,-t)dt=\frac{\mathrm{\Gamma }\left(\lambda \right)\mathrm{\Gamma }\left(a-\lambda \right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b-\lambda \right)},$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(iii), §13.10 and Ch.13

… ►

13.10.13 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{b-1-\frac{1}{2}\nu }𝐌(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=x^{-a+\frac{1}{2}\nu }{\mathrm{e}}^{-x}𝐌(\nu -b+1,\nu -a+1,x),$$ $x>0$, , $\mathrm{\Re }b>0$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(v), §13.10 and Ch.13

… ►

13.10.15 $${\int }_0^{\mathrm{\infty }}{t}^{\frac{1}{2}\nu }U(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{\mathrm{\Gamma }\left(\nu -b+2\right)}{\mathrm{\Gamma }\left(a\right)}{x}^{\frac{1}{2}\nu }U(\nu -b+2,\nu -a+2,x),$$ $x>0$, ,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(v), §13.10 and Ch.13

…

27: 9.14 Incomplete Airy Functions

… ►Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …

28: 31.12 Confluent Forms of Heun’s Equation

… ►

31.12.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+ϵ\right)\frac{dw}{dz}+\frac{\alpha z-q}{z(z-1)}w=0.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

§31.12

Permalink:

http://dlmf.nist.gov/31.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

… ►

31.12.2 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\delta }{z^2}+\frac{\gamma }{z}+1\right)\frac{dw}{dz}+\frac{\alpha z-q}{z^2}w=0.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

… ►

31.12.3 $$\frac{d^{2}w}{{dz}^2}-\left(\frac{\gamma }{z}+\delta +z\right)\frac{dw}{dz}+\frac{\alpha z-q}{z}w=0.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

Erratum (V1.0.7) for Equation (31.12.3)

Permalink:

http://dlmf.nist.gov/31.12.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.7):

Originally the sign in front of the second term in this equation was $+$. The correct sign is $-$.

Reported 2013-10-31 by Henryk Witek

See also:

Annotations for §31.12, §31.12 and Ch.31

… ►

31.12.4 $$\frac{d^{2}w}{{dz}^2}+\left(\gamma +z\right)z\frac{dw}{dz}+\left(\alpha z-q\right)w=0.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

§31.12

Permalink:

http://dlmf.nist.gov/31.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

…

29: 15.5 Derivatives and Contiguous Functions

… ►

15.5.1 $$\frac{d}{dz}F(a,b;c;z)=\frac{ab}{c}F(a+1,b+1;c+1;z),$$

ⓘ

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.E1

Encodings:

pMML, png, TeX

See also:

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

►

15.5.2 $$\frac{d^n}{{dz}^n}F(a,b;c;z)=\frac{{\left(a\right)}_n{\left(b\right)}_n}{{\left(c\right)}_n}F(a+n,b+n;c+n;z).$$

ⓘ

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

Permalink:

http://dlmf.nist.gov/15.5.E2

Encodings:

pMML, png, TeX

See also:

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

►

15.5.3 $${\left(z\frac{d}{dz}z\right)}^n\left(z^{a-1}F(a,b;c;z)\right)={\left(a\right)}_n{z}^{a+n-1}F(a+n,b;c;z).$$

ⓘ

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.E3

Encodings:

pMML, png, TeX

See also:

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

►

15.5.4 $$\frac{d^n}{{dz}^n}\left(z^{c-1}F(a,b;c;z)\right)={\left(c-n\right)}_n{z}^{c-n-1}F(a,b;c-n;z).$$

ⓘ

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.E4

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

…

30: 7.25 Software

… ►

§7.25(ii) $\mathrm{erf}x$, $\mathrm{erfc}x$, ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $x\in ℝ$

… ►

§7.25(iv) $C\left(x\right)$, $S\left(x\right)$, $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in ℝ$

… ►

§7.25(vi) $ℱ\left(x\right)$, $G\left(x\right)$, $𝖴(x,t)$, $𝖵(x,t)$, $x\in ℝ$

…