Euler beta integral

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31: 19.34 Mutual Inductance of Coaxial Circles

§19.34 Mutual Inductance of Coaxial Circles

… ►Application of (19.29.4) and (19.29.7) with $\alpha =1$, $a_{\beta }+b_{\beta }t=1-t$, $\delta =3$, and $a_{\gamma }+b_{\gamma }t=1$ yields ►

19.34.5 $$\frac{3c^2}{8\pi ab}M=3R_F(0,r_{+}^2,r_{-}^2)-2r_{-}^2{R}_D(0,r_{+}^2,r_{-}^2),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$

Permalink:

http://dlmf.nist.gov/19.34.E5

Encodings:

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See also:

Annotations for §19.34 and Ch.19

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19.34.6 $$\frac{c^2}{2\pi }M=(r_{+}^2+r_{-}^2)R_F(0,r_{+}^2,r_{-}^2)-4R_G(0,r_{+}^2,r_{-}^2).$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$

Referenced by:

§19.34

Permalink:

http://dlmf.nist.gov/19.34.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

… ►References for other inductance problems solvable in terms of elliptic integrals are given in Grover (1946, pp. 8 and 283).

32: 19.26 Addition Theorems

§19.26 Addition Theorems

… ►where …where for $\gamma =\alpha ,\beta ,\sigma$, except that ${\sigma }^2-\theta$ can be 0, and … ►

§19.26(iii) Duplication Formulas

… ►where …

33: 18.37 Classical OP’s in Two or More Variables

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18.37.8 $m\ne j$ and/or $n\ne \mathrm{\ell }$.

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Symbols:

$dx$: differential of $x$ , $P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)$: triangle polynomial, $y$: real variable, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.37.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(ii), §18.37(ii), §18.37 and Ch.18

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34: 31.9 Orthogonality

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31.9.1 $$w_m(z)=(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$

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Symbols:

$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.9(i), §31.9 and Ch.31

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31.9.2 $${\int }_{\zeta }^{(1+,0+,1-,0-)}{t}^{\gamma -1}{(1-t)}^{\delta -1}{(t-a)}^{ϵ-1}{w}_m(t)w_k(t)dt={\delta }_{m,k}{\theta }_m.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\zeta$: point and ${\theta }_m$: normalization constant

Permalink:

http://dlmf.nist.gov/31.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.9(i), §31.9 and Ch.31

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$$f_0(q_m,z)=H\mathrm{\ell }(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$

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$$f_1(q_m,z)=H\mathrm{\ell }(1-a,\alpha \beta -q_m;\alpha ,\beta ,\delta ,\gamma ;1-z),$$

… ►

31.9.6 $$\rho (s,t)=(s-t){(st)}^{\gamma -1}{\left((s-1)(t-1)\right)}^{\delta -1}{\left((s-a)(t-a)\right)}^{ϵ-1},$$

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Defines:

$\rho (s,t)$: weight function (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter and $a$: complex parameter

Permalink:

http://dlmf.nist.gov/31.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.9(ii), §31.9 and Ch.31

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35: 32.10 Special Function Solutions

… ►The solution (32.10.34) is an essentially transcendental function of both constants of integration since ${\text{P}}_{\text{VI}}$ with $\alpha =\beta =\gamma =0$ and $\delta =\frac{1}{2}$ does not admit an algebraic first integral of the form $P(z,w,w^{\prime },C)=0$, with $C$ a constant. …

36: 5.21 Methods of Computation

… ►An effective way of computing $\mathrm{\Gamma }\left(z\right)$ in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). … ►Similarly for $\ln \mathrm{\Gamma }\left(z\right)$, $\psi \left(z\right)$, and the polygamma functions. ►Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. … ►For the computation of the $q$-gamma and $q$-beta functions see Gabutti and Allasia (2008).

37: 36.6 Scaling Relations

§36.6 Scaling Relations

… ►

${\mathrm{\Psi }}_K(𝐱;k)=k^{{\beta }_K}{\mathrm{\Psi }}_K\left(𝐲(k)\right),$

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${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)=k^{{\beta }^{(\mathrm{U})}}{\mathrm{\Psi }}^{(\mathrm{U})}\left({𝐲}^{(\mathrm{U})}(k)\right),$

… ►

Indices for $k$-Scaling of Magnitude of ${\mathrm{\Psi }}_K$ or ${\mathrm{\Psi }}^{(\mathrm{U})}$ (Singularity Index)

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$\text{umbilics: }{\beta }^{(\mathrm{U})}={\displaystyle \frac{1}{3}}.$

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38: 15.9 Relations to Other Functions

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15.9.13 $$f(t)=\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\tilde{f}(\mathrm{i}\lambda ){\mathrm{\Phi }}_{\mathrm{i}\lambda }^{(\alpha ,\beta )}(t)\frac{\mathrm{\Gamma }\left(\frac{1}{2}(\alpha +\beta +1+\lambda )\right)\mathrm{\Gamma }\left(\frac{1}{2}(\alpha -\beta +1+\lambda )\right)}{\mathrm{\Gamma }\left(\alpha +1\right)\mathrm{\Gamma }\left(\lambda \right)2^{\alpha +\beta +1-\lambda }}d\lambda ,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $(a,b)$: open interval, $f(t)$: function and ${\mathrm{\Phi }}_{\lambda }^{(\alpha ,\beta )}(t)$: function

Permalink:

http://dlmf.nist.gov/15.9.E13

Encodings:

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See also:

Annotations for §15.9(ii), §15.9 and Ch.15

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39: 2.6 Distributional Methods

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2.6.4 $${\int }_0^{\mathrm{\infty }}\frac{t^{\alpha -1}}{{(x+t)}^{\alpha +\beta }}dt=\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }\left(\alpha +\beta \right)}\frac{1}{x^{\beta }},$$ $\mathrm{\Re }\alpha >0$, $\mathrm{\Re }\beta >0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Referenced by:

§2.6(i)

Permalink:

http://dlmf.nist.gov/2.6.E4

Encodings:

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See also:

Annotations for §2.6(i), §2.6 and Ch.2

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40: Guide to Searching the DLMF

… ►

Table 1: Query Examples

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Query	Matching records contain
…
int sin	the integral of the sin function
int_$^$ sin	any definite integral of sin
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Euler	the word ”Euler” or any of the various Euler terms such as Euler Gamma function $\mathrm{\Gamma }$, Euler Beta function $\mathrm{B}$, etc.
…
Gamma near/5 =	the terms $\mathrm{\Gamma }$ and = such that $\mathrm{\Gamma }$ is up to 5 terms before or after =.
…

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… ►

Table 2: Wildcard Examples

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Query	What it stands for
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int_$^$ sin	any definite integral of sin.

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