indefinite

SearchAdvancedHelp

(0.001 seconds)

1—10 of 18 matching pages

1: 19.13 Integrals of Elliptic Integrals

… ►For definite and indefinite integrals of complete elliptic integrals see Byrd and Friedman (1971, pp. 610–612, 615), Prudnikov et al. (1990, §§1.11, 2.16), Glasser (1976), Bushell (1987), and Cvijović and Klinowski (1999). ►For definite and indefinite integrals of incomplete elliptic integrals see Byrd and Friedman (1971, pp. 613, 616), Prudnikov et al. (1990, §§1.10.2, 2.15.2), and Cvijović and Klinowski (1994). …

2: 12.15 Generalized Parabolic Cylinder Functions

… ►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …

3: 22.14 Integrals

… ►

§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions

… ►

§22.14(ii) Indefinite Integrals of Powers of Jacobian Elliptic Functions

… ►The indefinite integral of the 3rd power of a Jacobian function can be expressed as an elementary function of Jacobian functions and a product of Jacobian functions. … ►

§22.14(iii) Other Indefinite Integrals

… ► …

4: 4.40 Integrals

… ►

§4.40(ii) Indefinite Integrals

… ►Extensive compendia of indefinite and definite integrals of hyperbolic functions include Apelblat (1983, pp. 96–109), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 139–160), Gröbner and Hofreiter (1950, pp. 160–167), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.4, 1.8, 2.4, 2.8).

5: 10.71 Integrals

… ►

§10.71(i) Indefinite Integrals

…

6: 11.7 Integrals and Sums

… ►

§11.7(i) Indefinite Integrals

… ►

11.7.6 $$f_{\nu +1}(z)=(2\nu +1)f_{\nu }(z)-z^{\nu +1}{𝐇}_{\nu }\left(z\right)+\frac{{(\frac{1}{2}{z}^2)}^{\nu +1}}{(\nu +1)\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)},$$ $\mathrm{\Re }\nu >-1$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: real or complex order

A&S Ref:

12.1.28 (The condition on $\nu$ has been corrected.)

Permalink:

http://dlmf.nist.gov/11.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(i), §11.7 and Ch.11

…

7: 4.26 Integrals

… ►

§4.26(ii) Indefinite Integrals

… ►Extensive compendia of indefinite and definite integrals of trigonometric and inverse trigonometric functions include Apelblat (1983, pp. 48–109), Bierens de Haan (1939), Gradshteyn and Ryzhik (2000, Chapters 2–4), Gröbner and Hofreiter (1949, pp. 116–139), Gröbner and Hofreiter (1950, pp. 94–160), and Prudnikov et al. (1986a, §§1.5, 1.7, 2.5, 2.7).

8: 9.10 Integrals

… ►

§9.10(i) Indefinite Integrals

… ►

§9.10(iii) Other Indefinite Integrals

…

9: 13.10 Integrals

… ►

§13.10(i) Indefinite Integrals

…

10: 9.11 Products

… ►

§9.11(iv) Indefinite Integrals

… ►

9.11.19 $${\int }_0^{\mathrm{\infty }}\frac{dt}{{\mathrm{Ai}}^2\left(t\right)+{\mathrm{Bi}}^2\left(t\right)}={\int }_0^{\mathrm{\infty }}\frac{tdt}{\mathrm{Ai}_{}^{\prime }{}_{}{}^2\left(t\right)+\mathrm{Bi}_{}^{\prime }{}_{}{}^2\left(t\right)}=\frac{{\pi }^2}{6}.$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $M\left(z\right)$: Airy modulus function, $N\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: Airy phase function, $\theta \left(z\right)$: Airy phase function and $x$: real variable

Proof sketch:

Extend the definitions of §9.8(i) to positive values of $x$, obtain the indefinite integrals of $1/M^2\left(x\right)$ and $x/N^2\left(x\right)$ via the first two of (9.8.14), then combine the values of $\theta \left(0\right)$ and $\varphi \left(0\right)$ given in §9.8(i) with $\theta \left(+\mathrm{\infty }\right)=\varphi \left(+\mathrm{\infty }\right)=0$ obtained from (9.8.4), (9.8.8), and §9.7(ii). (Communicated by M.E. Muldoon.)

Permalink:

http://dlmf.nist.gov/9.11.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(v), §9.11 and Ch.9

…