# incomplete integrals

(0.006 seconds)

## 1—10 of 78 matching pages

##### 1: 19.38 Approximations
Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …
##### 2: 19.1 Special Notation
All derivatives are denoted by differentials, not by primes. … of the first, second, and third kinds, respectively, and Legendre’s incomplete integralsThe first three functions are incomplete integrals of the first, second, and third kinds, and the $\operatorname{cel}$ function includes complete integrals of all three kinds.
##### 3: 6.11 Relations to Other Functions
###### Incomplete Gamma Function
6.11.1 $E_{1}\left(z\right)=\Gamma\left(0,z\right).$
##### 4: 19.15 Advantages of Symmetry
Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)). …
##### 5: 19.9 Inequalities
###### §19.9(ii) IncompleteIntegrals
Simple inequalities for incomplete integrals follow directly from the defining integrals19.2(ii)) together with (19.6.12):
19.9.11 $\phi\leq F\left(\phi,k\right)\leq\min(\phi/\Delta,{\operatorname{gd}^{-1}}% \left(\phi\right)),$
19.9.12 $\max(\sin\phi,\phi\Delta)\leq E\left(\phi,k\right)\leq\phi,$
19.9.17 $L\leq F\left(\phi,k\right)\leq\sqrt{UL}\leq\tfrac{1}{2}(U+L)\leq U,$
##### 6: 19.13 Integrals of Elliptic Integrals
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). …
##### 7: 8.14 Integrals
###### §8.14 Integrals
8.14.1 $\int_{0}^{\infty}e^{-ax}\frac{\gamma\left(b,x\right)}{\Gamma\left(b\right)}\,% \mathrm{d}x=\frac{(1+a)^{-b}}{a},$ $\Re a>0$, $\Re b>-1$,
8.14.2 $\int_{0}^{\infty}e^{-ax}\Gamma\left(b,x\right)\,\mathrm{d}x=\Gamma\left(b% \right)\frac{1-(1+a)^{-b}}{a},$ $\Re a>-1$, $\Re b>-1$.
8.14.3 $\int_{0}^{\infty}x^{a-1}\gamma\left(b,x\right)\,\mathrm{d}x=-\frac{\Gamma\left% (a+b\right)}{a},$ $\Re a<0$, $\Re\left(a+b\right)>0$,
8.14.4 $\int_{0}^{\infty}x^{a-1}\Gamma\left(b,x\right)\,\mathrm{d}x=\frac{\Gamma\left(% a+b\right)}{a},$ $\Re a>0$, $\Re\left(a+b\right)>0$,
##### 9: 8.6 Integral Representations
###### §8.6(iii) Compendia
For collections of integral representations of $\gamma\left(a,z\right)$ and $\Gamma\left(a,z\right)$ see Erdélyi et al. (1953b, §9.3), Oberhettinger (1972, pp. 68–69), Oberhettinger and Badii (1973, pp. 309–312), Prudnikov et al. (1992b, §3.10), and Temme (1996b, pp. 282–283).