# combined

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## 1—10 of 154 matching pages

##### 1: 20.11 Generalizations and Analogs

###### §20.11(v) Permutation Symmetry

… ►For $m=1,2,3,4$, $n=1,2,3,4$, and $m\ne n$, define twelve*combined theta functions*${\phi}_{m,n}(z,q)$ by ►

##### 2: 27.22 Software

Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below ${10}^{16}$. Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard $p-1$, and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

##### 3: 21.8 Abelian Functions

##### 4: 35.11 Tables

##### 5: 4.44 Other Applications

*Einstein functions*and

*Planck’s radiation function*are elementary combinations of exponentials, or exponentials and logarithms. …

##### 6: 34.10 Zeros

##### 7: 33.18 Limiting Forms for Large $\mathrm{\ell}$

##### 8: 33.21 Asymptotic Approximations for Large $|r|$

When $r\to \pm \mathrm{\infty}$ with $$, Equations (33.16.10)–(33.16.13) are combined with

Corresponding approximations for $s(\u03f5,\mathrm{\ell};r)$ and $c(\u03f5,\mathrm{\ell};r)$ as $r\to \mathrm{\infty}$ can be obtained via (33.16.17), and as $r\to -\mathrm{\infty}$ via (33.16.18).