# multiplication formula

(0.002 seconds)

## 1—10 of 27 matching pages

Next, …
##### 3: 3.8 Nonlinear Equations
###### §3.8(iv) Zeros of Polynomials
has $n$ zeros in $\mathbb{C}$, counting each zero according to its multiplicity. …No explicit general formulas exist when $n\geq 5$. … The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of $q(z)$. …
##### 4: Bibliography O
• F. W. J. Olver (1977a) Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.
• F. W. J. Olver (1977b) Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.
• ##### 5: 27.20 Methods of Computation: Other Number-Theoretic Functions
To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function $p\left(n\right)$ for $n. … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau\left(n\right)$, and the values can be checked by the congruence (27.14.20). …
##### 6: 2.8 Differential Equations with a Parameter
###### §2.8(v) Multiple and Fractional Turning Points
The approach used in preceding subsections for equation (2.8.1) also succeeds when $z_{0}$ is a multiple or fractional turning point. For the former $f(z)$ has a zero of multiplicity $\lambda=2,3,4,\dots$ and $g(z)$ is analytic. …For results, including error bounds, see Olver (1977c). For connection formulas for Liouville–Green approximations across these transition points see Olver (1977b, a, 1978). …
##### 7: Bibliography S
• D. M. Smith (1998) Algorithm 786: Multiple-precision complex arithmetic and functions. ACM Trans. Math. Software 24 (4), pp. 359–367.
• D. M. Smith (1989) Efficient multiple-precision evaluation of elementary functions. Math. Comp. 52 (185), pp. 131–134.
• D. M. Smith (1991) Algorithm 693: A FORTRAN package for floating-point multiple-precision arithmetic. ACM Trans. Math. Software 17 (2), pp. 273–283.
• D. Sornette (1998) Multiplicative processes and power laws. Phys. Rev. E 57 (4), pp. 4811–4813.
• A. H. Stroud (1971) Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, N.J..
• ##### 8: 18.18 Sums
###### §18.18(v) Linearization Formulas
Formula (18.18.27) is known as the Hille–Hardy formula. … Formula (18.18.28) is known as the Mehler formula. …
18.28.26 $\lim_{\lambda\to 0}r_{n}\left(\ifrac{x}{(2\lambda)};\lambda,qa\lambda^{-1},qc% \lambda^{-1},bc^{-1}\lambda\,|\,q\right)=P_{n}\left(x;a,b,c;q\right).$