# multiplication formula

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##### 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..
##### 9: 27.5 Inversion Formulas
###### §27.5 Inversion Formulas
The multiplicative functions are a subgroup of this group. …which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: … Special cases of Möbius inversion pairs are: … Other types of Möbius inversion formulas include: …
##### 10: 17.6 ${{}_{2}\phi_{1}}$ Function
17.6.1 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,\ifrac{c}{(ab)}\right)=\frac{\left(c/a,c/% b;q\right)_{\infty}}{\left(c,c/(ab);q\right)_{\infty}}.$
Related formulas are (17.7.3), (17.8.8) and … For similar formulas see Verma and Jain (1983). …
17.6.13 ${{}_{2}\phi_{1}}\left(a,b;c;q,q\right)+\frac{\left(q/c,a,b;q\right)_{\infty}}{% \left(c/q,aq/c,bq/c;q\right)_{\infty}}{{}_{2}\phi_{1}}\left(aq/c,bq/c;q^{2}/c;% q,q\right)=\frac{\left(q/c,abq/c;q\right)_{\infty}}{\left(aq/c,bq/c;q\right)_{% \infty}},$
where $|z|<1$, $|\operatorname{ph}\left(-z\right)|<\pi$, and the contour of integration separates the poles of $\left(q^{1+\zeta},cq^{\zeta};q\right)_{\infty}/\sin\left(\pi\zeta\right)$ from those of $1/\left(aq^{\zeta},bq^{\zeta};q\right)_{\infty}$, and the infimum of the distances of the poles from the contour is positive. …