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A function that can be defined as a Dirichlet series, i.e., is computed as an infinite sum of powers, F(n)=sum_(k=1)^infty[f(k)]^n, where f(k) can be interpreted as the set ...

About Eric Weisstein's World of Mathematics

Calculus II

e^(i(ntheta))=(e^(itheta))^n. (1) From the Euler formula it follows that cos(ntheta)+isin(ntheta)=(costheta+isintheta)^n. (2) A similar identity holds for the hyperbolic ...

There are several q-analogs of the cosine function. The two natural definitions of the q-cosine defined by Koekoek and Swarttouw (1998) are given by cos_q(z) = ...

D_q=1/(1-q)lim_(epsilon->0)(lnI(q,epsilon))/(ln(1/epsilon),) (1) where I(q,epsilon)=sum_(i=1)^Nmu_i^q, (2) epsilon is the box size, and mu_i is the natural measure. The ...

The q-analog of the factorial (by analogy with the q-gamma function). For k an integer, the q-factorial is defined by [k]_q! = faq(k,q) (1) = ...

A q-analog of the gamma function defined by Gamma_q(x)=((q;q)_infty)/((q^x;q)_infty)(1-q)^(1-x), (1) where (x,q)_infty is a q-Pochhammer symbol (Koepf 1998, p. 26; Koekoek ...

The q-analog of integration is given by int_0^1f(x)d(q,x)=(1-q)sum_(i=0)^inftyf(q^i)q^i, (1) which reduces to int_0^1f(x)dx (2) in the case q->1^- (Andrews 1986 p. 10). ...

The q-analog of pi pi_q can be defined by setting a=0 in the q-factorial [a]_q!=1(1+q)(1+q+q^2)...(1+q+...+q^(a-1)) (1) to obtain ...