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By a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge. For example, S = 1-1/2+1/3-1/4+1/5+... ...

Let f:D(z_0,r)\{z_0}->C be analytic and bounded on a punctured open disk D(z_0,r), then lim_(z->z_0)f(z) exists, and the function defined by f^~:D(z_0,r)->C f^~(z)={f(z) for ...

The value for zeta(2)=sum_(k=1)^infty1/(k^2) (1) can be found using a number of different techniques (Apostol 1983, Choe 1987, Giesy 1972, Holme 1970, Kimble 1987, Knopp and ...

The Riemann's moduli space gives the solution to Riemann's moduli problem, which requires an analytic parameterization of the compact Riemann surfaces in a fixed ...

A derivative of a complex function, which must satisfy the Cauchy-Riemann equations in order to be complex differentiable.

An Abelian differential is an analytic or meromorphic differential on a compact or closed Riemann surface.

Teichmüller's theorem asserts the existence and uniqueness of the extremal quasiconformal map between two compact Riemann surfaces of the same genus modulo an equivalence ...

The line R[s]=1/2 in the complex plane on which the Riemann hypothesis asserts that all nontrivial (complex) Riemann zeta function zeros lie. The plot above shows the first ...

The nontrivial zeros of the Riemann zeta function correspond to the eigenvalues of some Hermitian operator (Derbyshire 2004, pp. 277-278).

J_(nualphabeta)^mu=J_(nubetaalpha)^mu=1/2(R_(alphanubeta)^mu+R_(betanualpha)^mu), where R is the Riemann tensor.