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In particular, in these two identities an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero. When only finitely many of the angles are ...
The same criterion applies to products of arbitrary complex numbers (including negative reals) if the logarithm is understood as a fixed branch of logarithm which satisfies =, with the provision that the infinite product diverges when infinitely many a n fall outside the domain of , whereas finitely many such a n can be ignored in the sum.
Euler's formula states that, for any real number x, one has = + , where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x ("cosine plus i sine").
A uniformly recurrent word is a recurrent word in which for any given factor X in the sequence, there is some length n X (often much longer than the length of X) such that X appears in every block of length n X. [1] [6] [7] The terms minimal sequence [8] and almost periodic sequence (Muchnik, Semenov, Ushakov 2003) are also used.
The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory , on which most of modern mathematics can be developed, is the axiom of infinity , which guarantees the existence of infinite ...
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proven by Euclid in his work Elements . There are several proofs of the theorem.
One can also speak of "almost all" integers having a property to mean "all except finitely many", despite the integers not admitting a measure for which this agrees with the previous usage. For example, "almost all prime numbers are odd". There is a more complicated meaning for integers as well, discussed in the main article.
Actual infinity is completed and definite, and consists of infinitely many elements. Potential infinity is never complete: elements can be always added, but never infinitely many. "For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always ...