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Terms with infinitely many sine factors would necessarily be equal to zero. When only finitely many of the angles are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.
Many texts write φ = tan −1 y / x instead of φ = atan2(y, x), but the first equation needs adjustment when x ≤ 0. This is because for any real x and y , not both zero, the angles of the vectors ( x , y ) and (− x , − y ) differ by π radians, but have the identical value of tan φ = y / x .
The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root. It is closely related to Weierstrass factorization theorem , which does not restrict to entire functions with finite orders.
In fact, although Gauss also conjectured that there are infinitely many primes such that the ring of integers of () is a PID, it is not yet known whether there are infinitely many number fields (of arbitrary degree) such that is a PID. On the other hand, the ring of integers in a number field is always a Dedekind domain.
The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ(s) has a simple zero at each even negative integer s = −2n, known as the trivial zeros of ζ(s).
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 proviso 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.
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which generate infinitely many analogous formulas for when . Some formulas relating π and harmonic numbers are given here . Further infinite series involving π are: [ 15 ]