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In quantum field theory, a sum rule is a relation between a static quantity and an integral over a dynamical quantity. Therefore, they have a form such as: =where () is the dynamical quantity, for example a structure function characterizing a particle, and is the static quantity, for example the mass or the charge of that particle.
The technique of the previous example may also be applied to other Dirichlet series. If a n = μ ( n ) {\displaystyle a_{n}=\mu (n)} is the Möbius function and ϕ ( x ) = x − s {\displaystyle \phi (x)=x^{-s}} , then A ( x ) = M ( x ) = ∑ n ≤ x μ ( n ) {\displaystyle A(x)=M(x)=\sum _{n\leq x}\mu (n)} is Mertens function and
Instead, such a series must be interpreted by zeta function regularization. For this reason, Hardy recommends "great caution" when applying the Ramanujan sums of known series to find the sums of related series. [16]
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series.Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. [1] As an axiom, it offers a non-perturbative approach to quantum field theory. One example is the vertex operator algebra, which has been used to construct two-dimensional conformal field theories ...
In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σa n, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original ...
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum.It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus.