Such a behavior can be obtained from a theory once its beta function is known.

See the discussion of the beta function below.

(see beta function for more on integrals of this form).

Beta functions are usually computed in some kind of approximation scheme.

If a beta function is positive, the corresponding coupling increases with increasing energy.

The value of the beta function at an interaction point is referred to as beta star.

Which is a beta function, known as Veneziano amplitude.

Jacobi sum, the analogue of the beta function over finite fields.

In the opposite limit of low energies (infrared limit), the beta function is not known.

It was this beta function which was apparently found before full string theory was developed.

Here are some examples of beta functions computed in perturbation theory:

Here B is the beta function.

Each of the remaining integrals is now a particular value of the beta function:

One of the definitions of the Beta function reads:

If the beta functions of a quantum field theory vanish, then the theory is scale-invariant.

In this case, the non-zero beta function tells us that the classical scale-invariance is anomalous.

Another form for this same compound distribution, written more compactly in terms of the beta function, B, is as follows:

The beta functions can be rewritten in terms of gamma functions:

The cumulative distribution function can be written in terms of I, the regularized incomplete beta function.

Chapter one, covering the gamma and beta functions, is highly readable and definitive.

Here denotes the Beta function.

The beta function is typically adjusted to have a local minimum at such points (in order to minimize the beam size and thus the interaction rate).

The Nörlund-Rice integral is a contour integral involving the beta function.

The use of the incomplete beta function has the effect of introducing an exponential cutoff in the upper tail.

In 1973, it was discovered that a theory of interacting colored quarks, called quantum chromodynamics had a negative beta function.