functional equation

A simple form of functional equation is a recurrence relation.

In fact, this is no longer a recurrence relation but a functional equation.

This scaling function itself is solution to a functional equation.

The main method of solving elementary functional equations is substitution.

The approximate functional equation gives an estimate for the size of the error term.

The idea can even be generalised to situations where the variables don't refer to numbers at all, as in a functional equation.

This yields the functional equation on the generating function :

This in turn leads to remarkable functional equations satisfied by f(z).

These functional equations are satisfied by the gamma function.

When the domain is the real numbers, this is Cauchy's functional equation.

Solving functional equations can be very difficult but there are some common methods of solving them.

These relations may be derived by employing the functional equation together with Hurwitz's formula, given above.

The analytic continuation and functional equation then boil down to those of the Eisenstein series.

Euler's reflection formula is an important functional equation for the gamma function.

He contributed to the research areas of functional equations, iterative sequences and summability theory.

The resulting L-functions satisfy a number of analytic properties, including an important functional equation.

He did research on functional equations and on such practical problems as the filtration of groundwater.

Therefore use of the functional equation is basic, in order to study the zeta-function in the whole complex plane.

The functional equation also says all zeros (except the "obvious" ones) must be in the critical strip: real part is between 0 and 1.

If we use integration by parts, we see that the functional equation holds true for the gamma function.

The functional equation is the mathematical expression of the universality of period doubling.

It is straightforward to show that the Euler definition satisfies the functional equation (1) above.

Hypertranscendental functions usually arise as the solutions to functional equations, for example the Gamma function.

Jaynes cites Abel for the first known use of the associativity functional equation.

A property of the Riemann zeta function is its functional equation: