Abel transform

The Abelian integral equation is a special Volterra integral equation of first kind has the form:

Is being given and the unknown function is. The Volterra integral equation of first kind is more general than

Defined using a kernel function. Especially for core functions of the form

With, there is a general solution method by returning to the formula for the Euler beta function. It follows that:

Is In the Abelian integral equation.

The relationship expressed by the generalization of the Abelian integral equation for the relationship between functions and is also known as Abel transformation, ie is the Abel transform of. The information provided by the above-mentioned method for solving formula yields the inversion formula for the Abel transformation.

Application and History

Niels Henrik Abel examined in 1823 as one of the first integral equations, in conjunction with a mechanical problem. Until then, the mechanism was mainly determined by differential equations. Abel looked at a body moving under the influence of gravity along a located in a vertical plane curve according to ( 0.0 ).

Starting from the classical formula for speed

One is by integration over the distance the drop time

By the substitution to the final form:

Knowing the curve f ( y), one obtains the fall time. Abel also considered the inverse problem: the fall time is given, we obtain an Abelian integral equation for the unknown function f ( y).

Further applications of the Abelian integral equation or the Abel transformation there is in astrophysics, geophysics ( Herglotz -Wiechert method of determining the velocity distribution of arrival times of seismic waves ) and for example in the determination of the atmospheric data of planets by radio - occultation. As in the original application are the typical inverse problems.