In mathematics, Hadamard regularization (also called Hadamard finite part or Hadamard's partie finie) is a method of regularizing divergent integrals by dropping some divergent terms and keeping the finite part, introduced by . showed that this can be interpreted as taking the meromorphic continuation of a convergent integral.
If the Cauchy principal value integral
exists, then it may be differentiated with respect to to obtain the Hadamard finite part integral as follows:
Note that the symbols and are used here to denote Cauchy principal value and Hadamard finite-part integrals respectively.
The Hadamard finite part integral above (for ) may also be given by the following equivalent definitions:
The definitions above may be derived by assuming that the function is differentiable infinitely many times at , that is, by assuming that can be represented by its Taylor series about . For details, see . (Note that the term in the second equivalent definition above is missing in but this is corrected in the errata sheet of the book.)
Integral equations containing Hadamard finite part integrals (with unknown) are termed hypersingular integral equations. Hypersingular integral equations arise in the formulation of many problems in mechanics, such as in fracture analysis.
Consider the divergent integral
Its Cauchy principal value also diverges since
To assign a finite value to this divergent integral, we may consider
The inner Cauchy principal value is given by
Therefore,
Note that this value does not represent the area under the curve , which is clearly always positive. However, it can be seen where this comes from. Recall the Cauchy principal value of this integral, when evaluated at the endpoints, took the form
If one removes the infinite components, the pair of terms, that which remains, the finite part, is
which equals the value derived above.