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.

Description

If the Cauchy principal value exists, then it may be differentiated with respect to

are used here to denote Cauchy principal value and Hadamard finite-part integrals respectively.

The Hadamard finite part integral above (for

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.

Example

Its Cauchy principal value Note that this value does not represent the area under the curve

terms, that which remains, the finite part

which equals the value derived above.

References

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