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 Hadamard (1923, book III, chapter I, 1932). Riesz (1938, 1949) showed that this can be interpreted as taking the meromorphic continuation of a convergent integral.
If the Cauchy principal value integral 
formula_1 and formula_2 are used here to denote Cauchy principal value and Hadamard finite-part integrals respectively. 
The Hadamard finite part integral above (for "a" ; "x" ; "b") may also be given by the following equivalent definitions:
formula_3
formula_4 
The definitions above may be derived by assuming that the function "f" ("t") is differentiable infinitely many times at "t" 
 "x" for "a" ; "x" ; "b", that is, by assuming that "f" ("t") can be represented by its Taylor series about "t" 
 "x". For details, see Ang (2013). (Note that the term − ( − ) in the second equivalent definition above is missing in Ang (2013) but this is corrected in the errata sheet of the book.)
Integral equations containing Hadamard finite part integrals (with "f" ("t") unknown) are termed hypersingular integral equations. Hypersingular integral equations arise in the formulation of many problems in mechanics, such as in fracture analysis.
Example.
Consider the divergent integral
formula_5
Its Cauchy principal value also diverges since
formula_6
To assign a finite value to this divergent integral, we may consider
formula_7
The inner Cauchy principal value is given by
formula_8
Therefore,
formula_9
Note that this value does not represent the area under the curve "y"("t") 
 1/"t"2, 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
formula_10
If one removes the infinite components, the pair of formula_11 terms, that which remains is
formula_12
which equals the value derived above.