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
C
∫
a
b
f
(
t
)
t
−
x
d
t
(
for
a
<
x
<
b
)
{\displaystyle {\mathcal {C}\int _{a}^{b}{\frac {f(t)}{t-x}\,dt\quad ({\text{for }a<x<b)}
exists, then it may be differentiated with respect to
x to obtain the Hadamard finite part integral as follows:
d
d
x
(
C
∫
a
b
f
(
t
)
t
−
x
d
t
)
=
H
∫
a
b
f
(
t
)
(
t
−
x
)
2
d
t
(
for
a
<
x
<
b
)
.
{\displaystyle {\frac {d}{dx}\left({\mathcal {C}\int _{a}^{b}{\frac {f(t)}{t-x}\,dt\right)={\mathcal {H}\int _{a}^{b}{\frac {f(t)}{(t-x)^{2}\,dt\quad ({\text{for }a<x<b).}
Note that the symbols
C
{\displaystyle {\mathcal {C}
H
{\displaystyle {\mathcal {H}
The Hadamard finite part integral above (for a < x < b
H
∫
a
b
f
(
t
)
(
t
−
x
)
2
d
t
=
lim
ε
→
0
+
{
∫
a
x
−
ε
f
(
t
)
(
t
−
x
)
2
d
t
+
∫
x
+
ε
b
f
(
t
)
(
t
−
x
)
2
d
t
−
f
(
x
+
ε
)
+
f
(
x
−
ε
)
ε
}
,
{\displaystyle {\mathcal {H}\int _{a}^{b}{\frac {f(t)}{(t-x)^{2}\,dt=\lim _{\varepsilon \to 0^{+}\left\{\int _{a}^{x-\varepsilon }{\frac {f(t)}{(t-x)^{2}\,dt+\int _{x+\varepsilon }^{b}{\frac {f(t)}{(t-x)^{2}\,dt-{\frac {f(x+\varepsilon )+f(x-\varepsilon )}{\varepsilon }\right\},}
H
∫
a
b
f
(
t
)
(
t
−
x
)
2
d
t
=
lim
ε
→
0
+
{
∫
a
b
(
t
−
x
)
2
f
(
t
)
(
(
t
−
x
)
2
+
ε
2
)
2
d
t
−
π
f
(
x
)
2
ε
−
f
(
x
)
2
(
1
b
−
x
−
1
a
−
x
)
}
.
{\displaystyle {\mathcal {H}\int _{a}^{b}{\frac {f(t)}{(t-x)^{2}\,dt=\lim _{\varepsilon \to 0^{+}\left\{\int _{a}^{b}{\frac {(t-x)^{2}f(t)}{((t-x)^{2}+\varepsilon ^{2})^{2}\,dt-{\frac {\pi f(x)}{2\varepsilon }-{\frac {f(x)}{2}\left({\frac {1}{b-x}-{\frac {1}{a-x}\right)\right\}.}
The definitions above may be derived by assuming that the function f (t )t = x for a < x < b f (t )t = x 2013 ). (Note that the term − f (x )/ 2 1 / b − x 1 / a − x 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 )
Example Consider the divergent integral
∫
−
1
1
1
t
2
d
t
=
(
lim
a
→
0
−
∫
−
1
a
1
t
2
d
t
)
+
(
lim
b
→
0
+
∫
b
1
1
t
2
d
t
)
=
lim
a
→
0
−
(
−
1
a
−
1
)
+
lim
b
→
0
+
(
−
1
+
1
b
)
=
+
∞
{\displaystyle \int _{-1}^{1}{\frac {1}{t^{2}\,dt=\left(\lim _{a\to 0^{-}\int _{-1}^{a}{\frac {1}{t^{2}\,dt\right)+\left(\lim _{b\to 0^{+}\int _{b}^{1}{\frac {1}{t^{2}\,dt\right)=\lim _{a\to 0^{-}\left(-{\frac {1}{a}-1\right)+\lim _{b\to 0^{+}\left(-1+{\frac {1}{b}\right)=+\infty }
Its
Cauchy principal value also diverges since
C
∫
−
1
1
1
t
2
d
t
=
lim
ε
→
0
+
(
∫
−
1
−
ε
1
t
2
d
t
+
∫
ε
1
1
t
2
d
t
)
=
lim
ε
→
0
+
(
1
ε
−
1
−
1
+
1
ε
)
=
+
∞
{\displaystyle {\mathcal {C}\int _{-1}^{1}{\frac {1}{t^{2}\,dt=\lim _{\varepsilon \to 0^{+}\left(\int _{-1}^{-\varepsilon }{\frac {1}{t^{2}\,dt+\int _{\varepsilon }^{1}{\frac {1}{t^{2}\,dt\right)=\lim _{\varepsilon \to 0^{+}\left({\frac {1}{\varepsilon }-1-1+{\frac {1}{\varepsilon }\right)=+\infty }
To assign a finite value to this divergent integral, we may consider
H
∫
−
1
1
1
t
2
d
t
=
H
∫
−
1
1
1
(
t
−
x
)
2
d
t
|
x
=
0
=
d
d
x
(
C
∫
−
1
1
1
t
−
x
d
t
)
|
x
=
0
{\displaystyle {\mathcal {H}\int _{-1}^{1}{\frac {1}{t^{2}\,dt={\mathcal {H}\int _{-1}^{1}{\frac {1}{(t-x)^{2}\,dt{\Bigg |}_{x=0}={\frac {d}{dx}\left({\mathcal {C}\int _{-1}^{1}{\frac {1}{t-x}\,dt\right){\Bigg |}_{x=0}
The inner Cauchy principal value is given by
C
∫
−
1
1
1
t
−
x
d
t
=
lim
ε
→
0
+
(
∫
−
1
−
ε
1
t
−
x
d
t
+
∫
ε
1
1
t
−
x
d
t
)
=
lim
ε
→
0
+
(
ln
|
ε
+
x
1
+
x
|
+
ln
|
1
−
x
ε
−
x
|
)
=
ln
|
1
−
x
1
+
x
|
{\displaystyle {\mathcal {C}\int _{-1}^{1}{\frac {1}{t-x}\,dt=\lim _{\varepsilon \to 0^{+}\left(\int _{-1}^{-\varepsilon }{\frac {1}{t-x}\,dt+\int _{\varepsilon }^{1}{\frac {1}{t-x}\,dt\right)=\lim _{\varepsilon \to 0^{+}\left(\ln \left|{\frac {\varepsilon +x}{1+x}\right|+\ln \left|{\frac {1-x}{\varepsilon -x}\right|\right)=\ln \left|{\frac {1-x}{1+x}\right|}
Therefore,
H
∫
−
1
1
1
t
2
d
t
=
d
d
x
(
ln
|
1
−
x
1
+
x
|
)
|
x
=
0
=
2
x
2
−
1
|
x
=
0
=
−
2
{\displaystyle {\mathcal {H}\int _{-1}^{1}{\frac {1}{t^{2}\,dt={\frac {d}{dx}\left(\ln \left|{\frac {1-x}{1+x}\right|\right){\Bigg |}_{x=0}={\frac {2}{x^{2}-1}{\Bigg |}_{x=0}=-2}
Note that this value does not represent the area under the curve
y (t ) = 1/t 2 , which is clearly always positive.
References Ang, Whye-Teong (2013), Hypersingular Integral Equations in Fracture Analysis Woodhead Publishing , pp. 19–24, ISBN 978-0-85709-479-7 Ang, Whye-Teong, Errata Sheet for Hypersingular Integral Equations in Fracture Analysis (PDF) Blanchet, Luc; Faye, Guillaume (2000), "Hadamard regularization", Journal of Mathematical Physics 41 (11): 7675–7714, arXiv :gr-qc/0004008 Bibcode :2000JMP....41.7675B , doi :10.1063/1.1308506 , ISSN 0022-2488 , MR 1788597 , Zbl 0986.46024 Hadamard, Jacques (1923), Lectures on Cauchy's problem in linear partial differential equations ISBN 978-0-486-49549-1 JFM 49.0725.04 , MR 0051411 , Zbl 0049.34805 Hadamard, J. (1932), Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques (in French), Paris: Hermann & Cie., p. 542, Zbl 0006.20501 Riesz, Marcel (1938), "Intégrales de Riemann-Liouville et potentiels." , Acta Litt. Ac Sient. Univ. Hung. Francisco-Josephinae, Sec. Sci. Math. (Szeged )9 (1–1): 1–42, JFM 64.0476.03 , Zbl 0018.40704 , archived from the original on 2016-03-05, retrieved 2012-06-22 Riesz, Marcel (1938), "Rectification au travail "Intégrales de Riemann-Liouville et potentiels" , Acta Litt. Ac Sient. Univ. Hung. Francisco-Josephinae, Sec. Sci. Math. (Szeged )9 (2–2): 116–118, JFM 65.1272.03 , Zbl 0020.36402 , archived from the original on 2016-03-04, retrieved 2012-06-22 Riesz, Marcel (1949), "L'intégrale de Riemann-Liouville et le problème de Cauchy", Acta Mathematica 81 : 1–223, doi :10.1007/BF02395016 ISSN 0001-5962 , MR 0030102 , Zbl 0033.27601
