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}}}
and
H
{\displaystyle {\mathcal {H}}}
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:
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 ) 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 − 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 ) 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
∫
−
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 , Oxford:
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 , Dover Phoenix editions, Dover Publications, New York, p. 316,
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 ) (in French), 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 ) (in French), 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