The slide of a speech in the conference "ECMI2016" (The 19th European Conference on Mathematics for Industry) held at Santiago de Compostela, Spain in June 2016.

1. 1 / 24
An Application of the Hyperfunction Theory to Numerical Integration
ECMI2016
∗Hidenori Ogata (The University of Electro-Communications, Japan)
Hiroshi Hirayama (Kanagawa Institute of Technology, Japan)
17 June 2016

2. Contents
2 / 24
✓ ✏
We show an application of the hyperfunction theory, a generalized function
theory based on complex analysis, to numerical computations, in particular,
to numerical integrations.
✒ ✑

3. Contents
2 / 24
✓ ✏
We show an application of the hyperfunction theory, a generalized function
theory based on complex analysis, to numerical computations, in particular,
to numerical integrations.
✒ ✑
1. Hyperfunction theory
2. Hyperfunction method for numerical integrations
3. Hyperfunction method for Hadamard’s finite parts
4. Numerical examples
5. Summary

4. Contents
3 / 24
1. Hyperfunction theory
2. Hyperfunction method for numerical integrations
3. Hyperfunction method for Hadamard’s finite parts
4. Numerical examples
5. Summary

5. 1. Hyperfunction theory (M. Sato, 1958)
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b
a
φ(x)δ(x)dx = φ(0).
1
2πi C
φ(z)
z
dz = φ(0).
( a < 0 < b )
Dirac delta function Cauchy integral formula
O
C

6. 1. Hyperfunction theory (M. Sato, 1958)
4 / 24
b
a
φ(x)δ(x)dx = φ(0).
1
2πi C
φ(z)
z
dz = φ(0).
( a < 0 < b )
Dirac delta function Cauchy integral formula
O
C
Oa b
1
2πi C
φ(z)
z
dz = −
1
2πi
b
a
φ(x)
1
x + i0
−
1
x − i0
dx.

7. 1. Hyperfunction theory (M. Sato, 1958)
4 / 24
b
a
φ(x)δ(x)dx = φ(0).
1
2πi C
φ(z)
z
dz = φ(0).
( a < 0 < b )
Dirac delta function Cauchy integral formula
O
C
Oa b
b
a
φ(x)δ(x)dx =
1
2πi C
φ(z)
z
dz = −
1
2πi
b
a
φ(x)
1
x + i0
−
1
x − i0
dx.
∴ δ(x) = −
1
2πi
1
x + i0
−
1
x − i0
.

8. 1. Hyperfunction theory
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Hyperfunction theory (M. Sato, 1958)✓ ✏
• A generalized function theory based on complex analysis.
• A hyperfunction f(x) on an interval I is the difference of the boundary
values of an analytic function F(z).
f(x) = [F(z)] ≡ F(x + i0) − F(x − i0).
F(z) : the defining function of the hyperfunction f(x)
analytic in D I,
where D is a complex neighborhood of the interval I.
✒ ✑
D
I
R

9. 1. Hyperfunction theory: examples
6 / 24
• Dirac delta function
δ(x) = −
1
2πiz
= −
1
2πi
1
x + i0
−
1
x − i0
.
• Heaviside step function
H(x) =
1 ( x > 0 )
0 ( x < 0 )
= −
1
2πi
{log(−(x + i0)) − log(−(x − i0))} .
∗ log z is the principal value s.t. −π ≦ arg z < π
• Non-integral powers
x+
α
=
xα ( x > 0 )
0 ( x < 0 )
= −
(−(x + i0))α − (−(x − i0))α
2i sin(πα)
(α ∈ Z const.).
∗ zα
is the principal value s.t. −π ≦ arg z < π.

10. 1. Hyperfunction theory: examples
7 / 24
Heaviside step function
H(x) =
1 ( x > 0 )
0 ( x < 0 )
= F(x+i0)−F(x−i0), F(z) = −
1
2πi
log(−z).
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1Re z
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Im z
-1
-0.5
0
0.5
1
Re F(z)
The real part of the defining function F(z) = −
1
2πi
log(−z).

11. 1. Hyperfunction theory: examples
7 / 24
Heaviside step function
H(x) =
1 ( x > 0 )
0 ( x < 0 )
= F(x+i0)−F(x−i0), F(z) = −
1
2πi
log(−z).
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1Re z
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Im z
-1
-0.5
0
0.5
1
Re F(z)
Many functions with singularities are expressed by analytic functions in the
hyperfunction theory.

12. 1. Hyperfunction theory: integral
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Integral of a hyperfunction f(x) = F(x + i0) − F(x − i0)✓ ✏
I
f(x)dx ≡ −
C
F(z)dz
C : closed path which encircles I in the positive sense
and is included in D (F(z) is analytic in D I).
✒ ✑
D
C
I
• The integral in independent of the choise of C by the Cauchy integral theorem.

13. Contents
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1. Hyperfunction theory
2. Hyperfunction method for numerical integrations
3. Hyperfunction method for Hadamard’s finite parts
4. Numerical examples
5. Summary

14. 2. Hyperfunction method for numerical integrations
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We consider the evaluation of an integral
I
f(x)w(x)dx,
f(x) : analytic in a domain D s.t.
(I ⊂ D ⊂ C),
w(x) : weight function.
D
I
R

15. 2. Hyperfunction method for numerical integrations
10 / 24
We consider the evaluation of an integral
I
f(x)w(x)dx,
f(x) : analytic in a domain D s.t.
(I ⊂ D ⊂ C),
w(x) : weight function.
D
I
R
We regard the integrand as a hyperfunction.
✓ ✏
f(x)w(x)χI(x) = −
1
2πi
{f(x + i0)Ψ(x + i0) − f(x − i0)Ψ(x − i0)}
with χI(x) =
1 (x ∈ I)
0 (x ∈ I)
, Ψ(z) =
I
w(x)
z − x
dx.
✒ ✑

16. 2. hyperfunction method for numerical integrations
11 / 24
From the definition of hyperfunction integrals, we have
✓ ✏
I
f(x)w(x)dx =
1
2πi C
f(z)Ψ(z)dz.
=
1
2πi
uperiod
0
f(ϕ(u))Ψ(ϕ(u))ϕ′
(u)du,
C : z = ϕ(u) ( 0 ≦ u ≦ uperiod ) periodic function of period uperiod.
✒ ✑
D
C : z = ϕ(u)
I
Approximating the r.h.s. by the trapezoidal rule, we have ...

17. 2. Hyperfunction method for numerical integrations
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Hyperfunction method✓ ✏
I
f(x)w(x)dx ≃
h
2πi
N−1
k=0
f(ϕ(kh))Ψ(ϕ(kh))ϕ′
(kh),
with Ψ(z) =
b
a
w(x)
z − x
dx and h =
uperiod
N
.
✒ ✑
D
C : z = ϕ(u), 0 ≦ u ≦ uperiod
I

18. 2. Hyperfunction method for numerical integrations
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Hyperfunction method✓ ✏
I
f(x)w(x)dx ≃
h
2πi
N−1
k=0
f(ϕ(kh))Ψ(ϕ(kh))ϕ′
(kh),
with Ψ(z) =
b
a
w(x)
z − x
dx and h =
uperiod
N
.
✒ ✑
Ψ(z) for typical weight functions w(x)
I w(x) Ψ(z)
(a, b) 1 log
z − a
z − b
∗
(0, 1) xα−1(1 − x)β−1 B(α, β)z−1F(α, 1; α + β; z−1)∗∗
( α, β > 0 )
∗ log z is the principal value s.t. −π ≦ arg z < π.
∗∗ F(α, 1; α + β, z−1
) can be evaluated by the continued fraction.

19. 2. Hyperfunction method for numerical integrations
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The trapezoidal rule is efficient for integrals of periodic analytic functions.

20. 2. Hyperfunction method for numerical integrations
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The trapezoidal rule is efficient for integrals of periodic analytic functions.
Theoretical error estimate✓ ✏
If f(ϕ(w)) and ϕ(w) are analytic in | Im w| < d0,
|error| ≦ 2uperiod max
Im w=±d
|f(ϕ(w))Ψ(ϕ(w))ϕ′
(w)|
×
exp(−(2πd/uperiod)N)
1 − exp(−(2πd/uperiod)N)
( 0 < ∀d < d0 ).
. . . Geometric convergence.
✒ ✑

21. Contents
14 / 24
1. Hyperfunction theory
2. Hyperfunction method for numerical integrations
3. Hyperfunction method for Hadamard’s finite parts
4. Numerical examples
5. Summary

22. 3. Hadamard’s finite parts
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1
0
x−1
f(x)dx ( f(x) : finite as x → 0+ ) . . . divergent!

23. 3. Hadamard’s finite parts
15 / 24
1
0
x−1
f(x)dx ( f(x) : finite as x → 0+ ) . . . divergent!
Hadamard’s finite part✓ ✏
fp
1
0
x−1
f(x)dx ≡ lim
ǫ→0+
1
ǫ
x−1
f(x)dx + f(0) log ǫ .
✒ ✑

24. 3. Hadamard’s finite parts
15 / 24
1
0
x−1
f(x)dx ( f(x) : finite as x → 0+ ) . . . divergent!
Hadamard’s finite part✓ ✏
fp
1
0
x−1
f(x)dx ≡ lim
ǫ→0+
1
ǫ
x−1
f(x)dx + f(0) log ǫ .
✒ ✑
Hadamard’s finite part (n = 1, 2, . . .)✓ ✏
fp
1
0
x−n
f(x)dx
≡ lim
ǫ→+0
1
ǫ
x−n
f(x)dx +
n−2
k=0
ǫk+1−n
k!(k + 1 − n)
f(k)
(0) +
log ǫ
(n − 1)!
f(n−1)
(0) .
✒ ✑

25. 3. Hadamard’s finite parts
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Hadamard’s finite parts can be given by hyperfunction integrals.
fp
1
0
x−n
f(x)dx =
1
0
χ(0,1)x−n
f(x)dx
hyperfunction integral
+
n−2
k=0
f(k)
(0)
k!(k + 1 − n)
=
1
2πi C
z−n
f(z) log
z
z − 1
dz
approximated by the trapezoidal rule
+
n−2
k=0
f(k)
(0)
k!(k + 1 − n)

26. Contents
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1. Hyperfunction theory
2. Hyperfunction method for numerical integrations
3. Hyperfunction method for Hadamard’s finite parts
4. Numerical examples
5. Summary

27. 4. Example 1: numerical integration
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1
0
ex
xα−1
(1−x)β−1
dx = B(α, β)F(α; α+β; 1) with α = β = 10−4
.
We evaluated the integral by
• the hyperfunction method
• the DE formula (efficient for integrals with end-point singularities)
and compared the errors of the two methods.
• C++ programs, double precision.
• integral path for the hyperfunction method
z = 0.5 + 2.575 cos u + i2.425 sin u, 0 ≦ u ≦ 2π (ellipse).

28. 4. Example 1: numerical integrations
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-16
-14
-12
-10
-8
-6
-4
-2
0
0 5 10 15 20 25 30
log10(relativeerror)
N
hyperfunction rule
DE rule
relative errors
• the hyperfunction method error = O(0.024N ) (geometric convergence).
• The DE formula does not work for this integral.

29. 4. Example 1: Why the hyperfunction method works well?
20 / 24
integrand
e z
hyperfunction method
• (DE rule) The sampling points accumulate at the singularities.
• (hyperfunction method) The sampling points are distributed on a curve
in the complex plane where the integrand varies slowly.

30. 4. Example 2: Hadamard’s finite part
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fp
x
0
x−n
ex
dx =
∞
k=0(k=n−1)
1
k!(k − n + 1)
( n = 1, 2, . . . ).
We computed it by the hyperfunction method.
• C++ program & double precision
• integral path
z =
1
2
+
1
4
ρ +
1
ρ
cos u +
i
4
ρ −
1
ρ
sin u,
0 ≦ u < 2π ( ρ = 10, ellipse ).

31. 4. Example 2: Hadamard’s finite part
22 / 24
-16
-14
-12
-10
-8
-6
-4
-2
0
0 5 10 15 20
log10(relativeerror)
N
n=0
n=1
n=2
n=3
n=4
n=5
the relative errors of the hyperfunction method
n 1 2 3 4 5
error O(0.021N ) O(0.023N ) O(0.018N ) O(0.034N ) O(0.032N )
... geometric convergenc

32. Contents
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1. Hyperfunction theory
2. Hyperfunction method for numerical integrations
3. Hyperfunction method for Hadamard’s finite parts
4. Numerical examples
5. Summary

33. 5. Summary
24 / 24
• The hyperfunction theory is a generalized function theory based on complex
analysis.
• The hyperfunction method approximately computes desired integral by
evaluating the complex integrals which define them as hyperfunction
integrals
• Numerical examples show that the hyperfunction method is efficient for
integral with end-point singularities.

34. 5. Summary
24 / 24
• The hyperfunction theory is a generalized function theory based on complex
analysis.
• The hyperfunction method approximately computes desired integral by
evaluating the complex integrals which define them as hyperfunction
integrals
• Numerical examples show that the hyperfunction method is efficient for
integral with end-point singularities.
functions with singularities
(poles, discontinuities,
delta functions, ...)
←−←−←−
hyperfunction
analytic
functions

35. 5. Summary
24 / 24
• The hyperfunction theory is a generalized function theory based on complex
analysis.
• The hyperfunction method approximately computes desired integral by
evaluating the complex integrals which define them as hyperfunction
integrals
• Numerical examples show that the hyperfunction method is efficient for
integral with end-point singularities.
functions with singularities
(poles, discontinuities,
delta functions, ...)
←−←−←−
hyperfunction
analytic
functions
We expect that we can apply the hyperfunction theory to a wide range of
scientific computations.
!
Gracias!