Parametrizations of integrals

PARAMETRIZATIONS OF INTEGRALS

Grzegorz Biernat

Institute of Mathematics and Computer Science, Czestochowa University of Technology

Abstract. In the present paper I give parametric formulas of integrals of meromorphic forms in the case of C².

1. Parametrizations of integrals at finity

Integrals of meromorphic forms occur in the definition of residue. Let us remind then the definition of residue of holomorphic mapping at a point. To retain symmetry with second part of this paper we will limit to the case of C². Let f = (f₁, f₂) be holomorphic mapping in the neighbourhood of point α = (α₁, α₂) ∈ C² with zero isolated at this point; and g holomorphic function in the neighbourhood of point α. As residue of pair g, f at point a we define an integral of the form (s.[1, 2])

( )

( )

( ) ( )

∫

Γ ⋅

∧ π

=

a

z f z f

dz dz z g i f

a g

2 1

2 1

2 2

1 Res

where ^Γ_a ⁼

{

^{z: f}₁

( )

^{z =ε, f}₂

( )

^{z =ε}

}

is sufficiently small real two-cycle in the neighbourhood of point α with positive orientation given by nowhere not di- sappearing on Γα form d

(

arg f₁

)

∧d

(

argf₂

)

.

Calculating of residue we might then reduce to calculus of residues of mero- morphic functions of one variable. However if the germ of function f₁ in point α has reduced decomposition then

( )

z f

( )

z f

( )

z f₁ = ₁₁ K _1m for z from some neighbourhood of point α. Then (s. [3])

( )

j j m

j j

j

a f

f f

f g

g Φ

Φ ′ Φ

=

∑

Φ

≤

≤ o

o o

o

2 2 1

0 Jac res Res

where Φ_j is parametrization of the set of zeros of function f_1j defined in the neighbourhood of point 0 at C, Φ_j(0) = α, j = 1,..., m, a Jacf denote a Jacobian of the mapping f. Thus the integral of meromorphic two-form is reduced to the integrals of meromorphic functions

( )

( ) ( )

( ( ) ) ( ( ) )

( ( ) ) ( ( )

t

)

^dt

f t f t f

t i g

z f z f

dz dz z g

j j m

j C j

j

j a

Φ Φ ′ Φ

Φ π

=

⋅

∧

∑ ∫

∫

≤

Γ ≤ 2

2 1

2 1

2 1

Jac 2

where Cj are sufficiently small positively oriented circles with the center in point 0 at C. Similarly, if the germ of function f₂ at point α has reduced decomposition, then

( )

z f

( )

z f

( )

z f₂ = ₂₁ K _2n for z from some neighbourhood of point α, thus

( )

( ) ( )

( )

( )

( )

( )

( )

( )

( ) (

t

)

^dt

f t f t f

t i g

z f z f

dz dz z g

k k

n

k C k

k

k a

Ψ Ψ ′ Ψ

Ψ π

=

⋅

∧

∑ ∫

∫

≤

Γ ≤ 1

1 2 1

1

2 1

Jac 2

where Ψk is parametrization of the set of zeros of function f_2k defined in the neigh- bourhood of point 0 at C, Ψ_k(0) = α, k = 1,..., n.

Applying above parametric formulas we obtain the given relation between inte- grals of following two-forms (s. [4])

(*)

( )

( ) ( ) ( )

( )

( ) ( ) ( )

( )

( ) ( )

∫

∫

∫

Γ

σ Γ

σ Γ

σ

−

⋅

= ∧

⋅

−

− ∧

−

⋅

∧

a a

a f f z z a

dz dz z g z

f z f a z

dz dz z g z

f a z z f

dz dz z g

1 1 2 1

2 1 2

1 1 1

2 1 2

1 1 1

2

1

for σ≥0

2. Parametrizations of integrals at infinity

Integrals of rational forms occurs in definition of residue at infinity. At the be- ginning let us assume the following definitions. For polynomial h of two variables we define polynomial

( )

_







= 

1 2 1 deg 1 2

1 1 ,

~ ,

X X X h X X X

h ^h

and for point p=

(

0:1:y

)

∈P² its affine image ^~p=

(

^0,y

)

∈^C2.

Let f = (f₁, f₂) be polynomial defined on C² of components relatively prime and different then constants while g be arbitrary polynomial of two variables. Let us denote σ=degf₁+degf₂−degg−3. The residue of pair g, f at infinity we define by the formula (s. [4, 5])

( )

(

_{1 2}

)

~ 1

,~

~ ~ Res Res

2 1

f f X g f

g

l C C c

c σ

∩

∩

∈

∞

∑

∞

−

= for σ≥0

and

( ) ∑ ( )

∑

∞

∞ ∈ ∩

σ

− σ

−

∩

∈

∞ =− =−

l C b

b l

C a

a g f X f g X f f

f g

2 1

2 1

~ 1 2

1

~ 1

,~

~ ~

~ Res

~, Res ~

Res

for σ < 0

where l_∞ represents the line at infinity over C², while C₁ i C₂ are the closers at P² of curves f₁ = 0 and f₂ = 0, respectively. In the second part of definition we additional- ly assume that

(

^0:0:1

)

^∉C₁^∩l_∞ and

(

^0:0:1

)

^∉C₂^∩l_∞^, what in fact just simpli- fies the notation (s. [4]). The integrals of forms occurring in expression of residue at infinity we may now parametrize. Let σ ≤ 0 and let c∈

(

C₁∩C₂

)

∩l_∞^. If the germ of function ~f₁ at point c~ has a reduced decomposition, then

( )

x f

( )

x f

( )

x f₁ ~₁₁ ~_1p

~

= K for x from some neighbourhood of point .~

c Then

( )

( ) ( )

( ( ) ) ( ( ) )

( ( ) ) ( ( )

^t

)

^dt

f t f t f

t t i g

x f x f

dx dx x x g

j j p

j C j

j

j

j

c Φ

′ Φ Φ

Φ π

=

⋅

∧

∑ ∫

∫

≤

≤

σµ

Γ

σ

~ ~

~ ~

~ ~ Jac

~ ~

~ 2

~

~

2 2 2 1

1

2 1 1

~

where j

( )

^t

(

^t j

( )

^t

)

j ϕ

=

Φ ^µ ,

~ is parametrization of the set of zeros of function ~ in f_1j the neighbourhood of point 0 at C, ~ (0) ~, 1,..., ,

p j

j =c =

,Φ

C where ^~f =

(

^~f₁^,~f₂

)

^.

Similarly, if the germ of function ~₂

f at point c~ has reduced decomposition, then

( )

x f

( )

x f

( )

x f₂ ~₂₁ ~_2q

~

= K for x from some neighbourhood of point .~

c Then

( )

( ) ( )

( ( ) ) ( ( ) )

( ( ) ) ( ( )

^t

)

^dt

f t f t f

t t i g

x f x f

dx dx x x g

k k q

k C k

k k

k

c Ψ

′ Ψ Ψ

Ψ π

=

⋅

∧

∑ ∫

∫

≤

≤

σν

Γ

σ

~

~

~ ~

~

~ Jac

~ ~

~ 2

~

~

1 1 2 1

1

2 1 1

~

where ^Ψ~_k

( )

^{t =}

(

^tνk,ψ~_k

( )

^t

)

is parametrization of the set of zeros of function ~f_2k defined in the neighbourhood of point 0 at ~ (0) ~, 1,..., .

,Ψ_k =c k= q

C The residue at

point c∈

(

C ∩C

)

∩l_∞

2

1 at infinity we may, in this case, define as

( )

( )

( ) ( )

∫

Γ

σ

⋅

∧ π

−

=

c f x f x

dx dx x x g i f

cg

~ 1 2

2 1 1

2 ~ ~

~ 2

1 Res

Then

( )

∑

∩∞

∩

∈

∞ =

l C C c

cg f f

g

2 1

Res

Res for σ ≥ 0

Let σ < 0 and let .

1 ∩ ∞

∈C l

a If the germ of function ~₁

f at point has the reduced decomposition, then

( )

x f

( )

x f

( )

x f₁ ~₁₁ ~_1r

~

= K for x from some neighbourhood of point .~

a Then

( )

( ) ( )

( ( ) ) ( ( ) )

( ( ) )

( ( )

^t

)

^{t dt}

f t f t f

t i g

x f x x f

dx dx x

g j

j j r

j C I j

j

j

a 











 σµ

Φ − Φ ′ Φ

π Φ

⋅ =

∧

∑ ∫

∫

≤ Γ ≤

σ

− ~ ~

~ ~

~ ~ Jac

~ ~

~ 2

~

~

2 2 2 1

1 1

2 1

~

where j

( )

^t

(

^t j

( )

^t

)

j ϕ

=

Φ ^µ ~

,

~ is parametrization of the set of zeros of function ~ f_1j defined in the neighbourhood of point 0 at ~ (0) ~,

,Φ =α

C j and^~f_I =

(

^~f1^,X1^−σ~f2

)

^.

Then the residue at point a∈C ∩l_∞

1 at infinity we may define as

( )

( )

( ) ( )

∫

Γ

σ

−

⋅

∧ π

−

=

a f x x f x

dx dx x g i f

I g

a

~ 1 1 2

2 1

2 ~ ~

~ 2

Res 1

Then

∑

∩∞

∈

∞ =

l C a

I ag f f

g

1

Res

Res for σ<0

Similarly, let σ < 0 and let .

2 ∩ ∞

∈C l

b If the germ of function ~₂

f at point b~ has reduced decomposition, then

( )

x f

( )

x f

( )

x f₂ ~₂₁ ~_2s

~

= K

for x from some neighbourhood of point .~ b Then

( )

( ) ( )

( ( ) ) ( ( ) )

( ( ) )

( ( )

^t

)

^{t dt}

f t f t f

t i g

x f x f x

dx dx x

g k

k k

s

k C II k

k

b k















 σν

Ψ − Ψ ′ Ψ

π Ψ

⋅ =

∧

∑ ∫

∫

≤ Γ ≤

σ

− ~ ~

~

~

~ ~ Jac

~ ~

~ 2

~

~

1 1 2 1

1 1

2 1

~

where ^Ψ~_k

( )

^{t =}

(

^tνk,ψ~_k

( )

^t

)

is parametrization of the set of zeros of function ~f₂k

defined in the neighbourhood of point 0 in ~, 1,..., , )

0

~ (

, b k s

k = =

Ψ

C and

(

^~,~

)

^.

~

2 1

1 f f

X

f_II = ^−σ Then the residue at point b∈C ∩l_∞

2 at infinity we may define as

( )

( )

( ) ( )

∫

Γ σ

−

⋅

∧ π

−

=

bx f x f x

dx dx x g i f

II g

b

~ 1 1 2

2 1

2 ~ ~

~ 2

Res 1

Then

∑

∩∞

∈

∞ =

l C b

II b g f f

g

2

Res

Res for σ<0

Let us now observe that if a = b and σ < 0, then the residues I and II are con- nected by equation

(**)

( )

( )

∫ ( )

Γ

σ

−

⋅

∧ π

=

−

a f f x x

dx dx x g i f g f

g ^II_a

I a

~ 1 2 1

2 1

2 ~~

~ 2

1 Res

Res

References

[1] Arnold V.I., Singularities of Differentiable Maps, vol. I, Boston 1985.

[2] Griffiths P., Harris J., Principles of Algebraic Geometry, New York 1978.

[3] Biernat G., Représentation paramétrique d’un résidu multidimensional, Rev. Roum. Math. Pur.

Appl. 1991, 36, 5-6, 207-211.

[4] Biernat G., Théorème des rèsidus dans C², Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej 2002, 1(1), 19-24.

[5] Biernat G., On the Jacobi-Kronecker formula for a polynomial mapinng having zeros at infinity, Bull. Soc. Sci. Lettres Łódź 1992, 42 (29), XIV, 139.