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# The Equation y′ = wy and the Meromorphic Products

DOI link for The Equation y′ = wy and the Meromorphic Products

The Equation y′ = wy and the Meromorphic Products book

# The Equation y′ = wy and the Meromorphic Products

DOI link for The Equation y′ = wy and the Meromorphic Products

The Equation y′ = wy and the Meromorphic Products book

## ABSTRACT

Let K be a complete ultrametric algebraically closed field of characteristic zero and let D be a clopen bounded infraconnected set whose holes have diameter greater than a positive constant p.

Let H(D) be the Banach algebra of the Analytic Elements in D.

The Differential Equation
ℓ
(
w
)
y
1
=
w
y
(
w
∈
H
(
D
)
)
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has non zero solutions in H(D) if and only if w is of the form()
∑
i
=
1
q
u
i
x
−
c
i
+
∑
n
=
1
∞
(
1
x
−
a
n
−
1
x
−
b
n
)
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/025b03ff-bfc0-4e0d-a483-fa30d63d11a7/content/eq1938.tif"/>
where the c_{i}
belong to different holes T_{i}
, of D, each u_{i}
, is the residue of w in T_{i}
, and the sequences (a_{n}),(b_{n}) satisfy conditions like
lim
n
→
∞
(
a
n
−
b
n
)
=
0
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/025b03ff-bfc0-4e0d-a483-fa30d63d11a7/content/eq1939.tif"/>
. The solutions then are in the form
λ
∏
i
=
1
q
(
x
−
c
i
)
u
i
∏
n
=
1
∞
(
x
−
a
n
x
−
b
n
)
.
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/025b03ff-bfc0-4e0d-a483-fa30d63d11a7/content/eq1940.tif"/>
.

Then we characterize the w which have integrals in H(D) as the elements of the form 1 p h ∑ n = 1 ∞ 1 x − a n − 1 x − b n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/025b03ff-bfc0-4e0d-a483-fa30d63d11a7/content/eq1941.tif"/> and we define and study the quasi-integrable elements.

Let B be the open unit disk, and let θ ∈ H(B). We assume ℓ(θ) to have a non zero solution f analytic function in B (not supposed to belong to H(B)). Then
f
N
∈
H
(
B
)
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/025b03ff-bfc0-4e0d-a483-fa30d63d11a7/content/eq1942.tif"/>
for some N ∈ ℕ if and only if θ is of the form jjU) where uo is of the form
1
N
w
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where w (1) above. As a corollary we have an immediate demonstation to show that a function
1
+
x
N
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/025b03ff-bfc0-4e0d-a483-fa30d63d11a7/content/eq1944.tif"/>
does not belong to H(B). When K = ℂ_{p}, we apply these results to the search of a Frobenius Structure on ℓ(w).