Note 1
Differential Operator and Symbol
1
Complex variant partial differential operator 
z
z
Multi-index α = （α1, …, αn) ∈ 
0n
0nzα: = z1α1, …, znαn
α! = α1!...αn!
|α| = 
j
j
2
Topological space X
All of X’s open sets 
(X)
(X)
U ∈ (X)
(X)
Regular function f(z) : U → 
All of regular functions over U 
(U)
(U)
Sheaf (n) ∋ U 
(U)
(n) ∋ U (U)
3
P (z, z) = 
α(z)
zα, aα(z)∈
(U
cn)
z) = α(z)zα, aα(z)∈(Ucn)
P’s order Minimum of m
4
Sequence { aα(z)}α∈N0n⊂
(U
cn)
(Ucn)
5
{|α|
; z ∈ K } = 0
6
Differential operator of infinite order P (z, 
z) = 
α(z)
zα, aα(z)∈(Ucn)
z) = α(z)zα, aα(z)∈(Ucn)
7
ς= (ς1, …,ςn) ∈
n
n
Symbol P (z, ς) : = α(z)ςα
α(z)ςα
8
All of differential operators of infinite order Sheaf of 
linear space, expressed by D cn ∞
linear space, expressed by D cn ∞
[Reference]
<On infinite order, refer to the next.>
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