In function theory , the residual of a complex-valued function is an aid for calculating complex curve integrals using the residual theorem .
definition
Complex areas
Be one area , isolated in and holomorphic . Then for every point there exists a dotted neighborhood that is relatively compact in , with holomorphic. In this case, has to a Laurent development . Then one defines for the residual of in
D.
⊆
C.
{\ displaystyle D \ subseteq \ mathbb {C}}
D.
f
{\ displaystyle D_ {f}}
D.
{\ displaystyle D}
f
:
D.
∖
D.
f
→
C.
{\ displaystyle f \ colon D \ setminus D_ {f} \ to \ mathbb {C}}
a
∈
D.
f
{\ displaystyle a \ in D_ {f}}
U
: =
U
r
(
a
)
∖
{
a
}
⊂
D.
{\ displaystyle U: = U_ {r} (a) \ setminus \ {a \} \ subset D}
D.
{\ displaystyle D}
f
|
U
{\ displaystyle f | _ {U}}
f
{\ displaystyle f}
U
{\ displaystyle U}
f
|
U
(
z
)
=
∑
n
=
-
∞
∞
c
n
(
z
-
a
)
n
{\ displaystyle \ textstyle f | _ {U} (z) = \ sum _ {n = - \ infty} ^ {\ infty} c_ {n} (za) ^ {n}}
f
{\ displaystyle f}
a
{\ displaystyle a}
Res
a
(
f
)
: =
c
-
1
=
1
2
π
i
∮
∂
U
f
(
z
)
d
z
{\ displaystyle \ operatorname {Res} _ {a} (f): = c _ {- 1} = {\ frac {1} {2 \ pi \ mathrm {i}}} \ oint \ limits _ {\ partial U} f (z) dz}
.
Riemann number ball
The above definition can also be extended to the Riemann number sphere . Again let be a discrete set in and a holomorphic function. Then for all with the residual is also explained by the above definition. For one sets
P
1
=
C.
∪
{
∞
}
{\ displaystyle \ mathbb {P} _ {1} = \ mathbb {C} \ cup \ {\ infty \}}
D.
f
{\ displaystyle D_ {f}}
P
1
{\ displaystyle \ mathbb {P} _ {1}}
f
:
P
1
∖
D.
f
→
C.
{\ displaystyle f \ colon \ mathbb {P} _ {1} \ setminus D_ {f} \ to \ mathbb {C}}
a
∈
D.
f
{\ displaystyle a \ in D_ {f}}
a
≠
∞
{\ displaystyle a \ neq \ infty}
a
=
∞
∈
D.
f
{\ displaystyle a = \ infty \ in D_ {f}}
Res
∞
(
f
)
: =
-
c
-
1
=
1
2
π
i
∮
γ
f
(
z
)
d
z
,
{\ displaystyle \ operatorname {Res} _ {\ infty} (f): = - c _ {- 1} = {\ frac {1} {2 \ pi \ mathrm {i}}} \ oint \ limits _ {\ gamma } f (z) dz \ ,,}
where is a circle with a sufficiently large radius that is oriented clockwise, and is the -1 as above. Laurent series coefficient.
γ
{\ displaystyle \ gamma}
c
-
1
{\ displaystyle c _ {- 1}}
Features and Notes
Let be a domain and a holomorphic function in . Then Cauchy's integral theorem can be applied, from which it follows that the residual of in is zero.
D.
⊂
C.
{\ displaystyle D \ subset \ mathbb {C}}
f
:
D.
→
C.
{\ displaystyle f \ colon D \ to \ mathbb {C}}
a
{\ displaystyle a}
f
{\ displaystyle f}
a
{\ displaystyle a}
The integral representation shows that one can also speak of the residual of the differential form .
f
(
z
)
d
z
{\ displaystyle f (z) \ mathrm {d} z}
The residual theorem applies .
For rational functions , the so-called unity relation holds: . Here is the set of all poles of and the Riemann number sphere .
f
:
C.
^
→
C.
^
{\ displaystyle f: {\ hat {\ mathbb {C}}} \ to {\ hat {\ mathbb {C}}}}
∑
p
(
f
)
Res
p
(
f
)
=
0
{\ displaystyle \ sum _ {p (f)} \ operatorname {Res} _ {p} (f) = 0}
p
(
f
)
{\ displaystyle p (f)}
f
{\ displaystyle f}
C.
^
=
C.
∪
{
∞
}
{\ displaystyle {\ hat {\ mathbb {C}}} = \ mathbb {C} \ cup \ {\ infty \}}
Practical calculation
The following rules can be used in practice to calculate residuals of complex-valued functions in point :
f
,
G
{\ displaystyle f, g}
a
∈
C.
{\ displaystyle a \ in \ mathbb {C}}
The residual is -linear, i.e. H. for the following applies:
C.
{\ displaystyle \ mathbb {C}}
λ
,
μ
∈
C.
{\ displaystyle \ lambda, \ mu \ in \ mathbb {C}}
Res
a
(
λ
f
+
μ
G
)
=
λ
Res
a
f
+
μ
Res
a
G
{\ displaystyle \ operatorname {Res} _ {a} \ left (\ lambda f + \ mu g \ right) = \ lambda \ operatorname {Res} _ {a} f + \ mu \ operatorname {Res} _ {a} g}
Has in a pole first order, the following applies:
f
{\ displaystyle f}
a
{\ displaystyle a}
Res
a
f
=
lim
z
→
a
(
z
-
a
)
f
(
z
)
{\ displaystyle \ textstyle \ operatorname {Res} _ {a} f = \ lim _ {z \ rightarrow a} (za) f (z)}
Has in a pole first order and is in holomorphic, then:
f
{\ displaystyle f}
a
{\ displaystyle a}
G
{\ displaystyle g}
a
{\ displaystyle a}
Res
a
G
f
=
G
(
a
)
Res
a
f
{\ displaystyle \ operatorname {Res} _ {a} gf = g (a) \ operatorname {Res} _ {a} f}
Has in a zero first order, the following applies:
f
{\ displaystyle f}
a
{\ displaystyle a}
Res
a
1
f
=
1
f
′
(
a
)
{\ displaystyle \ operatorname {Res} _ {a} {\ tfrac {1} {f}} = {\ tfrac {1} {f '(a)}}}
Has in a zero first order and is in holomorphic, then:
f
{\ displaystyle f}
a
{\ displaystyle a}
G
{\ displaystyle g}
a
{\ displaystyle a}
Res
a
G
f
=
G
(
a
)
f
′
(
a
)
{\ displaystyle \ operatorname {Res} _ {a} {\ tfrac {g} {f}} = {\ tfrac {g (a)} {f '(a)}}}
Has in a pole -th order, the following applies:
f
{\ displaystyle f}
a
{\ displaystyle a}
n
{\ displaystyle n}
Res
a
f
=
1
(
n
-
1
)
!
lim
z
→
a
∂
n
-
1
∂
z
n
-
1
[
(
z
-
a
)
n
f
(
z
)
]
{\ displaystyle \ textstyle \ operatorname {Res} _ {a} f = {\ tfrac {1} {\ left (n-1 \ right)!}} \ lim _ {z \ rightarrow a} {\ frac {\ partial ^ {n-1}} {\ partial z ^ {n-1}}} [(za) ^ {n} f (z)]}
Has in a zero th order, then: .
f
{\ displaystyle f}
a
{\ displaystyle a}
n
{\ displaystyle n}
Res
a
f
′
f
=
n
{\ displaystyle \ operatorname {Res} _ {a} {\ tfrac {f '} {f}} = n}
Has in a zero th order and g in holomorphic, then: .
f
{\ displaystyle f}
a
{\ displaystyle a}
n
{\ displaystyle n}
a
{\ displaystyle a}
Res
a
G
f
′
f
=
G
(
a
)
n
{\ displaystyle \ operatorname {Res} _ {a} g {\ tfrac {f '} {f}} = g (a) n}
Has in a pole th order, then: .
f
{\ displaystyle f}
a
{\ displaystyle a}
n
{\ displaystyle n}
Res
a
f
′
f
=
-
n
{\ displaystyle \ operatorname {Res} _ {a} {\ tfrac {f '} {f}} = - n}
Has in a pole th order and g in holomorphic, then: .
f
{\ displaystyle f}
a
{\ displaystyle a}
n
{\ displaystyle n}
a
{\ displaystyle a}
Res
a
G
f
′
f
=
-
G
(
a
)
n
{\ displaystyle \ operatorname {Res} _ {a} g {\ tfrac {f '} {f}} = - g (a) n}
If the residual at the point is to be calculated, then applies . Because with applies
∞
{\ displaystyle \ infty}
Res
∞
f
=
Res
0
(
-
1
z
2
f
(
1
z
)
)
{\ displaystyle \ operatorname {Res} _ {\ infty} f = \ operatorname {Res} _ {0} \ left (- {\ tfrac {1} {z ^ {2}}} f ({\ tfrac {1} {z}}) \ right)}
w
=
1
z
{\ displaystyle w = {\ tfrac {1} {z}}}
f
(
w
)
d
w
=
f
(
1
z
)
d
1
z
=
-
1
z
2
f
(
1
z
)
d
z
{\ displaystyle f (w) \ mathrm {d} w = f ({\ tfrac {1} {z}}) \ mathrm {d} {\ tfrac {1} {z}} = - {\ tfrac {1} {z ^ {2}}} f ({\ tfrac {1} {z}}) \ mathrm {d} z}
The rules about the logarithmic derivative are also of theoretical interest in connection with the residual theorem.
f
′
f
{\ displaystyle {\ tfrac {f '} {f}}}
Examples
As mentioned earlier, when is on an open environment of holomorph.
Res
a
f
=
0
{\ displaystyle \ operatorname {Res} _ {a} f = 0}
f
{\ displaystyle f}
a
{\ displaystyle a}
If so has in a pole first order, and it is .
f
(
z
)
=
1
z
{\ displaystyle f (z) = {\ tfrac {1} {z}}}
f
{\ displaystyle f}
0
{\ displaystyle 0}
Res
0
f
=
1
{\ displaystyle \ operatorname {Res} _ {0} f = 1}
Res
1
z
z
2
-
1
=
1
2
{\ displaystyle \ operatorname {Res} _ {1} {\ tfrac {z} {z ^ {2} -1}} = {\ tfrac {1} {2}}}
As you can see immediately with the linearity and the rule of the logarithmic derivative, because it has a first order zero.
z
↦
z
2
-
1
{\ displaystyle z \ mapsto z ^ {2} -1}
1
{\ displaystyle 1}
The continued gamma function has in for poles of 1st order, and the residual there is .
-
n
{\ displaystyle -n}
n
∈
N
0
{\ displaystyle n \ in \ mathbb {N} _ {0}}
Res
-
n
Γ
=
(
-
1
)
n
n
!
{\ displaystyle \ operatorname {Res} _ {- n} \ Gamma = {\ tfrac {(-1) ^ {n}} {n!}}}
Algebraic view
Let there be a body and a connected regular actual curve over . Then there is a canonical map
for every closed point
K
{\ displaystyle K}
X
{\ displaystyle X}
K
{\ displaystyle K}
x
∈
X
{\ displaystyle x \ in X}
res
x
:
Ω
K
(
X
)
/
K
→
K
,
{\ displaystyle \ operatorname {res} _ {x} \ colon \ Omega _ {K (X) / K} \ to K,}
which assigns its residual in to every meromorphic differential form .
x
{\ displaystyle x}
If a -rational point and a local uniformizer, then the residual mapping can be specified explicitly as follows: Is a meromorphic differential form and a local representation, and is
x
{\ displaystyle x}
K
{\ displaystyle K}
t
{\ displaystyle t}
ω
{\ displaystyle \ omega}
ω
=
f
d
t
{\ displaystyle \ omega = f \, \ mathrm {d} t}
f
=
∑
k
=
-
N
∞
a
k
t
k
{\ displaystyle f = \ sum _ {k = -N} ^ {\ infty} a_ {k} t ^ {k}}
the Laurent series of , then applies
f
{\ displaystyle f}
res
x
ω
=
a
-
1
.
{\ displaystyle \ operatorname {res} _ {x} \ omega = a _ {- 1}.}
In particular, the algebraic residual in the case agrees with the function-theoretical one.
K
=
C.
{\ displaystyle K = \ mathbb {C}}
The analog of the residual theorem is correct: For every meromorphic differential form the sum of the residuals is zero:
ω
{\ displaystyle \ omega}
∑
x
∈
X
res
x
ω
=
0.
{\ displaystyle \ sum _ {x \ in X} \ operatorname {res} _ {x} \ omega = 0.}
swell
A construction of the algebraic residual mapping.
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