Goursat's Lemma, also known as the Goursat's Theorem, is a theorem in Complex analysis .
Goursat's lemma is a precursor to the Cauchy's integral theorem and is often used in its proof. It plays an important role in the development of complex analysis. Remarkably, the lemma only requires Complex differentiability but not continuous differentiability. The lemma was proved in its rectangular form by Édouard Goursat (1858 –1936 ) and published in 1884 . The triangular form commonly used today was introduced by Alfred Pringsheim .
Given the following assumptions:
(P1) Let
U
⊆
C
{\displaystyle {U}\subseteq \mathbb {C} }
(P2) Let
z
1
,
z
2
,
z
3
∈
C
{\displaystyle {z}_{1},{z}_{2},{z}_{3}\in \mathbb {C} }
Δ
(
z
1
,
z
2
,
z
3
)
:=
{
∑
k
=
1
3
λ
k
⋅
z
k
∣
(
∑
k
1
3
λ
k
=
1
)
∧
∀
k
∈
{
1
,
2
,
3
}
λ
k
∈
[
0
,
1
]
}
⊂
U
{\displaystyle \Delta {\left({z}_{1},{z}_{2},{z}_{3}\right)}:=\left\{\sum _{k=1}^{3}\lambda _{k}\cdot {z}_{k}{\mid }{\left({\sum _{{k}{1}}^{3}}\lambda _{k}={1}\right)}\wedge \forall {k}\in {\left\lbrace {1},{2},{3}\right\rbrace }\lambda _{k}\in [{0},{1}]\right\}\subset {U}}
(P3) Let
f
:
U
→
C
{\displaystyle {f}:{U}\to \mathbb {C} }
(P4) Let
⟨
z
1
,
z
2
,
z
3
⟩
:
[
0
,
3
]
→
C
{\displaystyle {\left\langle {z}_{1},{z}_{2},{z}_{3}\right\rangle }:{\left[{0},{3}\right]}\to \mathbb {C} }
Δ
(
z
1
,
z
2
,
z
3
)
{\displaystyle \Delta {\left({z}_{1},{z}_{2},{z}_{3}\right)}}
z
1
{\displaystyle {z}_{1}}
then the following statements hold:
(C1)
∫
⟨
z
1
,
z
2
,
z
3
⟩
f
(
z
)
d
z
=
0
{\displaystyle \int _{\left\langle {z}_{1},{z}_{2},{z}_{3}\right\rangle }{f{\left({z}\right)}}{d}{z}={0}}
Integration path along the triangle boundary Subdivision of the outer paths and insertion of additional paths between the midpoints of the sides, which cancel out in the line integral due to the reversed direction of the integration path, resulting in a sum of 0 and leaving the total integral unchanged. Inductive definition of the paths. The subtriangles are similar to the original triangle. By using the midpoints of the sides, the perimeter of a triangle
Δ
(
n
)
{\displaystyle \Delta ^{(n)}}
Δ
(
n
+
1
)
{\displaystyle \Delta ^{(n+1)}}
(S1) We define a sequence of triangular paths recursively as
γ
(
n
)
:=
⟨
z
1
(
n
)
,
z
2
(
n
)
,
z
3
(
n
)
⟩
{\displaystyle \gamma ^{(n)}:={\left\langle z_{1}^{(n)},z_{2}^{(n)},z_{3}^{(n)}\right\rangle }}
(S2) (DEF) For
n
=
0
{\displaystyle {n}={0}}
γ
(
0
)
:
[
0
,
3
]
→
C
{\displaystyle \gamma ^{(0)}:[0,3]\to \mathbb {C} }
γ
(
0
)
(
t
)
:=
⟨
z
1
,
z
2
,
z
3
⟩
(
t
)
:=
{
(
1
−
t
)
⋅
z
1
+
t
⋅
z
2
for
t
∈
[
0
,
1
]
(
2
−
t
)
⋅
z
2
+
(
t
−
1
)
⋅
z
3
for
t
∈
(
1
,
2
]
(
3
−
t
)
⋅
z
3
+
(
t
−
2
)
⋅
z
1
for
t
∈
(
2
,
3
]
{\displaystyle \gamma ^{(0)}(t):=\left\langle z_{1},z_{2},z_{3}\right\rangle (t):={\begin{cases}(1-t)\cdot z_{1}+t\cdot z_{2}&{\text{for }}t\in [0,1]\\(2-t)\cdot z_{2}+(t-1)\cdot z_{3}&{\text{for }}t\in (1,2]\\(3-t)\cdot z_{3}+(t-2)\cdot z_{1}&{\text{for }}t\in (2,3]\\\end{cases}}}
Furthermore, let
γ
(
n
)
{\displaystyle \gamma ^{(n)}}
γ
(
n
+
1
)
{\displaystyle \gamma ^{(n+1)}}
Justification: (P4,UT) (S3) (DEF) Definition: Triangle path
γ
1
(
n
)
:=
⟨
z
1
(
n
)
+
z
2
(
n
)
2
,
z
2
(
n
)
,
z
2
(
n
)
+
z
3
(
n
)
2
⟩
{\displaystyle {\gamma _{1}^{(n)}}:={\left\langle {\frac {z_{1}^{(n)}+z_{2}^{(n)}}{2}},z_{2}^{(n)},{\frac {z_{2}^{(n)}+z_{3}^{(n)}}{2}}\right\rangle }}
Justification: (S3,S4,S5) (S4) (DEF) Definition: Triangle path
γ
2
(
n
)
:=
⟨
z
2
(
n
)
+
z
3
(
n
)
2
,
z
3
(
n
)
,
z
1
(
n
)
+
z
3
(
n
)
2
⟩
{\displaystyle {\gamma _{2}^{(n)}}:={\left\langle {\frac {z_{2}^{(n)}+z_{3}^{(n)}}{2}},z_{3}^{(n)},{\frac {z_{1}^{(n)}+z_{3}^{(n)}}{2}}\right\rangle }}
(S5) (DEF) Definition: Triangle path
γ
3
(
n
)
:=
⟨
z
1
(
n
)
+
z
3
(
n
)
2
,
z
1
(
n
)
,
z
1
(
n
)
+
z
2
(
n
)
2
⟩
{\displaystyle {\gamma _{3}^{(n)}}:={\left\langle {\frac {z_{1}^{(n)}+z_{3}^{(n)}}{2}},z_{1}^{(n)},{\frac {z_{1}^{(n)}+z_{2}^{(n)}}{2}}\right\rangle }}
(S6) (DEF) Definition: Triangle path
γ
4
(
n
)
:=
⟨
z
1
(
n
)
+
z
2
(
n
)
2
,
z
2
(
n
)
+
z
3
(
n
)
2
,
z
1
(
n
)
+
z
3
(
n
)
2
⟩
{\displaystyle {\gamma _{4}^{(n)}}:={\left\langle {\frac {z_{1}^{(n)}+z_{2}^{(n)}}{2}},{\frac {z_{2}^{(n)}+z_{3}^{(n)}}{2}},{\frac {z_{1}^{(n)}+z_{3}^{(n)}}{2}}\right\rangle }}
(S7) (DEF) Let
i
∈
{
1
,
2
,
3
,
4
}
{\displaystyle {i}\in {\left\lbrace {1},{2},{3},{4}\right\rbrace }}
∀
k
∈
{
1
,
,
2
,
3
,
4
}
:
|
∫
γ
k
(
n
)
f
(
z
)
d
z
|
≤
|
∫
γ
i
(
n
)
f
(
z
)
d
z
|
{\displaystyle \forall _{{k}\in {\left\lbrace {1},,{2},{3},{4}\right\rbrace }}:{\left|\int _{\gamma _{k}^{(n)}}{f{\left({z}\right)}}{\left.{d}{z}\right.}\right|}\leq {\left|\int _{\gamma _{i}^{(n)}}{f{\left({z}\right)}}{\left.{d}{z}\right.}\right|}}
γ
(
n
+
1
)
:=
γ
i
(
n
)
{\displaystyle \gamma ^{\left({n}+{1}\right)}:={\gamma _{i}^{(n)}}}
(S8)
⇒
{\displaystyle \Rightarrow }
∫
γ
(
n
)
f
(
z
)
d
z
=
∑
k
=
1
4
∫
γ
k
(
n
)
f
(
z
)
d
z
{\displaystyle \int _{\gamma ^{(n)}}f(z)\,dz=\sum _{k=1}^{4}\int _{\gamma _{k}}^{(n)}f(z)\,dz}
(S9)
⇒
{\displaystyle \Rightarrow }
|
∫
γ
n
f
(
z
)
d
z
|
=
|
∑
k
=
1
4
∫
γ
k
(
n
)
f
(
z
)
d
z
|
≤
∑
k
=
1
4
|
∫
γ
k
(
n
)
f
(
z
)
d
z
|
≤
4
⋅
|
∫
γ
k
(
n
)
f
(
z
)
d
z
|
{\displaystyle {\left|\int _{\gamma ^{n}}{f{\left({z}\right)}}{\left.{d}{z}\right.}\right|}={\left|{\sum _{{k}={1}}^{4}}\int _{\gamma _{k}^{(n)}}{f{\left({z}\right)}}{\left.{d}{z}\right.}\right|}\leq {\sum _{{k}={1}}^{4}}{\left|\int _{\gamma _{k}^{(n)}}{f{\left({z}\right)}}{\left.{d}{z}\right.}\right|}\leq {4}\cdot {\left|\int _{\gamma _{k}^{(n)}}{f{\left({z}\right)}}{\left.{d}{z}\right.}\right|}}
n
∈
N
{\displaystyle {n}\in \mathbb {N} }
Justification: (S7,WG4,DU) (S10)
⇒
{\displaystyle \Rightarrow }
0
≤
|
∫
⟨
z
1
,
z
2
,
z
3
⟩
f
(
z
)
d
z
|
=
|
∫
γ
(
0
)
f
(
z
)
d
z
|
≤
4
⋅
|
∫
γ
(
1
)
f
(
z
)
d
z
|
≤
…
≤
4
n
⋅
|
∫
γ
i
(
n
)
f
(
z
)
d
z
|
=
4
n
⋅
|
∫
γ
(
n
+
1
)
f
(
z
)
d
z
|
{\displaystyle {0}\leq {\left|\int _{\left\langle {z}_{1},{z}_{2},{z}_{3}\right\rangle }{f{\left({z}\right)}}{d}{z}\right|}={\left|\int _{\gamma ^{(0)}}f(z){\left.{d}{z}\right.}\right|}\leq {4}\cdot {\left|\int _{\gamma ^{\left({1}\right)}}{f{\left({z}\right)}}{\left.{d}{z}\right.}\right|}\leq \ldots \leq {4}^{n}\cdot {\left|\int _{\gamma _{i}^{(n)}}{f{\left({z}\right)}}{\left.{d}{z}\right.}\right|}={4}^{n}\cdot {\left|\int _{\gamma ^{\left({n}+{1}\right)}}{f{\left({z}\right)}}{\left.{d}{z}\right.}\right|}}
(S11) The nested definition of the sub-triangles yields for all
n
∈
N
{\displaystyle {n}\in \mathbb {N} }
Δ
(
z
1
(
n
)
,
z
2
(
n
)
,
z
3
(
n
)
)
⊃
Δ
(
z
1
(
n
+
1
)
,
z
2
(
n
+
1
)
,
z
3
(
n
+
1
)
)
{\displaystyle \Delta {\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}\supset \Delta {\left({{z}_{1}^{\left({n}+{1}\right)}},{{z}_{2}^{\left({n}+{1}\right)}},{{z}_{3}^{\left({n}+{1}\right)}}\right)}}
lim
n
→
∞
diam
(
Δ
(
z
1
(
n
)
,
z
2
(
n
)
,
z
3
(
n
)
)
)
=
0
{\displaystyle \lim _{n\to \infty }\,{\text{diam}}\left(\Delta {\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}\right)=0}
(S12)
⇒
{\displaystyle \Rightarrow }
∃
z
0
∈
U
∀
n
∈
N
:
z
0
∈
Δ
(
n
)
:=
Δ
(
z
1
(
n
)
,
z
2
(
n
)
,
z
3
(
n
)
)
{\displaystyle \exists _{{z}_{0}\in {U}}\forall _{{n}\in \mathbb {N} }:{z}_{0}\in \Delta ^{(n)}:=\Delta {\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}}
{
z
0
}
=
⋂
n
∈
N
Δ
(
n
)
{\displaystyle {\left\lbrace {z}_{0}\right\rbrace }=\bigcap _{{n}\in \mathbb {N} }\Delta ^{(n)}}
(S13) We use the holomorphism of
f
{\displaystyle f}
z
0
∈
U
{\displaystyle z_{0}\in U}
f
(
z
)
:=
f
(
z
0
)
+
f
′
(
z
0
)
⋅
(
z
−
z
0
)
+
r
(
z
)
{\displaystyle f(z):=f(z_{0})+f'(z_{0})\cdot (z-z_{0})+r(z)}
lim
z
→
z
0
r
(
z
)
z
−
z
0
=
0
{\displaystyle \lim _{z\to z_{0}}{\frac {r(z)}{z-z_{0}}}=0}
Justification: (P3) (S14)
⇒
{\displaystyle \Rightarrow }
h
:
U
→
C
{\displaystyle {h}:{U}\to \mathbb {C} }
h
(
z
)
:=
f
(
z
0
)
+
f
′
(
z
0
)
⋅
(
z
−
z
0
)
{\displaystyle h(z):=f(z_{0})+f'(z_{0})\cdot (z-z_{0})}
H
(
z
)
:=
f
(
z
0
)
+
f
′
(
z
0
)
⋅
1
2
⋅
(
z
−
z
0
)
2
{\displaystyle H(z):=f(z_{0})+f'(z_{0})\cdot {\frac {1}{2}}\cdot (z-z_{0})^{2}}
Justification: since
h
(
z
)
{\displaystyle h(z)}
(S15)
⇒
{\displaystyle \Rightarrow }
γ
(
n
)
{\displaystyle \gamma ^{(n)}}
h
:
U
→
C
{\displaystyle {h}:{U}\to \mathbb {C} }
∫
γ
k
(
n
)
h
(
z
)
=
0
{\displaystyle \int _{\gamma _{k}^{(n)}}{h}{\left({z}\right)}={0}}
Justification: (SF) (S16)
⇒
{\displaystyle \Rightarrow }
γ
(
n
)
{\displaystyle \gamma ^{(n)}}
f
:
U
→
C
{\displaystyle {f}:{U}\to \mathbb {C} }
∫
γ
k
(
n
)
f
(
z
)
d
z
=
∫
γ
k
(
n
)
h
(
z
)
+
r
(
z
)
d
z
=
∫
γ
k
(
n
)
r
(
z
)
d
z
{\displaystyle \int _{\gamma _{k}^{(n)}}{f{{\left({z}\right)}{\left.{d}{z}\right.}}}=\int _{\gamma _{k}^{(n)}}{h}{\left({z}\right)}+{r}{\left({z}\right)}{\left.{d}{z}\right.}=\int _{\gamma _{k}^{(n)}}{r}{\left({z}\right)}{\left.{d}{z}\right.}}
r
(
z
)
{\displaystyle r(z)}
[ edit | edit source ] (S17)
⇒
{\displaystyle \Rightarrow }
lim
z
→
z
0
r
(
z
)
z
−
z
0
=
0
{\displaystyle \lim _{z\to z_{0}}{\frac {r(z)}{z-z_{0}}}=0}
ϵ
>
0
{\displaystyle \epsilon >{0}}
δ
>
0
{\displaystyle \delta >{0}}
|
z
−
z
0
|
<
δ
⇒
|
r
(
z
)
z
−
z
0
|
<
ϵ
{\displaystyle |z-z_{0}|<\delta \Rightarrow \left|{\frac {r(z)}{z-z_{0}}}\right|<\epsilon }
Justification:
ϵ
{\displaystyle \epsilon }
δ
{\displaystyle \delta }
g
(
z
)
:=
r
(
z
)
z
−
z
0
{\displaystyle g(z):={\frac {r(z)}{z-z_{0}}}}
g
{\displaystyle g}
z
0
{\displaystyle z_{0}}
(S18)
⇒
{\displaystyle \Rightarrow }
ϵ
>
0
{\displaystyle \epsilon >{0}}
δ
>
0
{\displaystyle \delta >0}
|
z
−
z
0
|
<
δ
⇒
|
r
(
z
)
|
<
ϵ
⋅
|
z
−
z
0
|
{\displaystyle |z-z_{0}|<\delta \Rightarrow |r(z)|<\epsilon \cdot |z-z_{0}|}
0
≤
|
∫
⟨
z
1
,
z
2
,
z
3
⟩
f
(
z
)
d
z
|
≤
4
n
⋅
|
∫
γ
k
(
n
)
f
(
z
)
d
z
|
=
4
n
⋅
|
∫
γ
k
(
n
)
r
(
z
)
d
z
|
≤
4
n
⋅
∫
γ
k
(
n
)
|
r
(
z
)
|
d
z
≤
4
n
⋅
∫
γ
k
(
n
)
ϵ
⋅
|
z
−
z
0
|
d
z
{\displaystyle 0\leq \left|\int _{\left\langle z_{1},z_{2},z_{3}\right\rangle }f(z)\,dz\right|\leq 4^{n}\cdot \left|\int _{\gamma _{k}^{(n)}}f(z)\,dz\right|=4^{n}\cdot \left|\int _{\gamma _{k}^{(n)}}r(z)\,dz\right|\leq 4^{n}\cdot \int _{\gamma _{k}^{(n)}}|r(z)|\,dz\leq 4^{n}\cdot \int _{\gamma _{k}^{(n)}}\epsilon \cdot |z-z_{0}|\,dz}
Justification: (S2) (S20) From the condition
lim
n
→
∞
diam
(
Δ
(
n
)
)
=
0
{\displaystyle \lim _{n\to \infty }{\text{diam}}\left(\Delta ^{(n)}\right)=0}
ϵ
>
0
{\displaystyle \epsilon >0}
n
δ
∈
N
{\displaystyle n_{\delta }\in \mathbb {N} }
Δ
(
n
)
⊆
D
δ
(
z
0
)
{\displaystyle \Delta ^{(n)}\subseteq {D}_{\delta }(z_{0})}
n
>
n
δ
{\displaystyle n>n_{\delta }}
(S21)
⇒
{\displaystyle \Rightarrow }
|
z
−
z
0
|
<
L
(
γ
(
n
)
)
=
1
2
n
⋅
L
(
γ
)
{\displaystyle |z-z_{0}|<L\left(\gamma ^{(n)}\right)={\frac {1}{2^{n}}}\cdot L\left(\gamma \right)}
n
∈
N
{\displaystyle n\in \mathbb {N} }
z
∈
Δ
(
n
)
{\displaystyle z\in \Delta ^{(n)}}
Justification: The factor
1
2
n
{\displaystyle {\frac {1}{2^{n}}}}
Δ
(
n
)
{\displaystyle \Delta ^{(n)}}
0
≤
|
∫
⟨
z
1
,
z
2
,
z
3
⟩
f
(
z
)
d
z
|
≤
4
n
⋅
∫
γ
k
(
n
)
ϵ
⋅
|
z
−
z
0
|
d
z
≤
4
n
⋅
ϵ
⋅
∫
γ
k
(
n
)
1
2
n
⋅
L
(
γ
)
d
z
=
4
n
⋅
ϵ
⋅
1
2
n
⋅
L
(
γ
)
≤
∫
γ
k
(
n
)
1
d
z
⏟
L
(
γ
k
(
n
)
)
{\displaystyle 0\leq \left|\int _{\langle z_{1},z_{2},z_{3}\rangle }f(z)\,dz\right|\leq 4^{n}\cdot \int _{\gamma _{k}^{(n)}}\epsilon \cdot |z-z_{0}|\,dz\leq 4^{n}\cdot \epsilon \cdot \int _{\gamma _{k}^{(n)}}{\frac {1}{2^{n}}}\cdot {\mathcal {L}}(\gamma )\,dz=4^{n}\cdot \epsilon \cdot {\frac {1}{2^{n}}}\cdot {\mathcal {L}}(\gamma )\underbrace {\leq \int _{\gamma _{k}^{(n)}}1\,dz} _{{\mathcal {L}}(\gamma _{k}^{(n)})}}
≤
4
n
⋅
ϵ
⋅
1
2
n
⋅
L
(
γ
)
⋅
L
(
γ
k
(
n
)
)
≤
4
n
⋅
ϵ
⋅
L
(
γ
)
4
n
=
ϵ
⋅
L
(
γ
)
{\displaystyle \leq 4^{n}\cdot \epsilon \cdot {\frac {1}{2^{n}}}\cdot L(\gamma )\cdot {\mathcal {L}}(\gamma _{k}^{(n)})\leq 4^{n}\cdot \epsilon \cdot {\frac {{\mathcal {L}}(\gamma )}{4^{n}}}=\epsilon \cdot {\mathcal {L}}(\gamma )}
ϵ
>
0
{\displaystyle \epsilon >0}
Justification: (S19,LIW,IAL) (C1)
⇒
{\displaystyle \Rightarrow }
∫
⟨
z
1
,
z
2
,
z
3
⟩
f
(
z
)
d
z
=
0
{\displaystyle \int _{\left\langle {z}_{1},{z}_{2},{z}_{3}\right\rangle }{f{\left({z}\right)}}{d}{z}={0}}
(DU)
∀
a
,
b
∈
C
:
|
a
+
b
|
≤
|
a
|
+
|
b
|
{\displaystyle \forall _{a,b\in \mathbb {C} }:|a+b|\leq |a|+|b|}
(DI) Definition: Let
M
⊂
C
{\displaystyle M\subset {C}}
diam
(
M
)
:=
sup
{
|
b
−
a
|
:
a
,
b
∈
M
}
{\displaystyle {\text{diam}}(M):={\text{sup}}\lbrace |b-a|\,:\,a,b\in M\rbrace }
(WE) Definition (Path): Let
U
⊆
C
{\displaystyle {U}\subseteq \mathbb {C} }
a
,
b
∈
R
{\displaystyle a,b\in \mathbb {R} }
a
<
b
{\displaystyle a<b}
γ
{\displaystyle \gamma }
U
⊆
C
{\displaystyle U\subseteq \mathbb {C} }
γ
:
[
a
,
b
]
→
U
{\displaystyle \gamma :[a,b]\to U}
(SPU) Definition (Trace): Let
γ
:
[
a
,
b
]
→
U
{\displaystyle \gamma :[a,b]\to U}
U
⊆
C
{\displaystyle {U}\subseteq \mathbb {C} }
γ
{\displaystyle \gamma }
Spur
(
γ
)
:=
{
γ
(
t
)
∈
C
∣
t
∈
[
a
,
b
]
}
{\displaystyle {\text{Spur}}(\gamma ):=\lbrace \gamma (t)\in \mathbb {C} \,{\mid }\,t\in [a,b]\rbrace }
(WZ) Definition (Path-connected): Let
U
⊆
C
{\displaystyle {U}\subseteq \mathbb {C} }
U
{\displaystyle {U}}
γ
:
[
a
,
b
]
→
U
{\displaystyle \gamma :[a,b]\to U}
U
⊆
C
{\displaystyle U\subseteq \mathbb {C} }
γ
(
a
)
=
z
1
{\displaystyle \gamma (a)=z_{1}}
γ
(
b
)
=
z
2
{\displaystyle \gamma (b)=z_{2}}
Spur
(
γ
)
⊆
U
{\displaystyle {\text{Spur}}(\gamma )\subseteq U}
(GE) Definition (Domain): A subset
G
⊆
C
{\displaystyle {G}\subseteq \mathbb {C} }
G
{\displaystyle {G}}
G
≠
∅
{\displaystyle {G}\neq \emptyset }
G
{\displaystyle {G}}
(WG1) Definition (Smooth path): A path
γ
:
[
a
,
b
]
→
C
{\displaystyle \gamma :[a,b]\to \mathbb {C} }
(UT) Definition (Subdivision): Let
[
a
,
b
]
{\displaystyle [a,b]}
n
∈
N
{\displaystyle n\in \mathbb {N} }
a
=
u
0
<
…
<
u
n
=
b
{\displaystyle {a}={u}_{0}<{\ldots }<{u}_{n}={b}}
(
u
0
,
…
,
u
n
)
∈
R
n
+
1
{\displaystyle {\left({u}_{0},\ldots ,{u}_{n}\right)}\in \mathbb {R} ^{n+1}}
[
a
,
b
]
{\displaystyle {\left[{a},{b}\right]}}
(WG2) Definition (Path subdivision): Let
γ
:
[
a
,
b
]
→
C
{\displaystyle \gamma :[a,b]\to \mathbb {C} }
U
⊆
C
{\displaystyle {U}\subseteq \mathbb {C} }
n
∈
N
{\displaystyle {n}\in \mathbb {N} }
(
u
0
,
…
,
u
n
)
{\displaystyle {\left({u}_{0},\ldots ,{u}_{n}\right)}}
[
a
,
b
]
{\displaystyle [a,b]}
γ
k
:
[
u
k
−
1
,
u
k
]
→
C
{\displaystyle \gamma _{k}:{\left[{u}_{{k}-{1}},{u}_{k}\right]}\to \mathbb {C} }
k
∈
{
1
,
…
,
n
}
{\displaystyle {k}\in {\left\lbrace {1},\ldots ,{n}\right\rbrace }}
U
{\displaystyle {U}}
(
γ
1
,
…
,
γ
n
)
{\displaystyle {\left(\gamma _{1},\ldots ,\gamma _{n}\right)}}
γ
{\displaystyle \gamma }
γ
n
(
b
)
=
γ
(
b
)
{\displaystyle \gamma _{n}{\left({b}\right)}=\gamma {\left({b}\right)}}
∀
k
∈
{
1
,
…
,
n
}
∀
t
∈
[
u
k
−
1
,
u
k
)
:
γ
k
(
t
)
=
γ
(
t
)
∧
γ
k
(
u
k
−
1
)
=
γ
k
−
1
(
u
k
)
{\displaystyle \forall _{{k}\in {\left\lbrace {1},\ldots ,{n}\right\rbrace }}\forall _{{t}\in {\left[{u}_{{k}-{1}},{u}_{k}\right)}}:\gamma _{k}{\left({t}\right)}=\gamma {\left({t}\right)}\wedge \gamma _{k}{\left({u}_{{k}-{1}}\right)}=\gamma _{{k}-{1}}{\left({u}_{k}\right)}}
(WG3) Definition (Piecewise smooth path): A path
γ
:
[
a
,
b
]
→
C
{\displaystyle \gamma :{\left[{a},{b}\right]}\to \mathbb {C} }
(
γ
1
,
…
γ
n
)
{\displaystyle {\left(\gamma _{1},\ldots \gamma _{n}\right)}}
γ
{\displaystyle \gamma }
γ
k
{\displaystyle \gamma _{k}}
k
∈
{
1
,
…
,
n
}
{\displaystyle {k}\in {\left\lbrace {1},\ldots ,{n}\right\rbrace }}
(WG4) Definition (Path integral): Let
f
:
U
→
C
{\displaystyle f:U\to \mathbb {C} }
γ
:
[
a
,
b
]
→
U
{\displaystyle \gamma :[a,b]\to U}
∫
γ
f
:=
∫
γ
f
(
z
)
d
z
:=
∫
a
b
f
(
γ
(
t
)
)
⋅
γ
′
(
t
)
d
t
{\displaystyle \int _{\gamma }f:=\int _{\gamma }f(z)\,dz:=\int _{a}^{b}f(\gamma (t))\cdot \gamma '(t)\,dt}
γ
{\displaystyle \gamma }
(
γ
1
,
…
,
γ
n
)
{\displaystyle (\gamma _{1},\ldots ,\gamma _{n})}
∫
γ
f
(
z
)
d
z
:=
∑
k
=
1
n
∫
γ
k
f
(
z
)
d
z
{\displaystyle \int _{\gamma }f(z)\,dz:=\sum _{k=1}^{n}\int _{\gamma _{k}}f(z)\,dz}
(SF) Theorem (Primitive with closed paths): If a continuous function
f
:
U
→
C
{\displaystyle f:U\to \mathbb {C} }
F
:
U
→
C
{\displaystyle F:U\to \mathbb {C} }
γ
:
[
a
,
b
]
→
U
{\displaystyle \gamma :[a,b]\to U}
∫
γ
f
(
z
)
d
z
=
F
(
b
)
−
F
(
a
)
{\displaystyle \int _{\gamma }f(z)\,dz=F(b)-F(a)}
(LIW) Length of the integration path: Let
γ
:
[
a
,
b
]
→
C
{\displaystyle \gamma :{\left[{a},{b}\right]}\to \mathbb {C} }
L
(
γ
)
{\displaystyle {\mathcal {L}}(\gamma )}
L
(
γ
)
:=
∫
a
b
|
γ
′
(
t
)
|
d
t
{\displaystyle {\mathcal {L}}(\gamma ):=\int _{a}^{b}|\gamma '(t)|\,dt}
If
γ
:
[
a
,
b
]
→
C
{\displaystyle \gamma :{\left[{a},{b}\right]}\to \mathbb {C} }
(
γ
1
,
…
γ
n
)
{\displaystyle {\left(\gamma _{1},\ldots \gamma _{n}\right)}}
γ
k
{\displaystyle \gamma _{k}}
L
(
γ
)
{\displaystyle {\mathcal {L}}(\gamma )}
γ
k
{\displaystyle \gamma _{k}}
L
(
γ
)
:=
∑
k
=
1
n
L
(
γ
k
)
{\displaystyle {\mathcal {L}}(\gamma ):=\sum _{k=1}^{n}{\mathcal {L}}(\gamma _{k})}
(IAL) Integral estimate over the length of the integration path: Let
γ
:
[
a
,
b
]
→
G
{\displaystyle \gamma :{\left[{a},{b}\right]}\to \mathbb {G} }
G
⊆
C
{\displaystyle G\subseteq \mathbb {C} }
f
{\displaystyle f}
Spur
(
γ
)
{\displaystyle {\text{Spur}}(\gamma )}
|
∫
γ
f
(
z
)
d
z
|
≤
max
z
∈
Spur
(
γ
)
|
f
(
z
)
|
⋅
L
(
γ
)
{\displaystyle \left|\int _{\gamma }f(z)\,dz\right|\leq \max _{z\in {\text{Spur}}(\gamma )}|f(z)|\cdot {\mathcal {L}}(\gamma )}
Eberhard Freitag & Rolf Busam: Funktionentheorie 1 , Springer-Verlag, Berlin
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![{\displaystyle \Delta {\left({z}_{1},{z}_{2},{z}_{3}\right)}:=\left\{\sum _{k=1}^{3}\lambda _{k}\cdot {z}_{k}{\mid }{\left({\sum _{{k}{1}}^{3}}\lambda _{k}={1}\right)}\wedge \forall {k}\in {\left\lbrace {1},{2},{3}\right\rbrace }\lambda _{k}\in [{0},{1}]\right\}\subset {U}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f78e379c02101d5c04e8b465df2d10b6acbc878)
![{\displaystyle {\left\langle {z}_{1},{z}_{2},{z}_{3}\right\rangle }:{\left[{0},{3}\right]}\to \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/127c097c04d4a02f73f0f5529751284899cec282)
![{\displaystyle \gamma ^{(0)}:[0,3]\to \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5082ba01da5dfb93d129dba1301a0252c144158)
![{\displaystyle \gamma ^{(0)}(t):=\left\langle z_{1},z_{2},z_{3}\right\rangle (t):={\begin{cases}(1-t)\cdot z_{1}+t\cdot z_{2}&{\text{for }}t\in [0,1]\\(2-t)\cdot z_{2}+(t-1)\cdot z_{3}&{\text{for }}t\in (1,2]\\(3-t)\cdot z_{3}+(t-2)\cdot z_{1}&{\text{for }}t\in (2,3]\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba04b29c64aa2f78cc75d33216432f06b59debf1)
![{\displaystyle \gamma :[a,b]\to U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d032547cae79cddd62ad7e267d9073b29a5a89d0)
![{\displaystyle {\text{Spur}}(\gamma ):=\lbrace \gamma (t)\in \mathbb {C} \,{\mid }\,t\in [a,b]\rbrace }](https://wikimedia.org/api/rest_v1/media/math/render/svg/99742c04ca75ecf8df6e428216f829ee5973716f)
![{\displaystyle \gamma :[a,b]\to \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba8e67ece48ed58462781818669d29170ae4809)
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle {\left[{a},{b}\right]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/218c5a550234daaeebb7fe581a9d9c5d8ccfe692)
![{\displaystyle \gamma _{k}:{\left[{u}_{{k}-{1}},{u}_{k}\right]}\to \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e68d17150031bf3855b5789c04bb6eca3fd166c3)
![{\displaystyle \gamma :{\left[{a},{b}\right]}\to \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/54fd4de339dd0c75918fcb2fa4ca67e88ce1bf7c)
![{\displaystyle \gamma :{\left[{a},{b}\right]}\to \mathbb {G} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7445c5302ec68ba931509d8b346990c7345c22d9)
