Smooth paths and path subdivision
edit
The following definitions were abbreviated with acronyms and are used as justifications for transformations or conclusions in proofs.
(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
}
{\displaystyle {k}\in {\left\lbrace {1},\ldots ,{n}\right\rbrace }}
t
∈
[
u
k
−
1
,
u
k
]
{\displaystyle {t}\in {\left[{u}_{{k}-{1}},{u}_{k}\right]}}
γ
k
(
t
)
=
γ
(
t
)
∧
γ
k
(
u
k
−
1
)
=
γ
k
−
1
(
u
k
)
{\displaystyle \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}
Definition (Integration path): An integration path is a piecewise smooth (piecewise continuously differentiable) path.
The following path is piecewise continuously differentiable (smooth) and for the vertices
z
1
,
z
2
,
z
3
∈
Spur
(
γ
)
{\displaystyle z_{1},z_{2},z_{3}\in {\text{Spur}}(\gamma )}
γ
:
[
0
,
3
]
→
C
{\displaystyle \gamma :[0,3]\to \mathbb {C} }
[
0
,
3
]
{\displaystyle [0,3]}
γ
(
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 (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}}}
Paths from convex combinations
edit
The piecewise continuously differentiable path is formed from convex combination .The sub-paths
γ
1
:=
⟨
z
1
,
z
2
⟩
{\displaystyle \gamma _{1}:=\left\langle z_{1},z_{2}\right\rangle }
γ
1
:
[
0
,
1
]
→
C
,
(
1
−
t
)
⋅
z
1
+
t
⋅
z
2
{\displaystyle \gamma _{1}:[0,1]\to \mathbb {C} ,\ (1-t)\cdot z_{1}+t\cdot z_{2}}
γ
2
:=
⟨
z
2
,
z
3
⟩
{\displaystyle \gamma _{2}:=\left\langle z_{2},z_{3}\right\rangle }
γ
2
:
[
1
,
2
]
→
C
,
(
2
−
t
)
⋅
z
2
+
(
t
−
1
)
⋅
z
3
{\displaystyle \gamma _{2}:[1,2]\to \mathbb {C} ,\ (2-t)\cdot z_{2}+(t-1)\cdot z_{3}}
γ
3
:=
⟨
z
3
,
z
1
⟩
{\displaystyle \gamma _{3}:=\left\langle z_{3},z_{1}\right\rangle }
γ
3
:
[
2
,
3
]
→
C
,
(
3
−
t
)
⋅
z
3
+
(
t
−
2
)
⋅
z
1
{\displaystyle \gamma _{3}:[2,3]\to \mathbb {C} ,\ (3-t)\cdot z_{3}+(t-2)\cdot z_{1}}
are continuously differentiable.
![{\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 {t}\in {\left[{u}_{{k}-{1}},{u}_{k}\right]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65b609fcbc93b5afc68b7dbdedbbe0076acaf0a1)
![{\displaystyle \gamma :{\left[{a},{b}\right]}\to \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/54fd4de339dd0c75918fcb2fa4ca67e88ce1bf7c)
![{\displaystyle \gamma :[a,b]\to U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d032547cae79cddd62ad7e267d9073b29a5a89d0)
![{\displaystyle \gamma :[0,3]\to \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ae39234b02b016151a7d879b74d236d7765bd0)
![{\displaystyle [0,3]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c9e70f7d437509d4ebedb0eaf7ada946e91a79)
![{\displaystyle \gamma (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/eefa3ab713229c860bb4a6dcbd5087378df38157)
![{\displaystyle \gamma _{1}:[0,1]\to \mathbb {C} ,\ (1-t)\cdot z_{1}+t\cdot z_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a93d43a4f0592b505c774b9cf7cfe49cd8a483)
![{\displaystyle \gamma _{2}:[1,2]\to \mathbb {C} ,\ (2-t)\cdot z_{2}+(t-1)\cdot z_{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1315423bd28ff002e4c1ab6012f6513915440ac5)
![{\displaystyle \gamma _{3}:[2,3]\to \mathbb {C} ,\ (3-t)\cdot z_{3}+(t-2)\cdot z_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b77d2d72edc1c1b9a960b9cc82be6e886e67d8bd)
