Given a subset
U
⊂
C
{\displaystyle U\subset \mathbb {C} }
U
{\displaystyle U}
γ
:
[
a
,
b
]
→
U
{\displaystyle \gamma \colon [a,b]\rightarrow U}
a
<
b
{\displaystyle a<b}
a
,
b
∈
R
{\displaystyle a,b\in \mathbb {R} }
The Trace of a path
γ
:
[
a
,
b
]
→
U
{\displaystyle \gamma \colon [a,b]\rightarrow U}
U
⊂
C
{\displaystyle U\subset \mathbb {C} }
γ
{\displaystyle \gamma }
S
p
u
r
(
γ
)
:=
{
γ
(
t
)
∈
C
|
t
∈
[
a
,
b
]
}
{\displaystyle \mathrm {Spur} (\gamma ):=\{\gamma (t)\in \mathbb {C} \ |\ t\in [a,b]\}}
There is a way
γ
:
[
a
,
b
]
→
U
{\displaystyle \gamma \colon [a,b]\rightarrow U}
U
⊂
C
{\displaystyle U\subset \mathbb {C} }
γ
{\displaystyle \gamma }
γ
(
a
)
=
γ
(
b
)
{\displaystyle \gamma (a)=\gamma (b)}
Be
U
⊂
C
{\displaystyle U\subset \mathbb {C} }
C
{\displaystyle \mathbb {C} }
U
{\displaystyle U}
Be
U
⊂
C
{\displaystyle U\subset \mathbb {C} }
U
{\displaystyle U}
:⟺
∀
z
1
,
z
2
∈
U
∃
γ
:
[
a
,
b
]
→
U
:
γ
(
a
)
=
z
1
∧
γ
(
b
)
=
z
2
∧
S
p
u
r
(
γ
)
⊆
U
{\displaystyle :\Longleftrightarrow \ \forall _{z_{1},z_{2}\in U}\exists _{\gamma \colon [a,b]\rightarrow U}:\ \gamma (a)=z_{1}\wedge \gamma (b)=z_{2}\wedge Spur(\gamma )\subseteq U}
Be
G
⊂
C
{\displaystyle G\subset \mathbb {C} }
C
{\displaystyle \mathbb {C} }
G
{\displaystyle G}
G
{\displaystyle G}
than you call
G
{\displaystyle G}
C
{\displaystyle \mathbb {C} }
Let
z
o
∈
C
{\displaystyle z_{o}\in \mathbb {C} }
r
>
0
{\displaystyle r>0}
γ
z
o
,
r
:
[
0
,
2
π
]
→
C
{\displaystyle \gamma _{z_{o},r}\colon [0,2\pi ]\rightarrow \mathbb {C} }
z
o
∈
C
{\displaystyle z_{o}\in \mathbb {C} }
γ
z
o
,
r
(
t
)
:=
z
o
+
r
⋅
e
i
⋅
t
{\displaystyle \gamma _{z_{o},r}(t):=z_{o}+r\cdot e^{i\cdot t}}
Let
z
o
∈
C
{\displaystyle z_{o}\in \mathbb {C} }
a
,
b
>
0
{\displaystyle a,b>0}
γ
z
o
,
a
,
b
:
[
0
,
2
π
]
→
C
{\displaystyle \gamma _{z_{o},a,b}\colon [0,2\pi ]\rightarrow \mathbb {C} }
z
o
∈
C
{\displaystyle z_{o}\in \mathbb {C} }
γ
z
o
,
a
,
b
(
t
)
:=
z
o
+
a
⋅
cos
(
t
)
+
i
⋅
b
⋅
sin
(
t
)
{\displaystyle \gamma _{z_{o},a,b}(t):=z_{o}+a\cdot \cos(t)+i\cdot b\cdot \sin(t)}
Let
z
1
,
z
2
∈
C
{\displaystyle z_{1},z_{2}\in \mathbb {C} }
t
∈
[
0
,
1
]
{\displaystyle t\in [0,1]}
γ
z
1
,
z
2
:
[
0
,
1
]
→
C
{\displaystyle \gamma _{z_{1},z_{2}}\colon [0,1]\rightarrow \mathbb {C} }
z
1
,
z
2
∈
C
{\displaystyle z_{1},z_{2}\in \mathbb {C} }
γ
z
1
,
z
2
(
t
)
:=
(
1
−
t
)
⋅
z
1
+
t
⋅
z
2
{\displaystyle \gamma _{z_{1},z2}(t):=(1-t)\cdot z_{1}+t\cdot z_{2}}
Such a path is called a convex combination of the first order (see also Higher-Order Convex Combinations ).
Animation of a Convex Combination of Two Vectors as Mapping [ edit | edit source ] Convex Combination as Mapping in an Animated GIF Let
G
⊂
C
{\displaystyle G\subset \mathbb {C} }
G
{\displaystyle G}
γ
:
[
a
,
b
]
→
U
{\displaystyle \gamma \colon [a,b]\rightarrow U}
a
<
b
{\displaystyle a<b}
a
,
b
∈
R
{\displaystyle a,b\in \mathbb {R} }
An integration path can, for example, be expressed piecewise as convex combinations between multiple points
z
1
,
…
z
n
∈
C
{\displaystyle z_{1},\ldots z_{n}\in \mathbb {C} }
z
1
,
…
z
n
∈
C
{\displaystyle z_{1},\ldots z_{n}\in \mathbb {C} }
Ellipse
Convex Combination
Paths in Topological Vector Spaces
This learning resource can be presented as a Wiki2Reveal Slide Set
This Wiki2Reveal Slide Set Course: Function Theory Wiki2Reveal Slides was generated with the Wiki2Reveal Link Generator .
![{\displaystyle \gamma \colon [a,b]\rightarrow U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e64ae78f386a9c5e428536d26cbc1d52f9bd7082)
![{\displaystyle \mathrm {Spur} (\gamma ):=\{\gamma (t)\in \mathbb {C} \ |\ t\in [a,b]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f9495134d22b1aba136f6d4a877c83854ec5acc)
![{\displaystyle :\Longleftrightarrow \ \forall _{z_{1},z_{2}\in U}\exists _{\gamma \colon [a,b]\rightarrow U}:\ \gamma (a)=z_{1}\wedge \gamma (b)=z_{2}\wedge Spur(\gamma )\subseteq U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63e24363688fd1b4d24fc1461b12ce881b93c21e)
![{\displaystyle \gamma _{z_{o},r}\colon [0,2\pi ]\rightarrow \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ea69dae2350897859f84f440a9916973f02adb5)
![{\displaystyle \gamma _{z_{o},a,b}\colon [0,2\pi ]\rightarrow \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f36d881b6f2e490ed3696eea651cdbe5f8d1b4)
![{\displaystyle t\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31a5c18739ff04858eecc8fec2f53912c348e0e5)
![{\displaystyle \gamma _{z_{1},z_{2}}\colon [0,1]\rightarrow \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d17989c4d0e56a79611d406f2a685114de84641)
