Ungleichungen sind ein wesentliches Hilfsmittel um zentrale Aussage in der Funktionentheorie zu zeigen. Da
C
{\displaystyle \mathbb {C} }
vollständige/totale Ordnung besitzt, muss man für Abschätzungen auf den Betrag der Funktionen übergehen.
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{\displaystyle f:[a,b]\rightarrow \mathbb {C} }
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R
{\displaystyle f_{1}:[a,b]\rightarrow \mathbb {R} }
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{\displaystyle f_{2}:[a,b]\rightarrow \mathbb {R} }
f
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{\displaystyle f=f_{1}+i\cdot f_{2}}
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{\displaystyle \left|\int _{a}^{b}f(t)\,dt\right|\leq \int _{a}^{b}|f_{1}(t)|\,dt+\int _{a}^{b}|f_{2}(t)|\,dt}
Führen Sie den Beweis für UG.RI aus. Der Beweis erfolgt durch Zerlegung in Realteilfunktion und Imaginärteilfunktion, Linearität des Integrals und der Anwendung der Dreiecksungleichung.
Bearbeiten
Sei
f
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{\displaystyle f:[a,b]\rightarrow \mathbb {C} }
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{\displaystyle \left|\int _{a}^{b}f(t)\,dt\right|\leq \int _{a}^{b}|f(t)|\,dt}
Der Beweis wird über eine Fallunterscheidung geführt mit:
(BI-1)
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{\displaystyle \int _{a}^{b}f(t)\,dt=0}
(BI-2)
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{\displaystyle \int _{a}^{b}f(t)\,dt\not =0}
Mit
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{\displaystyle \int _{a}^{b}f(t)\,dt=0}
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{\displaystyle \left|\int _{a}^{b}f(t)\,dt\right|=0}
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{\displaystyle |f(t)|\geq 0}
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{\displaystyle \int _{a}^{b}|f(t)|\,dt\geq 0}
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{\displaystyle \left|\int _{a}^{b}f(t)\,dt\right|=0\leq \int _{a}^{b}|f(t)|\,dt}
Das Integral
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{\displaystyle \beta =\int _{a}^{b}f(t)\,dt\in \mathbb {C} }
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{\displaystyle \beta \not =0}
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β
¯
{\displaystyle |\beta |={\sqrt {\beta \cdot {\overline {\beta }}}}}
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{\displaystyle |\beta |={\frac {|\beta |^{2}}{|\beta |}}={\frac {\beta \cdot {\overline {\beta }}}{|\beta |}}=\underbrace {\frac {\overline {\beta }}{|\beta |}} _{\alpha :=}\cdot \beta =\alpha \cdot \beta }
Mit
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{\displaystyle \beta \not =0}
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{\displaystyle |\beta |=\alpha \cdot \beta =\alpha \cdot \int _{a}^{b}f(t)\,dt=\int _{a}^{b}\alpha \cdot f(t)\,dt}
Sei
g
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⋅
f
{\displaystyle g:=\alpha \cdot f}
g
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→
C
{\displaystyle g:[a,b]\rightarrow \mathbb {C} }
g
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R
{\displaystyle g_{1}:[a,b]\rightarrow \mathbb {R} }
g
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R
{\displaystyle g_{2}:[a,b]\rightarrow \mathbb {R} }
g
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{\displaystyle g=g_{1}+i\cdot g_{2}}
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{\displaystyle {\begin{array}{rcl}{\mathfrak {Re}}{\bigg (}\int _{a}^{b}g(t)\,dt{\bigg )}&=&{\mathfrak {Re}}{\bigg (}\underbrace {\int _{a}^{b}g_{1}(t)\,dt} _{\in \mathbb {R} }+i\cdot \underbrace {\int _{a}^{b}g_{2}(t)\,dt} _{\in \mathbb {R} }{\bigg )}\\&=&{\mathfrak {Re}}{\bigg (}\underbrace {\int _{a}^{b}g_{1}(t)\,dt} _{\in \mathbb {R} }{\bigg )}=\int _{a}^{b}g_{1}(t)\,dt\\&=&\int _{a}^{b}{\mathfrak {Re}}(g_{1}(t))\,dt\\\end{array}}}
Weil
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{\displaystyle |\beta |=\int _{a}^{b}\alpha \cdot f(t)\,dt\in \mathbb {R} }
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{\displaystyle |\beta |={\mathfrak {Re}}{\bigg (}\underbrace {\int _{a}^{b}\alpha \cdot f(t)\,dt} _{\in \mathbb {R} }{\bigg )}=\int _{a}^{b}\underbrace {{\mathfrak {Re}}\left(\alpha \cdot f(t)\right)} _{\in \mathbb {R} }\,dt}
Die folgende Realteilabschätzung gegen den Betrag einer komplexen Zahl
z
{\displaystyle z}
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{\displaystyle {\mathfrak {Re}}(z)=z_{1}\leq |z_{1}|={\sqrt {z_{1}^{2}}}\leq {\sqrt {z_{1}^{2}+z_{2}^{2}}}=|z|}
für
z
=
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+
i
⋅
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2
{\displaystyle z=z_{1}+i\cdot z_{2}}
R
e
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⏟
∈
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{\displaystyle \underbrace {{\mathfrak {Re}}\left(\alpha \cdot f(t)\right)} _{\in \mathbb {R} }}
Die folgende Abschätzung ergibt analog zu Schritt 5 über die Linearität des Integrals
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{\displaystyle |\beta |=\int _{a}^{b}{\mathfrak {Re}}\left(\alpha \cdot f(t)\right)\,dt\leq \int _{a}^{b}\left|\alpha \cdot f(t)\right|\,dt=|\alpha |\cdot \int _{a}^{b}\left|f(t)\right|\,dt}
Da
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{\displaystyle |\alpha |=\left|{\frac {\overline {\beta }}{|\beta |}}\right|={\frac {|{\overline {\beta }}|}{|\beta |}}=1}
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{\displaystyle \left|\int _{a}^{b}f(t)\,dt\right|=\alpha \cdot \int _{a}^{b}|f(t)|\,dt\leq |\alpha |\cdot \int _{a}^{b}|f(t)|\,dt=\int _{a}^{b}|f(t)|\,dt}
q.e.d.
Bearbeiten
Sei
γ
:
[
a
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b
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→
C
{\displaystyle \gamma :[a,b]\rightarrow \mathbb {C} }
Integrationsweg und
f
:
U
→
C
{\displaystyle f:U\rightarrow \mathbb {C} }
γ
{\displaystyle \gamma }
S
p
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r
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γ
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:=
{
γ
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:
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[
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}
⊂
U
{\displaystyle Spur(\gamma ):=\{\gamma (t)\,:\,t\in [a,b]\}\subset U}
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{\displaystyle \left|\int _{\gamma }f(z)\,dz\right|\leq \max _{z\in Spur(\gamma )}|f(z)|\cdot {\mathcal {L}}(\gamma )}
Dabei ist
L
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)
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{\displaystyle {\mathcal {L}}(\gamma )=\int _{a}^{b}|\gamma '(t)|\,dt}
Durch Verwendung der obigen Abschätzung nach oben durch
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{\displaystyle \left|f(z)\,dz\right|\leq \max _{z\in Spur(\gamma )}|f(z)|}
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U
G
−
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M
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{\displaystyle {\begin{array}{rcl}\displaystyle \left|\int _{\gamma }f(z)\,dz\right|&{\stackrel {UG-BI}{\leq }}&\displaystyle \int _{a}^{b}\left|f(\gamma (t))\cdot \gamma {\,}'(t)\right|\,dt=\int _{a}^{b}\left|f(\gamma (t))\right|\cdot \left|\gamma {\,}'(t)\right|\,dt\\&\leq &\displaystyle \int _{a}^{b}\underbrace {\max _{z\in Spur(\gamma )}|f(z)|} _{M:=}\cdot |\gamma {\,}'(t)|\,dz=M\cdot \int _{\gamma }|\gamma {\,}'(t)|\,dz\\&=&\displaystyle M\cdot {\mathcal {L}}(\gamma )\\\end{array}}}
↑ Funktionentheorie, Fischer, W., Lieb, W. (1988) Vieweg, S. 37
![{\displaystyle f:[a,b]\rightarrow \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/808e07bf306bb74c974003e5e32cd5dc306fd093)
![{\displaystyle f_{1}:[a,b]\rightarrow \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df0b18b6ad6edaf3ecd6b3b7535c236f05a1b849)
![{\displaystyle f_{2}:[a,b]\rightarrow \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/195234869081ea96612e5c856d95934b8d7454e6)
![{\displaystyle g:[a,b]\rightarrow \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d83fdf44c36d1d1b29c40fcb8099b2ef2958be3e)
![{\displaystyle g_{1}:[a,b]\rightarrow \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd83219d7e4faefa8fce23877bcc8f05bc29d902)
![{\displaystyle g_{2}:[a,b]\rightarrow \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f7ae99967030ebd812fe113ed68a87b801fed23)
![{\displaystyle \gamma :[a,b]\rightarrow \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/90453f67f4a2b31d84e5f1a2a705a23d2049d4f4)
![{\displaystyle Spur(\gamma ):=\{\gamma (t)\,:\,t\in [a,b]\}\subset U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6200c6cf0c599b1deeac094332c9d53622a2fd14)
