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Formelsammlung Mathematik: Bestimmte Integrale: Form R(x,Ci)

1周前 (05-04) 5次浏览

2.1Bearbeiten
{displaystyle int _{0}^{infty }{text{Ci}}(ax),{text{Ci}}(bx),dx={frac {1}{max{a,b}}}cdot {frac {pi }{2}}qquad a,b>0}
Beweis

In der Formel

{displaystyle int {text{Ci}}(ax),{text{Ci}}(bx),dx=x,{text{Ci}}(ax),{text{Ci}}(bx)-{frac {sin ax}{a}},{text{Ci}}(bx)-{frac {sin bx}{b}},{text{Ci}}(ax)+{frac {1}{2a}}{Big (}{text{Si}}(ax+bx)+{text{Si}}(ax-bx){Big )}+{frac {1}{2b}}{Big (}{text{Si}}(ax+bx)-{text{Si}}(ax-bx){Big )}}

setze {displaystyle 0,} und {displaystyle infty } als Integrationsgrenzen ein.

Asymptotisch verhalten sich {displaystyle {text{Ci}}(ax)} und {displaystyle {text{Ci}}(bx)} für {displaystyle xto 0+} wie {displaystyle log x} und für {displaystyle xto infty ,} wie {displaystyle {frac {cos x}{x}}}.

Also sind {displaystyle {Big [}x,{text{Ci}}(ax),{text{Ci}}(bx){Big ]}_{0}^{infty },,,,,{Big [}{frac {sin ax}{a}},{text{Ci}}(bx){Big ]}_{0}^{infty },,,,,{Big [}{frac {sin bx}{b}},{text{Ci}}(ax){Big ]}_{0}^{infty }} jeweils gleich {displaystyle 0-0=0}.

Der übrige Term {displaystyle left[{frac {1}{2a}}{Big (}{text{Si}}(ax+bx)+{text{Si}}(ax-bx){Big )}+{frac {1}{2b}}{Big (}{text{Si}}(ax+bx)-{text{Si}}(ax-bx){Big )}right]_{0}^{infty }} verschwindet für {displaystyle x=0}.

Für {displaystyle xto infty } geht der Term gegen

{displaystyle bullet quad {frac {1}{2a}}left({frac {pi }{2}}+{frac {pi }{2}}right)+{frac {1}{2b}}left({frac {pi }{2}}-{frac {pi }{2}}right)={frac {1}{a}}cdot {frac {pi }{2}}} falls {displaystyle a>b}.

{displaystyle bullet quad {frac {1}{2a}}left({frac {pi }{2}}+0right)+{frac {1}{2b}}left({frac {pi }{2}}+0right)={frac {1}{a}}cdot {frac {pi }{2}}={frac {1}{b}}cdot {frac {pi }{2}}} falls {displaystyle a=b}.

{displaystyle bullet quad {frac {1}{2a}}left({frac {pi }{2}}-{frac {pi }{2}}right)+{frac {1}{2b}}left({frac {pi }{2}}+{frac {pi }{2}}right)={frac {1}{b}}cdot {frac {pi }{2}}} falls {displaystyle a<b}.