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

2.1Bearbeiten

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

Beweis

In der Formel

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

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

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

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

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

Für {displaystyle x o infty } geht der Term gegen

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

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

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