Parameterintegral mit Singularitäten [ ] Sei
F
(
ϵ
)
=
∫
r
0
(
ϵ
)
r
1
(
ϵ
)
d
r
r
(
r
−
r
2
(
ϵ
)
)
(
r
−
r
0
(
ϵ
)
)
(
r
1
(
ϵ
)
−
r
)
{\displaystyle F(\epsilon)=\int\limits_{r_0(\epsilon)}^{r_1(\epsilon)}\frac{dr} {\sqrt{r(r-r_2(\epsilon))(r-r_0(\epsilon))(r_1(\epsilon)-r)}}
}
wobei
0
≤
r
2
(
ϵ
)
<
r
0
(
ϵ
)
<
r
1
(
ϵ
)
{\displaystyle 0\leq r_2(\epsilon)<r_0(\epsilon)<r_1(\epsilon)}
r
0
{\displaystyle r_0\,}
r
1
{\displaystyle r_1\,}
r
2
{\displaystyle r_2\,}
0
{\displaystyle 0\,}
r
2
(
0
)
=
0
{\displaystyle r_2(0)=0}
Berechne
F
(
0
)
{\displaystyle F(0)\,}
F
′
(
0
)
{\displaystyle F'(0)\,}
Tipps [ ] Bearbeite die Aufgabe Allgemeines Parameterintegral mit Singularitaeten
∫
d
r
r
(
r
−
r
0
)
(
r
1
−
r
)
=
2
r
0
r
1
arctan
(
r
−
r
0
)
r
1
(
r
1
−
r
)
r
0
{\displaystyle \int\frac{dr}{r\sqrt{(r-r_0)(r_1-r)}}=\frac2{\sqrt{r_0r_1}}\arctan\sqrt{\frac{(r-r_0)r_1}{(r_1-r)r_0}}}
r
0
<
r
<
r
1
{\displaystyle r_0<r<r_1\,}
∫
d
r
r
2
(
r
−
r
0
)
(
r
1
−
r
)
=
(
r
−
r
0
)
(
r
1
−
r
)
r
0
r
1
r
+
r
0
+
r
1
(
r
0
r
1
)
3
/
2
arctan
(
r
−
r
0
)
r
1
(
r
1
−
r
)
r
0
{\displaystyle \int\frac{dr}{r^2\sqrt{(r-r_0)(r_1-r)}}=\frac{\sqrt{(r-r_0)(r_1-r)}}{r_0r_1r}+\frac{r_0+r_1}{(r_0r_1)^{3/2}} \arctan\sqrt{\frac{(r-r_0)r_1}{(r_1-r)r_0}}}
r
0
<
r
<
r
1
{\displaystyle r_0<r<r_1\,}
Lösung [ ] Die
ϵ
{\displaystyle \epsilon\,}
F
(
0
)
=
∫
r
0
r
1
d
r
r
(
r
−
r
0
)
(
r
1
−
r
)
=
[
2
r
0
r
1
arctan
(
r
−
r
0
)
r
1
(
r
1
−
r
)
r
0
]
r
0
r
1
=
π
r
0
r
1
.
{\displaystyle F(0)=\int\limits_{r_0}^{r_1}\frac{dr}{r\sqrt{(r-r_0)(r_1-r)}} =\left[\frac{2}{\sqrt{r_0r_1}}\arctan\sqrt{\frac{(r-r_0)r_1}{(r_1-r)r_0}}\right]_{r_0}^{r_1} =\frac{\pi}{\sqrt{r_0r_1}}.}
Allgemeines Parameterintegral mit Singularitaeten (
c
=
−
1
{\displaystyle c=-1\,}
n
=
4
{\displaystyle n=4\,}
r
3
=
0
{\displaystyle r_3=0\,}
F
′
(
0
)
=
r
1
′
−
r
0
′
r
1
−
r
0
F
(
0
)
+
(
r
1
r
0
′
−
r
0
r
1
′
r
1
−
r
0
−
r
2
′
2
)
∫
r
0
r
1
d
r
r
2
(
r
−
r
0
)
(
r
1
−
r
)
{\displaystyle F'(0)=\frac{r_1'-r_0'}{r_1-r_0}F(0)+\left(\frac{r_1r_0'-r_0r_1'}{r1-r_0}-\frac{r_2'}2\right) \int\limits_{r_0}^{r_1}\frac{dr}{r^2\sqrt{(r-r_0)(r_1-r)}}}
=
r
1
′
−
r
0
′
r
1
−
r
0
π
r
0
r
1
+
(
r
1
r
0
′
−
r
0
r
1
′
r
1
−
r
0
−
r
2
′
2
)
[
r
0
+
r
1
(
r
0
r
1
)
3
/
2
arctan
(
r
−
r
0
)
r
1
(
r
1
−
r
)
r
0
]
r
0
r
1
{\displaystyle =\frac{r_1'-r_0'}{r_1-r_0}\frac{\pi}{\sqrt{r_0r_1}}+\left(\frac{r_1r_0'-r_0r_1'}{r1-r_0}-\frac{r_2'}2\right) \left[\frac{r_0+r_1}{(r_0r_1)^{3/2}} \arctan\sqrt{\frac{(r-r_0)r_1}{(r_1-r)r_0}}\right]_{r_0}^{r_1}
}
=
(
r
1
′
−
r
0
′
r
1
−
r
0
+
(
r
1
r
0
′
−
r
0
r
1
′
r
1
−
r
0
−
r
2
′
2
)
r
0
+
r
1
r
0
r
1
)
π
r
0
r
1
{\displaystyle =\left(\frac{r_1'-r_0'}{r_1-r_0}+\left(\frac{r_1r_0'-r_0r_1'}{r1-r_0}-\frac{r_2'}2\right) \frac{r_0+r_1}{r_0r_1} \right)\frac{\pi}{\sqrt{r_0r_1}}
}
=
r
1
2
r
0
′
−
r
0
2
r
1
′
−
1
2
r
2
′
(
r
1
2
−
r
0
2
)
(
r
1
−
r
0
)
r
0
r
1
π
r
0
r
1
{\displaystyle =\frac{r_1^2r_0'-r_0^2r_1'-\frac12r_2'(r_1^2-r_0^2)}{(r_1-r_0)r_0r_1}\frac{\pi}{\sqrt{r_0r_1}}
}
=
π
r
1
2
(
2
r
0
′
−
r
2
′
)
−
r
0
2
(
2
r
1
′
−
r
2
′
)
2
(
r
1
−
r
0
)
(
r
0
r
1
)
3
/
2
{\displaystyle =\pi\frac{r_1^2(2r_0'-r_2')-r_0^2(2r_1'-r_2')}{2(r_1-r_0)(r_0r_1)^{3/2}}
}
Suchbegriffe [ ] Parameterintegral, integrierbare Singularitäten, Ableitung
ähnliche Aufgaben [ ] Niemanden würde diese Aufgabe interessieren, wenn man sie nicht zur Berechnung der Perihel-Präzession des Merkur in der allgemeinen Relativitätstherie gebrauchen könnte.
![{\displaystyle F(0)=\int \limits _{r_{0}}^{r_{1}}{\frac {dr}{r{\sqrt {(r-r_{0})(r_{1}-r)}}}}=\left[{\frac {2}{\sqrt {r_{0}r_{1}}}}\arctan {\sqrt {\frac {(r-r_{0})r_{1}}{(r_{1}-r)r_{0}}}}\right]_{r_{0}}^{r_{1}}={\frac {\pi }{\sqrt {r_{0}r_{1}}}}.}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/617187f82a1046f88ce0011e3a579307c95ba31d)
![{\displaystyle ={\frac {r_{1}'-r_{0}'}{r_{1}-r_{0}}}{\frac {\pi }{\sqrt {r_{0}r_{1}}}}+\left({\frac {r_{1}r_{0}'-r_{0}r_{1}'}{r1-r_{0}}}-{\frac {r_{2}'}{2}}\right)\left[{\frac {r_{0}+r_{1}}{(r_{0}r_{1})^{3/2}}}\arctan {\sqrt {\frac {(r-r_{0})r_{1}}{(r_{1}-r)r_{0}}}}\right]_{r_{0}}^{r_{1}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/85d5e2bb5f52b373b5e6bf05b5da601ff25198ad)
