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Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/7/Makefile.inc | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/7/bch.tex | 76 | ||||
| -rw-r--r-- | vorlesungen/slides/7/chapter.tex | 1 |
3 files changed, 78 insertions, 0 deletions
diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc index aac9585..8b2b9e0 100644 --- a/vorlesungen/slides/7/Makefile.inc +++ b/vorlesungen/slides/7/Makefile.inc @@ -19,6 +19,7 @@ chapter5 = \ ../slides/7/liealgbeispiel.tex \ ../slides/7/vektorlie.tex \ ../slides/7/kommutator.tex \ + ../slides/7/bch.tex \ ../slides/7/dg.tex \ ../slides/7/zusammenhang.tex \ ../slides/7/quaternionen.tex \ diff --git a/vorlesungen/slides/7/bch.tex b/vorlesungen/slides/7/bch.tex new file mode 100644 index 0000000..3fad600 --- /dev/null +++ b/vorlesungen/slides/7/bch.tex @@ -0,0 +1,76 @@ +% +% bch.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Baker-Campbell-Hausdorff-Formel} +$g(t),h(t)\in G +\uncover<2->{\Rightarrow \exists A,B\in LG\text{ mit } +g(t)=\exp At, h(t)=\exp Bt}$ +\uncover<3->{% +\begin{align*} +g(t) +&= +I + At + \frac{A^2t^2}{2!} + \frac{A^3t^3}{3!} + \dots, +& +h(t) +&= +I + Bt + \frac{B^2t^2}{2!} + \frac{B^3t^3}{3!} + \dots +\end{align*}} +\uncover<5->{% +\begin{block}{Kommutator in G: $c(t) = g(t)h(t)g(t)^{1}h(t)^{-1}$} +\begin{align*} +\uncover<6->{c(t) +&= +\biggl( + {\color<7,9-11,13-15,19-21>{red}I} + + {\color<8,16-19>{red}A}t + + \frac{{\color<12>{red}A^2}t^2}{2!} + + \dots +\biggr) +\biggl( + {\color<7,8,10-12,14-15,17-18,21>{red}I} + + {\color<9,16,19-20>{red}B}t + + \frac{{\color<13>{red}B^2}t^2}{2!} + + \dots +\biggr) +\exp(-{\color<10,14,17,19,21>{red}A}t) +\exp(-{\color<11,15,18,20-21>{red}B}t) +} +\\ +&\uncover<7->{={\color<7>{red}I}} +\uncover<8->{+t( + \uncover<8->{ {\color<8>{red}A}} + \uncover<9->{+ {\color<9>{red}B}} + \uncover<10->{- {\color<10>{red}A}} + \uncover<11->{- {\color<11>{red}B}} +)} +\uncover<12->{+\frac{t^2}{2!}( + \uncover<12->{ {\color<12>{red}A^2}} + \uncover<13->{+ {\color<13>{red}B^2}} + \uncover<14->{+ {\color<14>{red}A^2}} + \uncover<15->{+ {\color<15>{red}B^2}} +)} +\\ +&\phantom{\mathstrut=I} +\uncover<12->{+t^2( + \uncover<16->{ {\color<16>{red}AB}} + \uncover<17->{- {\color<17>{red}A^2}} + \uncover<18->{- {\color<18>{red}AB}} + \uncover<19->{- {\color<19>{red}BA}} + \uncover<20->{- {\color<20>{red}B^2}} + \uncover<21->{+ {\color<21>{red}AB}} +)} +\uncover<22->{+t^3(\dots)+\dots} +\\ +&\uncover<23->{= +I + \frac{t^2}{2}[A,B] + o(t^3) +} +\end{align*}} +\end{block} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex index 668c2a7..0068cf4 100644 --- a/vorlesungen/slides/7/chapter.tex +++ b/vorlesungen/slides/7/chapter.tex @@ -18,6 +18,7 @@ \folie{7/liealgbeispiel.tex} \folie{7/vektorlie.tex} \folie{7/kommutator.tex} +\folie{7/bch.tex} \folie{7/dg.tex} \folie{7/zusammenhang.tex} \folie{7/quaternionen.tex} |