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* Page found: Stabilität und Langzeitverhalten (eq math.2045.47)

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TeX (original user input): 

\begin{align}
  & \bar{\varpi }{{*}^{(1)}}:{{\varpi }_{1}}=\varpi ,{{\varpi }_{2}}=0,{{\varpi }_{3}}=0 \\
 & 0=\det (A-\lambda 1)=\left| \begin{matrix}
   -\lambda  & 0 & 0  \\
   0 & -\lambda  & {{k}_{2}}{{\omega }_{{}}}  \\
   0 & -{{k}_{3}}\omega  & -\lambda   \\
\end{matrix} \right|=-\lambda \left( {{\lambda }^{2}}+{{k}_{2}}{{k}_{3}}{{\omega }^{2}} \right) \\
 & \Rightarrow {{\lambda }_{1}}^{(1)}=0,{{\lambda }_{2/3}}^{(1)}=\pm i\omega \sqrt{{{k}_{2}}{{k}_{3}}} \\
\end{align}

TeX (checked): 

{\begin{aligned}&{\bar {\varpi }}{{*}^{(1)}}:{{\varpi }_{1}}=\varpi ,{{\varpi }_{2}}=0,{{\varpi }_{3}}=0\\&0=\det(A-\lambda 1)=\left|{\begin{matrix}-\lambda &0&0\\0&-\lambda &{{k}_{2}}{{\omega }_{}}\\0&-{{k}_{3}}\omega &-\lambda \\\end{matrix}}\right|=-\lambda \left({{\lambda }^{2}}+{{k}_{2}}{{k}_{3}}{{\omega }^{2}}\right)\\&\Rightarrow {{\lambda }_{1}}^{(1)}=0,{{\lambda }_{2/3}}^{(1)}=\pm i\omega {\sqrt {{{k}_{2}}{{k}_{3}}}}\\\end{aligned}}

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MathML (35.784 KB / 4.151 KB) :

ϖ ¯ * ( 1 ) : ϖ 1 = ϖ , ϖ 2 = 0 , ϖ 3 = 0 0 = det ⁡ ( A - λ ⁢ 1 ) = | - λ 0 0 0 - λ k 2 ⁢ ω 0 - k 3 ⁢ ω - λ | = - λ ⁢ ( λ 2 + k 2 ⁢ k 3 ⁢ ω 2 ) ⇒ λ 1 ( 1 ) = 0 , λ 2 / 3 ( 1 ) = ± i ⁢ ω ⁢ k 2 ⁢ k 3 missing-subexpression fragments ¯ italic-ϖ superscript 1 : subscript italic-ϖ 1 ϖ , subscript italic-ϖ 2 0 , subscript italic-ϖ 3 0 missing-subexpression 0 𝐴 𝜆 1 𝜆 0 0 0 𝜆 subscript 𝑘 2 𝜔 0 subscript 𝑘 3 𝜔 𝜆 𝜆 superscript 𝜆 2 subscript 𝑘 2 subscript 𝑘 3 superscript 𝜔 2 missing-subexpression formulae-sequence ⇒ absent superscript subscript 𝜆 1 1 0 superscript subscript 𝜆 2 3 1 plus-or-minus 𝑖 𝜔 subscript 𝑘 2 subscript 𝑘 3 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\bar{\varpi}{{*}^{(1% )}}:{{\varpi}_{1}}=\varpi,{{\varpi}_{2}}=0,{{\varpi}_{3}}=0\\ &\displaystyle 0=\det(A-\lambda 1)=\left|\begin{matrix}-\lambda&0&0\\ 0&-\lambda&{{k}_{2}}{{\omega}}\\ 0&-{{k}_{3}}\omega&-\lambda\\ \end{matrix}\right|=-\lambda\left({{\lambda}^{2}}+{{k}_{2}}{{k}_{3}}{{\omega}^% {2}}\right)\\ &\displaystyle\Rightarrow{{\lambda}_{1}}^{(1)}=0,{{\lambda}_{2/3}}^{(1)}=\pm i% \omega\sqrt{{{k}_{2}}{{k}_{3}}}\\ \end{aligned}}}

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ϖ¯*(1):ϖ1=ϖ,ϖ2=0,ϖ3=00=det⁡(A−λ1)=|−λ000−λk2ω0−k3ω−λ|=−λ(λ2+k2k3ω2)⇒λ1(1)=0,λ2/3(1)=±iωk2k3

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  • ω
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