Mapping the imaginary axis of the complex (s) plane - Mathematica Stack Exchange most recent 30 from mathematica.stackexchange.com 2019-09-21T18:27:43Z https://mathematica.stackexchange.com/feeds/question/43800 https://creativecommons.org/licenses/by-sa/4.0/rdf https://mathematica.stackexchange.com/q/43800 0 Mapping the imaginary axis of the complex (s) plane Pipe https://mathematica.stackexchange.com/users/1209 2014-03-11T14:16:00Z 2014-03-12T10:09:11Z <p>In this problem is very difficult to analyze the roots s. Firstly I thought that the problem is in precision, but I am not sure. The criterion which I want to obtain is that one of these roots has a positive real part. Characteristic equation has the following form:</p> <p>$$\frac{2 \pi }{\int_{-\infty }^{\infty } \frac{1}{\text{D1}(k,s)} \, dk}+\text{a6} s^2+\text{k0}=0$$</p> <p>The imaginary axis of the complex s-plane can be parametrized by conidering s=i*r (i=sqrt(-1)). So the equation has new form for consideration </p> <pre><code> k0 = a6 r^2 - (1/(2 Pi) NIntegrate[1/( a1 a3 k^4 - a2 a4 (-k a5 + r)^2), {k, -\[Infinity], \[Infinity]}])^-1 </code></pre> <p>where </p> <pre><code>a1 = 2*10^11; a2 = 7.687*10^-3; a3 = 3.055*10^-5; a4 = 7849; a5 = 500; a6 = 70; \[Alpha] = Sqrt[(a1 a3)/(a4 a2)]; c = Sqrt[k0/a6]; a5 &gt; &lt; Sqrt[4 \[Alpha] c]; </code></pre> <p>The mapping rule follows from the characteristic equation, which should be rewritten to express the chosen parameter explicitly. Once the mapping is accomplished, a mapped line is obtained, which divides the parameter plane into domains with different number of roots s* possessing a positive real part. Every line is normally shaded from the side, which is related to the right-hand side of the imaginary axis of the (s)-plane now. Obviously, crossing a decomposition curve in the direction of the shading, one extra root with a positive real part is gained. This curve (rule) should be obtained for parameter r from -Infinity to +Infinity for discuss. So I need two separately diagrams for a5 > Sqrt[4 [Alpha] c] or a5 &lt; Sqrt[4 [Alpha] c].</p> <p>I am completely not sure how to obtain curves from the equation</p> <pre><code> k0 = a6 r^2 - (1/(2 Pi) NIntegrate[1/( a1 a3 k^4 - a2 a4 (-k a5 + r)^2), {k, -\[Infinity], \[Infinity]}])^-1 </code></pre> <p>presented here </p> <p><img src="https://i.stack.imgur.com/urHhH.jpg" alt="enter image description here"></p> <p>where N is number of roots in the region</p>