I would like to find a simple differential equation that, on taking the Laplace transform, results in essential singularities. I am looking, when teaching, to find examples of differential equations that results in each of the possible singularities on the complex plane.
An example with poles is easy; any linear differential equation with constant coefficients will do. The Laplace transform of the homogeneous solution will give poles. The second order differential equation is a good simple candidate.
Along these lines I have started with the equation in the Laplace domain of
$L_t=-e^{-\frac{1}{s+\epsilon -i}}-e^{-\frac{1}{s+\epsilon +i}}$
where s is the Laplace variable and $\epsilon$ a parameter. I want real results from the inverse Laplace transform so have included the complex conjugate of the singularity located at $s=-\epsilon +i$ . A plot of the absolute value is
Which shows the singularities clearly.
Using Mathematica to get the inverse Laplace transform I get
$f(t)=-e^{t (-\epsilon -i)} \left(\delta (t)-\frac{J_1\left(2 \sqrt{t}\right)}{\sqrt{t}}\right)-e^{t (-\epsilon +i)} \left(\delta (t)-\frac{J_1\left(2 \sqrt{t}\right)}{\sqrt{t}}\right)$
Where $J_1$ is the Bessel function of the first kind. Plotting this result for $\epsilon =0.001$ gives
Which looks complicated.
What differential equation would have this as the homogeneous solution? Perhaps this question can be made simpler by alternative forms in the Laplace plane. Any suggestions?