I remember it was in the textbook, I was studying the textbook at the time, it had the usual problems, let me look for it.

In studying the hydrodynamics of extrusion flows, It is usually necessary to convert a physical quantity from the frequency domain(such as the Fourier transform) back to the time domain. This process is called inverse transformation. In the process of inverse transformation, we need to solve an integral, and the integrand of this integral contains a constant whose value is determined by the boundary conditions.

For the Laplace equation in the extrusion flow problem, we can use the separation of variables method to obtain a solution of the form X(x)Y(y)Z(z), where X(x) is the solution along the direction of the extrusion flow, and Y(y) and Z(z) are the solutions perpendicular to the direction of the extrusion flow. When taking the inverse transformation, we need to expand X(x) into a linear combination of orthogonal basis functions that satisfy the boundary conditions and can be solved by Laplace's inverse transformation formula.

For the boundary conditions in the extrusion flow problem, the velocity at the boundary of the extrusion plate is usually zero. In this case, we need to discretize the X-axis and then solve an eigenvalue problem for y and z at each x point to get a set of eigenfunctions and eigenvalues. And in this process, the eigenvalue tanx equals x comes up, because it's symmetric about x equals 0, and it's singular at x equals 0. This special eigenvalue is very important for solving the integral in inverse transformation, because it can eliminate the singularity and allow the integral to converge.

So, when you take the inverse Laplace transform, you use the eigenvalue tanx is equal to x.