## 2 Materials and Methods
### Approximation of the Hyperbolic Sine Term  
In the following, the weighting function \(W_{1}^{*}\) will be examined, but it is important to note that the procedure for \(W_{2}^{*}\) is analogous to that for \(W_{1}^{*}\). This can be inferred from Equation (12), as only the poles of a function and its zeros, located at the same positions as the poles, are relevant for an inverse Laplace transform (ILT). Since \(\cosh\) does not introduce additional poles or zeros, all formulas applied to \(W_{1}^{*}\) can also be used for \(W_{2}^{*}\). \[W_{2}^{*}(\zeta)=W_{1}^{*}(\zeta)\,\cosh\!\left[\sqrt{\frac{\text{I}_{0}(\sqrt{ \zeta})}{\text{I}_{2}(\sqrt{\zeta})}}\zeta\text{D}n\right] \tag{30}\]  
With:  
\[W_{1}^{*}(\zeta)=\ \frac{8\,Dn\sqrt{\text{I}_{2}(\sqrt{\zeta})}}{\sinh\!\left[ \sqrt{\frac{\text{I}_{0}(\sqrt{\zeta})}{\text{I}_{2}(\sqrt{\zeta})}}\zeta \text{D}n\right]\!\zeta\sqrt{\text{I}_{0}(\sqrt{\zeta})}}. \tag{31}\]  
The order of the poles of a function is crucial when calculating the residues needed for an ILT. To determine the order of the poles, we use an approximation for \(\sinh\) from [PERSON] [34], as shown in Equation (32). This approximation is ideal because it is expressed as a product, preserving the poles' properties and the function's zeros. In contrast, Taylor series expansions do not maintain the poles and are thus unsuitable [35]. Increasing \(n\) in Equation (32) corresponds to increasing the number of complex zeros of the \(\sinh\) term. \[\sinh(x)\approx\sinh_{\text{approx}}(n,x)=x\left[\prod_{k=1}^{n}\left(1+\frac {x^{2}}{k^{2}*\pi^{2}}\right)\right] \tag{32}\]  
By inserting \(n=5\) and \(x=\left[\sqrt{\frac{\text{I}_{0}(\sqrt{\zeta})}{\text{I}_{2}(\sqrt{\zeta})}} \zeta\text{D}n\right]\), Equation (32) becomes:  
\[\sinh_{\text{approx}}\left(5,\left[\sqrt{\frac{\text{I}_{0}( \sqrt{\zeta})}{\text{I}_{2}(\sqrt{\zeta})}}\zeta\text{D}n\right]\right)=\sqrt{ \frac{\text{I}_{0}(\sqrt{\zeta})}{\text{I}_{2}(\sqrt{\zeta})}}\zeta\text{D}n \left(1+\frac{\text{I}_{0}(\sqrt{\zeta})\zeta^{2}\text{D}n^{2}}{\left(\text{I }_{2}(\sqrt{\zeta})\pi^{2}\right)}\right) \tag{33}\]  
It is important to note that the approximation \(\sinh_{\text{approx}}(5,x)\) is equivalent to the standard \"small angle\" approximation of \(\sinh(x)=x\), which is derived from a truncated Taylor series and multiplied by the first five complex zeros of \(\sinh(x)\). \[W_{\text{1 app}}^{*}(\zeta)=\ \frac{8\,Dn\sqrt{\text{I}_{2}(\sqrt{\zeta})}}{\sinh_{ \text{approx}}(5,x)\zeta\sqrt{\text{I}_{0}(\sqrt{\zeta})}} \tag{34}\]  
With this approximation, the weighting function can be separated into the incompressible part \(W_{\text{inc}}^{*}(\zeta)\), which is derived using the small angle approximation of \(\sinh\) for \(Dn\to 0\), and additional terms from the \(\sinh\) approximation. \[W_{\text{1,app}}^{*}(\zeta)= \frac{8\,Dn\sqrt{\text{I}_{2}(\sqrt{\zeta})}}{\sinh_{\text{approx }}(5,x)\zeta\sqrt{\text{I}_{0}(\sqrt{\zeta})}} \tag{35}\] \[= W_{\text{inc}}^{*}(\zeta)\frac{1}{\left(1+\frac{\text{I}_{0}( \sqrt{\zeta})\zeta^{2}\text{D}n^{2}}{\left(12\sqrt{\zeta}\right)\pi^{2}} \right)\left(1+\frac{\text{I}_{0}(\sqrt{\zeta})\zeta^{2}\text{D}n^{2}}{\left( 4\text{I}_{2}(\sqrt{\zeta})\pi^{2}\right)}\right)\left(1+\frac{\text{I}_{0}( \sqrt{\zeta})\zeta^{2}\text{D}n^{2}}{\left(9\text{I}_{2}(\sqrt{\zeta})\pi^{2} \right)}\right)}\]where \(W^{*}_{\rm inc}(\zeta)=\frac{8\mathrm{I}_{2}(\sqrt{\zeta})}{\zeta^{2}\mathrm{I}_{0}( \sqrt{\zeta})}\). Note that \(W^{*}_{\rm inc}(\zeta)\) can be obtained by taking the limit of \(\lim_{\mathrm{D}n\to 0}W^{*}_{1}\) using \(\lim_{x\to 0}\sinh(x)=\lim_{x\to 0}\tanh(x)=x\), which are the Taylor series approximations of the hyperbolic terms. Although \(W^{*}_{\rm 1,app}\) will not be used to compute the residues, it demonstrates that the complex poles of \(\sinh\) are simple. With this information, the ILT can be calculated from the residues of the function, which requires knowledge of the order and location of the poles.