I am estimating the scattering field from these below integrals by asymptotic approximation (saddle point method) for H and E polarizations, respectively. \begin{align} I_{\text{H}} &= 2jka\int_{-\pi/2}^{\pi/2}\cos({\varphi + \phi_0 - \phi}) e^{jka[\cos{\varphi} + \cos({\varphi + \phi_0 - \phi})]} \ d\varphi. \\ I_{\text{E}} &= -2jka\int_{-\pi/2}^{\pi/2}\cos{\varphi}~e^{jka[\cos{\varphi} + \cos({\varphi + \phi_0 - \phi})]} \ d\varphi. \end{align} $\phi_0$ is already known. Then the results are compared with one by using Simpson rule. The difference between two methods is calculated in the range of $\phi=[0^{\circ},360^{\circ}]$ as following \begin{align} \text{Error} = \big|\text{uI} - \text{nI}\big|^2, \end{align} in which $uI$ is the result estimated by asymptotic solution, $nI$ is the result estimated by Simpson rule. The $\text{Error}$ values for H and E cases as in the figure. The figure shows the average value for the range $\phi=[0^{\circ},360^{\circ}]$ with each $ka$ \begin{align} \text{Error}_{average} = \frac{\text{Error}_{0^{\circ}}+\text{Error}_{1^{\circ}}+\text{Error}_{2^{\circ}}+...+\text{Error}_{360^{\circ}}}{361} \end{align} I have no idea why different integrals give us the same $\text{Error}$ as calculating following the third formulation $\text{Error}$ as calculating following the third equation.
I have checked the data and see that for every observation angle $\phi$, the results of $I_{\text{H}}$ and $I_{\text{E}}$ are different for H and E polarizations, respectively. However, the $\text{Error}$ between two polarizations are always opposite in sign. For example \begin{align} \text{Error}_{H} &= \text{uI}_{H} - \text{nI}_{H} = a + bj, \\ \text{Error}_{E} &= \text{uI}_{E} - \text{nI}_{E} = -a - bj. \end{align} Could someone please explain for me why the \text{Error} between H and E polarizations are always like that even when $\text{I}_{H}$ and $\text{I}_{E}$ are different integrals.
Thank you very much!
