I was working on my project, and I came across this integration. After googling, I realized that this is an Exponential Integral. And I'm pretty much new to this. Here is the integration
\begin{equation}
\int_{R_{min}}^{R_{max}} \frac{e^{-\alpha r}}{r}dr
\end{equation}
Here $R_{max}$ & $R_{min}$ is a real numbers like 160 and 50 respectively.
$\alpha$ is a complex number $\alpha = i \frac{2 \pi}{\lambda}$ , here $i$ is $\sqrt{-1}$ and $\lambda$ is a real number .
I have no idea how to evaluate this integral. In my project, I've expressed all the results in terms of $\alpha$. Can you give me any result that can be represented as $\alpha$.
Like
\begin{equation}
\int_{R_{min}}^{R_{max}} e^{-\alpha r}dr
\\
=\dfrac{\mathrm{e}^{-\alpha R_{min}}}{\alpha}-\dfrac{\mathrm{e}^{-\alpha R_{max}}}{\alpha}
\end{equation}
This above integration is represented in terms of $\alpha$, and I can simply write the code in MATLAB.
Can you give me some results like this one?
Best Answer
By definition of the exponential integral: $$ \int_{x_1}^{x_2}\frac{e^{-at}}tdt=E_1(a x_1)-E_1(a x_2). $$