this is my first time asking a question on here so please forgive me if I have made any formatting mistakes.
I have the integral $f(x) = \int_0^\infty \frac{e^{-t}}{x + t} \; dt$ and I have shown the relationship between $f(x)$ and the exponential integral, $E_i(x) = \int_x^\infty \frac{e^{-t}}{t} \; dt$.
My assignment requires me to find an application for the integral, $f(x)$, but since I have shown the relationship between $f(x)$ and $E_i(x)$ I am only required to find an application for the exponential integral. I've tried to look into time-dependent heat transfer and reservoir engineering, however, I'm finding these two applications quite hard to follow since I cannot explicitly see a version of this integral online.
Could anyone possibly help me out with an application and how the integral is used in the application? If not, some suggestions of websites or books would be most helpful.
To add context to the question, I have used an asymptotic approximation and compared the approximation to the integral through a graphic representation. Furthermore, I have also investigated the error between the approximation and the integral.
Thank you in advance. :)
A fundamental theorem of communication theory is the Channel Coding Theorem, which states that the maximum rate of communication (transmission speed, in bits per time unit, if you will) for which the decoding error can be made arbitrarily small, is given by the Shannon capacity.
Now, for one of the simplest channels, which is the additive white Gaussian noise (AWGN) channel, the complex-valued channel input $X$ has bounded average transmit power $\mathsf{E}[|X|^2] \leq P$, and the complex-valued channel output $Y$ is given by $$ Y = X + N $$ where the additive noise $N$ is complex standard Gaussian $\mathcal{N}_\mathbb{C}(0,1)$.
The receiver tries to decode the message encoded by the transmitter in the signal $X$ from observing the output $Y$. The capacity $C$ of this channel is given by a very well-known formula: $$ C = \log_2(1+P) \quad \text{bits/time unit} $$
Now, in wireless communications, the random attenuation of the channel is modelled by a random variable (due to multi-path propagation and random fluctuations of constructive and destructive multi-path interference). This phenomenon is called fading. This is often modelled by extending the system equation to $$ Y = HX + N $$ where the fading coefficient $H \sim \mathcal{N}_\mathbb{C}(0,1)$ is standard complex Gaussian. In this case, assuming the receiver knows $H$, the capacity is given by $$ \begin{align} C_\text{fading} &= \mathsf{E}\left[\log_2\left(1+P|H|^2\right)\right] = \int_0^\infty \log_2(1+P\xi) e^{-\xi} \mathrm{d}\xi \\ &= \frac{1}{\ln(2)}e^{\frac{1}{P}}E_1\left(\frac{1}{P}\right) \end{align} $$ where $E_1(x) = \int_x^\infty \frac{e^{-t}}{t}\mathrm{d}t$. Note that by Jensen's inequality, $C \geq C_\text{fading}$. So you can say that fading, in a certain sense, can be a hindrance to communication.
I have omitted many details, but I hope you get the rough idea.