Which of the following best describes the function $Ei\left(x\right)$?

• A. It is the exponential integral, defined as $Ei\left(x\right)={\int }_{-\infty }^x\frac{e^t}{t}dt$.
• B. It is the error function, defined as $erf\left(x\right)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}dt$.
• C. It is the natural exponential function, defined as $e^x$.
• D. It is the inverse of the exponential function, defined as $ln\left(x\right)$.

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the problem by identifying the function in question, which is \operatorname{Ei}(x). This is referred to as the exponential integral.
2. Step 2: Recall the definition of the exponential integral \operatorname{Ei}(x). It is defined mathematically as \operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} dt.
3. Step 3: Compare the given options to the definition of \operatorname{Ei}(x). The first option matches the definition of the exponential integral, \operatorname{Ei}(x).
4. Step 4: Review the other options to ensure they do not describe \operatorname{Ei}(x). For example, \operatorname{erf}(x) is the error function, e^x is the natural exponential function, and \ln(x) is the natural logarithm, which is the inverse of the exponential function.
5. Step 5: Conclude that the correct description of \operatorname{Ei}(x) is the first option: 'It is the exponential integral, defined as \operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} dt.'