Exponential and logarithmic integrals Simplified

Exponential and logarithmic integrals are specialized functions often encountered in various fields of mathematics, physics, and engineering. Let’s explore each in detail.

Exponential Integrals

Definition:

The exponential integral, typically denoted as \( \text{Ei}(x) \), is a function defined as the integral of the function \( \frac{e^t}{t} \) with respect to \( t \). The most common definition is:

\( \text{Ei}(x) = -\int_{-x}^{\infty} \frac{e^{-t}}{t} \, dt \)

for \( x > 0 \). For negative values of \( x \), the definition involves a principal value to handle the singularity at \( t = 0 \).

Properties:

1. Asymptotic Behavior: As \( x \) approaches infinity, \( \text{Ei}(x) \) behaves asymptotically like \( \frac {e^x}{x} \).
2. Special Cases: When \( x = 0 \), the exponential integral is undefined.
3. Relation to Other Functions: The exponential integral is related to incomplete Gamma functions and can be expressed in terms of them.

Logarithmic Integrals

Definition:

The logarithmic integral, often denoted as \( \text{li}(x) \), is defined for positive ( x ) and is related to the prime counting function in number theory. The function is defined as:

\( \text{li}(x) = \int_{0}^{x} \frac{dt}{\ln(t)} \)

where ( \ln(t) ) is the natural logarithm of \( t \). For \( x > 1 \), the definition usually involves a Cauchy principal value to deal with the singularity at \( t = 1 \).

Properties:

1. Relation to Prime Numbers: The logarithmic integral is significant in number theory, particularly in estimating the distribution of prime numbers.
2. Asymptotic Behavior: For large \( x \), \( \text{li}(x) \) grows logarithmically.
3. Special Values: \( \text{li}(2) \) is often taken as the starting point for calculating \( \text{li}(x) \) for \( x > 2 \).

Example:

Let’s consider an example involving the logarithmic integral, \( \text{li}(x) \). We’ll calculate \( \text{li}(10) \), which is the integral of \( \frac{1}{\ln(t)} \) from \(0\) to \(10\). As mentioned earlier, the logarithmic integral is significant in number theory, particularly in the context of prime numbers.

The integral we want to evaluate is:

\( \text{li}(10) = \int_{0}^{10} \frac{dt}{\ln(t)} \)

However, we face a challenge with the singularity at \( t = 1 \) (where \( \ln(t) = 0 )\). To handle this, we can split the integral into two parts: from 0 to just below 1, and from just above 1 to 10. In practice, we would actually start the integral from a small positive number rather than 0 to avoid the singularity at \( t = 0 \) as well.

Let’s calculate \( \text{li}(10) \) using this approach.

The value of the logarithmic integral \( \text{li}(10) \) is approximately \( 6.166 \). This calculation involved splitting the integral to manage the singularity at \( t = 1 \) and starting from a very small positive number instead of zero to avoid the singularity at \( t = 0 \). This method is a common approach for handling integrals with such singularities, particularly in the context of logarithmic integrals.