Question

The integral \(\int_{x}^{\infty} t^{p-1} e^{-t} d t=\Gamma(p, x)\) is called an incomplete \(\Gamma\) function. [Note that if \(x=0, \text { this integral is } \Gamma(p) .]\) By repeated integration by parts, find several terms of the asymptotic series for \(\Gamma(p, x)\).

Short answer

The asymptotic series for \Gamma(p, x)\ is \ x^{p-1} e^{-x} + (p-1) x^{p-2} e^{-x} + \frac{(p-1)(p-2)}{2!} x^{p-3} e^{-x} + \ldots.

Step-by-step solution

Setup Integration by Parts

Integration by parts formula is \(\int u dv = uv - \int v du\). Choose \(u = t^{p-1}\) and \(dv = e^{-t} dt\).

Determine du and v

Compute \(du = (p-1) t^{p-2} dt\) and integrate \(dv = e^{-t} dt\) to obtain \(v = -e^{-t}\).

Apply Integration by Parts

Substitute into the integration by parts formula: \(\int_{x}^{\infty} t^{p-1} e^{-t} dt = \left[-t^{p-1} e^{-t}\right]_{x}^{\infty} + (p-1) \int_{x}^{\infty} t^{p-2} e^{-t}dt\). As \(t \to \infty\), the term \([-t^{p-1} e^{-t}]\) tends to zero, so it simplifies to: \([-x^{p-1} e^{-x}] + (p-1) \int_{x}^{\text{inf}} t^{p-2} e^{-t} dt\).

Iterate the Process

Repeat the integration by parts on the remaining integral. Now set \(u = t^{p-2}\) and \(dv = e^{-t} dt\). Derive \(du = (p-2) t^{p-3} dt\) and \(v = -e^{-t}\). Apply the formula again. Each repetition will generate another term in the series.

Generalize the Series

The general form of the asymptotic series is \[\Gamma(p,x) \approx x^{p-1} e^{-x} + (p-1) x^{p-2} e^{-x} + \frac{(p-1)(p-2)}{2!} x^{p-3} e^{-x} + \ldots\]. Each term takes the form \( \frac{(p-1)(p-2)...(p-n)}{n!} x^{p-n-1} e^{-x}\).

Key concepts

Asymptotic Series

Understanding the concept of asymptotic series can be a game-changer. An asymptotic series gives an approximation to describe complex functions. These series provide a way to estimate the behavior of a function in a limit, typically as a variable approaches infinity. For the incomplete Gamma function, asymptotic series help in understanding how the function behaves for large values of its parameters.
In our exercise, we derived an asymptotic series for the incomplete Gamma function \(\text{Γ}(p, x)\). The final form of the series looks like this: \[\text{Γ}(p,x) \approx x^{p-1} e^{-x} + (p-1) x^{p-2} e^{-x} + \frac{(p-1)(p-2)}{2!} x^{p-3} e^{-x} + \ldots\] Each term in this series becomes smaller and smaller as x increases. Because of this, after a few terms, additional terms contribute less and less, making it easier to approximate the function precisely. This is incredibly useful in computations because calculating the Gamma function directly for large parameters can be inefficient or impractical.

Integration by Parts

The method of integration by parts simplifies complex integrals by transforming them into simpler ones. The integration by parts formula is \[\text{∫} u \ dv = uv - \text{∫} v \ du\] Here’s how it helps:
1. Choose which parts of the integral will be u and dv.
2. Differentiate u to get du.
3. Integrate dv to get v.
4. Substitute into the formula to simplify the original integral.
Notice in our example, we repeatedly used integration by parts to break down the incomplete Gamma function. At each step, the integral became simpler. This step-by-step reduction made it possible to identify a pattern, eventually leading us to the asymptotic series. This method not only simplifies the individual integral computations but also helps to recognize series patterns crucial for approximations.

Special Functions

Special functions are mathematical functions that have specific names and notations because they frequently appear in various problems, such as those in physics and engineering. The Gamma function is one such special function. It generalizes the factorial function to non-integer values. The incomplete Gamma function \(\text{Γ}(p, x)\) gives the integral of the function from x to infinity, providing solutions to problems involving integrals that can't be solved by elementary functions.
Understanding these special functions is crucial in advanced mathematics and applied science.
Here are key highlights:

• The Gamma function, denoted \(\text{Γ}(p)\), extends the concept of factorial to the continuous domain.
• The incomplete Gamma function looks at partial integrals, which can help approximate behavior over specific intervals.
• Special functions like these often come with useful properties and relationships making complex computations tractable.

Knowing how to work with these functions and their properties can open up solving a wide range of practical and theoretical problems.