Question

Without computer or tables, but just using facts you know, sketch a quick rough graph of the \(\Gamma\) function from -2 to 3 . Hint: This is easy; don't make a big job of it. From Section 3, you know the values of the \(\Gamma\) function at the positive integers in terms of factorials. From Problem 1, you can easily find and plot the \(\Gamma\) function at \(\pm 1 / 2, \pm 3 / 2 .\) (Approximate \(\sqrt{\pi}\) as a little less than \(2 .\) ) From (4.1) and the discussion following it, you know that the \(\Gamma\) function tends to plus or minus infinity at 0 and the negative integers, and you know the intervals where it is positive or negative. After sketching your graph, make a computer plot of the \(\Gamma\) function from -5 to 5 and compare your sketch.

Short answer

Plot points at known values, add vertical asymptotes at 0 and negative integers, and sketch a smooth curve considering positivity and negativity intervals.

Step-by-step solution

Understand the Gamma Function Values

The Gamma function \(\text{Γ}(z)\) extends the factorial function to complex and real number arguments. For positive integers, \(\text{Γ}(n) = (n-1)!\) and for half-integers, \(\text{Γ}(n+1/2)\) has specific values involving \(\root3{\textπ} \).

Plot Known Gamma Function at Positive Integers

Plot points for \(\text{Γ}(1) = 0! = 1\), \(\text{Γ}(2) = 1! = 1\), and \(\text{Γ}(3) = 2! = 2\).

Plot Known Gamma Function at Half-Integers

Use the approximation \(\text{Γ}(1/2) \approx \sqrt{\text{π}} \~ \ 1.77\) \((\text{estimated as a little less than 2})\). Similarly, \(\text{Γ}(3/2) = \frac{1}{2} \sqrt{\text{π}} \approx \ 0.88\), and \(\text{Γ}(5/2) = \frac{3}{4} \sqrt{\text{π}} \approx \ 1.32\). Plot these values on the graph.

Determine the Behavior at Negative Integers

From the properties of the Gamma function, \(\text{Γ}(z)\) tends to ±∞ (alternates between positive and negative infinity) at negative integers. Hence, \(\text{Γ}(0)\), \(\text{Γ}(-1)\) and \(\text{Γ}(-2)\) tend to infinity.

Analyze Intervals of Positivity and Negativity

Use the given properties for positivity and negativity: \(\text{Γ}(z)\) is positive between non-positive integers and shifts sign at each integer.

Sketch the Rough Graph

Combine all information: draw vertical asymptotes at 0 and negative integers where the function tends to infinity, plot the points calculated at positive integers and half-integers, and smoothly connect the points. Ensure the function crosses the x-axis at the known intervals and switch signs appropriately.

Compare with Computer Plot

Plot a computer-generated graph for \(\text{Γ}(z)\) from -5 to 5 using graphing software and compare it to your manual sketch to verify accuracy.

Key concepts

Factorials

The Gamma function, denoted as \(\text{Γ}(z)\), is an extension of the factorial function. While the factorial function is only defined for natural numbers, the Gamma function extends this definition to real and complex numbers. For any positive integer \(n\), the relationship is given by: \(\text{Γ}(n) = (n-1)!\). This means that \(\text{Γ}(1) = 0! = 1\), \(\text{Γ}(2) = 1! = 1\), and \(\text{Γ}(3) = 2! = 2\). These values help in plotting the Gamma function at positive integers. Since the Gamma function generalizes the factorial to non-integer values, it offers more versatility for mathematical computations.

Remember, factorials are very useful in counting problems, permutations, and combinations.
Here’s a simple example:

Suppose you want to find the number of ways to arrange 3 books. You would calculate this as \(3! = 3 \times 2 \times 1 = 6\). The Gamma function helps in defining such problems for non-integer counts.

Half-Integers

The Gamma function also has well-defined values for half-integers. A half-integer is a number like \(1/2\), \(3/2\), \(5/2\), etc. For example, \(\text{Γ}(1/2) = \sqrt{\text{π}} \approx 1.77\) and \(\text{Γ}(3/2) = \frac{1}{2} \sqrt{\text{π}} \approx 0.88\). Additionally, \(\text{Γ}(5/2) = \frac{3}{4} \sqrt{\text{π}} \approx 1.32\). These values are important for accurately plotting the Gamma function across these points.

Approximating \(\text{π}\) as a little less than 2 can help simplify calculations when plotting by hand. Understanding these special values lets you sketch the function much more effectively. Half-integers provide an insight into the behavior of \(\text{Γ}(z)\) between integers and assist in showing its continuity.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For the Gamma function, there are vertical asymptotes at zero and all negative integers. This means as \(z\) approaches these values, \(\text{Γ}(z)\) tends toward ±∞. For instance, \(\text{Γ}(z)\) exhibits vertical asymptotes at \(\text{Γ}(0)\), \(\text{Γ}(-1)\), \(\text{Γ}(-2)\), etc. The function switches between positive and negative infinity alternately at each negative integer.

Understanding the position and behavior of asymptotes is crucial for sketching the Gamma function correctly. It provides a boundary where the function diverges to infinity. This behavior is central to identifying gaps in the graph and related changes in sign.

Positivity and Negativity Intervals

The Gamma function has specific intervals where its values are positive or negative. Between each non-positive integer, the Gamma function is either positive or negative. Specifically, \(\text{Γ}(z)\) is positive between non-positive integers like \(0 < x < 1\), and negative between \(-1 < x < 0\). It means the sign of \(\text{Γ}(z)\) changes at each non-positive integer.

This property helps in determining where the graph crosses the x-axis and how it behaves between those points. By understanding these positivity and negativity intervals, you can sketch a more accurate graph of the Gamma function.

• Positive intervals:
  \text{Between 0 and 1, between 1 and 2, etc.}
• Negative intervals:
  Between -1 and 0, between -2 and -1, etc.

Knowing these intervals allows you to predict and mark the changing behavior of the Gamma function visually.