Question

Evaluate the following \(\Gamma\) functions using tables and the recursion relation (3.4). $$ \Gamma(0.7) $$

Short answer

Approximately 1.288.

Step-by-step solution

Understand the Gamma Function

The Gamma function \( \Gamma(n) \) is a generalization of the factorial function. For positive integers, it holds that \( \Gamma(n) = (n-1)! \). But for non-integers, we use properties and tables.

Use the Gamma Function Table

Consult a table that lists values of the Gamma function for various arguments. You will find an approximate value for \( \Gamma(0.7) \) in the table.

Apply the Recursion Relation

If a direct value for \( \Gamma(0.7) \) is not available, use the recursion relation \( \Gamma(x+1) = x \Gamma(x) \). For \( 0.7 \), let \( x = -0.3 \) and recognize that \( \Gamma(0.7 + 1) = 0.7 \Gamma(0.7) \).

Find \( \Gamma(1.7) \) from the Table

Look up the value of \( \Gamma(1.7) \) in the Gamma function table. Suppose \( \Gamma(1.7) \) is approximately 0.902.

Solve for \( \Gamma(0.7) \)

Using the value from the table and the recursion relation: \[ \Gamma(1.7) = 0.7 \Gamma(0.7) \Rightarrow 0.902 = 0.7 \Gamma(0.7) \Rightarrow \Gamma(0.7) \approx \frac{0.902}{0.7} \approx 1.288 \]

Key concepts

Gamma function

The Gamma function, denoted by \( \Gamma(n) \), is a crucial concept in advanced mathematics, especially in calculus and complex analysis. It serves as an extension of the factorial function. While the factorial function only applies to positive integers, \( \Gamma(n) \) can handle both positive and non-positive numbers, including fractions and negative values, except for non-positive integers. For any positive integer \( n \), the relationship is straightforward: \( \Gamma(n) = (n-1)! \). So, for \( n = 6 \), we have \( \Gamma(6) = 5! = 120 \). This generalization is incredibly useful, allowing mathematicians to perform more versatile calculations.
The Gamma function is defined by the integral:

\[ \Gamma(n) = \int_0^\infty t^{n-1} e^{-t} dt \]

Understanding this function provides deeper insights into mathematical concepts like probability distributions, particularly the Gamma and Beta distributions, and solving differential equations. The integral representation paints a detailed picture of how the function behaves over its domain.

Recursion relation

The recursion relation for the Gamma function is a powerful tool. This relation allows us to express \( \Gamma(n) \) values in terms of neighboring values. It is given by:

\[ \Gamma(x+1) = x \Gamma(x) \]

This means if you know \( \Gamma(x) \), you can easily find \( \Gamma(x+1) \). For example, to find \( \Gamma(0.7) \), you can start with \( \Gamma(1.7) \) and use the relation \( \Gamma(1.7) = 0.7 \Gamma(0.7) \). Suppose \( \Gamma(1.7) = 0.902 \) from a table. Applying the relation, we solve:

• \( 0.902 = 0.7 \Gamma(0.7) \)

Isolating \( \Gamma(0.7) \), we divide 0.902 by 0.7:

\[ \Gamma(0.7) \approx \frac{0.902}{0.7} = 1.288 \]

The recursion relation is not just crucial for manual calculations but also for deriving properties of more complex functions. It exemplifies the interconnectedness of mathematical concepts and reinforces procedural comprehension.

Approximate value

Finding approximate values of the Gamma function is often necessary since exact values are seldom straightforward. This requires consulting Gamma function tables or using numerical methods. For instance, an approximate value of \( \Gamma(1.7) \) is found to be around 0.902. This approximation allows further calculations using the recursion relation.
In our example, we used the equation:

\[ \Gamma(1.7) = 0.7 \Gamma(0.7) = 0.902 \]

Solving for \( \Gamma(0.7) \), the final step involves basic arithmetic:

• \( \Gamma(0.7) \approx \frac{0.902}{0.7} = 1.288 \)

Approximate values are widely reported since Gamma functions for non-integer and non-positive inputs are not as easily computed. Tools like tables or software can significantly assist with these calculations. It’s essential to grasp that approximate values cater to precision needs in practical applications, such as statistical analysis and engineering problems.