Question

Use Stirling's formula to show that $\underset{n\to \mathrm{\infty }}{lim}{n}^{x}B(x,n)=\mathrm{\Gamma }(x)$.

Short answer

$\underset{n\to \mathrm{\infty }}{lim}{n}^{x}B(x,n)=\mathrm{\Gamma }(x)$

Step-by-step solution

- Recall Stirling's formula

Stirling's formula is an asymptotic approximation for factorials, given by $$n!\approx \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n.$$

- Understand Beta function

The Beta function, denoted as $B(x,n)$, is defined by the integral $$B(x,n)={\int }_0^1{t}^{x-1}(1-t{)}^{n-1}dt.$$

- Express Beta function in terms of Gamma functions

Using the relationship between the Beta and Gamma functions, we have $$B(x,n)=\frac{\mathrm{\Gamma }(x)\mathrm{\Gamma }(n)}{\mathrm{\Gamma }(x+n)}.$$

- Apply Stirling's formula to Gamma functions

Applying Stirling's formula to $\mathrm{\Gamma }(n)$ and $\mathrm{\Gamma }(x+n)$, we get:$$\mathrm{\Gamma }(n)\approx \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n$$$$\mathrm{\Gamma }(x+n)\approx \sqrt{2\pi (x+n)}{\left(\frac{x+n}{e}\right)}^{x+n}.$$

- Simplify the expression

Substitute the Stirling approximations into the Beta function expression:$$B(x,n)\approx \frac{\mathrm{\Gamma }(x)\cdot \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n}{\sqrt{2\pi (x+n)}{\left(\frac{x+n}{e}\right)}^{x+n}}.$$

- Analyze the limit

To find the limit, we first simplify the fraction, especially focusing on the significant terms (those that have n): $$n^{x}B(x,n)\approx n^x\frac{\mathrm{\Gamma }(x)\cdot \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n}{\sqrt{2\pi (x+n)}{\left(\frac{x+n}{e}\right)}^{x+n}}.$$

- Approximate the fraction

For large n, $x+n\approx n$, so our expression simplifies, and we consider higher order terms negligible:$$n^{x}B(x,n)\approx n^x\frac{\mathrm{\Gamma }(x)}{(n^x)}\approx \mathrm{\Gamma }(x).$$

- Conclude the limit

Thus, by taking the limit as n approaches infinity, we finally get:$$\underset{n\to \mathrm{\infty }}{lim}{n}^{x}B(x,n)=\mathrm{\Gamma }(x).$$

Key concepts

Beta function

The Beta function, represented as $B(x,n)$, is a special function in mathematics. It is defined by an integral,
$$B(x,n)={\int }_0^1{t}^{x-1}(1-t{)}^{n-1}dt.$$
This integral expression shows how the Beta function relates to the values of $x$ and $n$. It's used extensively in calculus and probability theory.

• The Beta function is symmetric, meaning $B(x,n)=B(n,x)$.
• It can be expressed using Gamma functions, as \ $B(x,n)=\frac{\mathrm{\Gamma }(x)\mathrm{\Gamma }(n)}{\mathrm{\Gamma }(x+n)}$. This relation simplifies many calculations.
• It's particularly useful in problems involving continuous probability distributions, such as the Beta distribution.

Understanding and using the Beta function helps in solving complex problems in both pure and applied mathematics.

Gamma function

The Gamma function is denoted by $\mathrm{\Gamma }(x)$ and is an extension of the factorial function for real and complex numbers. For a positive integer $x$, the Gamma function is defined as:
$$\mathrm{\Gamma }(x)=(x-1)!$$
More generally, for any $x>0$, it is given by:
$$\mathrm{\Gamma }(x)={\int }_0^{\mathrm{\infty }}{t}^{x-1}{e}^{-t}dt.$$

• The Gamma function is used to generalize the concept of factorial to non-integer values.
• It satisfies the relation \ $\mathrm{\Gamma }(x+1)=x\mathrm{\Gamma }(x)$, analogous to the factorial property \ $n!=n\times (n-1)!$.
• The Gamma function has applications in various fields, including complex analysis, statistics, and combinatorics.

Its connection to the Beta function simplifies many integrals and series. When used in conjunction with Stirling's approximation, it provides powerful tools for asymptotic analysis.

Asymptotic approximation

Asymptotic approximation is a method used in mathematics to describe the behavior of functions as the argument tends toward a certain limit, often infinity. In the context of factorials and Gamma functions, Stirling's formula is a commonly used asymptotic approximation:
$$n!\approx \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n$$

• Stirling's formula helps approximate large factorials, making calculations more manageable.
• This approximation is useful in statistical physics, number theory, and probabilistic methods.
• It also simplifies expressions involving the Gamma function, as seen in the exercise's solution.

Asymptotic approximations like Stirling's provide insightful approaches to understanding the behavior of complex functions in their limits.

factorials in calculus

Factorials, denoted as $n!$, are products of all positive integers up to $n$. In calculus, they appear in many contexts such as series expansions, integrals, and combinatorics. For example, in the Taylor series, factorials are used to express the coefficients:
$$e^x=\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$$

• Factorials help evaluate the convergence and behavior of series.
• They are key components in binomial coefficients, combinations, and permutations.
• Understanding factorials is essential for solving problems in integral and differential calculus.

Factorials also play a crucial role in explaining the Gamma function, providing a bridge between discrete mathematics and continuous functions.