Behaviour of Gamma function when z = -n

In summary, the conversation is about the behavior of the Gamma function at z = -n and the expression (-1)^n/(n!z) + O(1). The speaker recognizes the first term as the residue of f when z = -n, but is unsure about the rest. They are looking for an explanation for this expression and wondering why it involves both n and z. They also mention that the Gamma function diverges at negative integers and ask if the expression is an expansion of the Gamma function when z is close to a negative integer.
  • #1
naaa00
91
0
Hi there,

I'm actually trying to understand why the behaviour of the Gamma function at z = -n is

(-1)^n/(n!z) + O(1)

The first term (although without the z) I recognized it as the residue of f when z= -n. But the rest, no idea. Any explanation is very appreciated.
 
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  • #2
naaa00 said:
Hi there,

I'm actually trying to understand why the behaviour of the Gamma function at z = -n is

(-1)^n/(n!z) + O(1)

The first term (although without the z) I recognized it as the residue of f when z= -n. But the rest, no idea. Any explanation is very appreciated.

Where are you getting that asymptotic expression? Why does it involve both n and z? The Gamma function diverges at the negative integers; are you looking for an expansion of the Gamma function when z is close to a negative integer?
 

1. What is the Gamma function?

The Gamma function, denoted by Γ(z), is a mathematical function that extends the factorial function to complex and real numbers. It is defined as the integral of the exponential function with the variable in the exponent.

2. What is the behavior of the Gamma function when z = -n?

When z = -n, where n is a positive integer, the Gamma function has a singularity (a point where the function is undefined) at z = 0. This means that the function has a vertical asymptote at z = 0 and is undefined at that point.

3. How does the Gamma function behave for negative non-integer values of z?

For negative non-integer values of z, the Gamma function is undefined. This is because the function requires z to be a positive integer for it to be defined. However, it is possible to extend the Gamma function to negative non-integer values using complex analysis techniques.

4. What is the relationship between the Gamma function and the factorial function?

The Gamma function is an extension of the factorial function, which is only defined for positive integers. The Gamma function can be thought of as a continuous version of the factorial function, as it can be used to calculate factorials for non-integer values of z.

5. How does the behavior of the Gamma function change for different values of z?

The behavior of the Gamma function changes depending on the value of z. For positive values of z, the function increases rapidly and tends to infinity as z approaches infinity. For negative values of z, the function is undefined. For complex values of z, the behavior of the Gamma function is more intricate and can be studied using techniques from complex analysis.

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