What are the two definitions of gamma function and how are they related?

In summary, the gamma function is a mathematical function that extends the factorial function to complex and real numbers. Its purpose is to allow for the generalization of the factorial function to non-integer values, making it useful in areas such as statistics, physics, and engineering. The gamma function is closely related to the factorial function, with the latter being equivalent to the former for positive integer values. It is defined for all complex numbers except for non-positive integers, and its range includes all complex numbers except for negative real numbers and zero. The gamma function can be calculated using numerical methods or specialized algorithms, and it is also available as a built-in function in many mathematical software programs.
  • #1
LHS1
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I spend some time studying special functions recently. I found two definitions of gamma function, one in form of integral and the other in form of infinite products, and I cannot prove of their equivalence. I found the definition in infinite product form important in proofing many properties of gamma function such as gamma reflection formula, therefore I eager to learn it. Could anyone help me, please !
 
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  • #3
Thank you very much!
 

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 t^(z-1)e^(-t)dt from 0 to infinity.

2. What is the purpose of the gamma function?

The gamma function is primarily used in mathematical analysis, particularly in areas such as statistics, physics, and engineering. It allows for the generalization of the factorial function to non-integer values, providing a way to calculate values such as π and the volume of a hypersphere.

3. How is the gamma function related to the factorial function?

The gamma function is a generalization of the factorial function, as it can be used to calculate the factorial of non-integer values. In fact, for positive integer values, the gamma function is equivalent to the factorial function.

4. What is the domain and range of the gamma function?

The gamma function is defined for all complex numbers except for non-positive integers, where it has poles. Its range includes all complex numbers except for negative real numbers and zero.

5. How is the gamma function calculated?

The gamma function can be approximated using numerical methods, but for more accurate calculations, special algorithms such as the Lanczos approximation can be used. It is also available as a built-in function in many mathematical software programs.

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