Trouble understanding the Gamma function

In summary, the Gamma function is an integral given by \Gamma(z)=\int_{0}^{\infty}{dt\, t^{z-1}e^{-t}}, which has poles for integers of z less than 1 and is finite everywhere else. However, for non-integer values of z less than 0, the integral seems to have an infinite contribution right after t=0. This can be seen by expanding the integrand into a Laurent series. To extend the function to the rest of the complex plane, analytic continuation or enforcing the property \Gamma(z+1)=z\Gamma(z) is necessary.
  • #1
vcdfrexzaswq
5
0
The http://en.wikipedia.org/wiki/Gamma_function" is the integral [tex]\Gamma(z)=\int_{0}^{\infty}{dt\, t^{z-1}e^{-t}}[/tex] . It has poles for integers of z less than 1 and is finite everywhere else. But to me it seems like it should be infinite for non integer values of z less than 0.

My reasoning: when t is close to 0, the function t[tex]^{z-1}[/tex]e[tex]^{-t}[/tex] is approximately equal to t[tex]^{z-1}[/tex].

The integral [tex]\int_{t_1}^{t_2}{dt\, t^{z-1}}[/tex] is equal to ([tex]t_2^{z}[/tex] - [tex]t_1^{z}[/tex])/z. When z<0 and [tex]t_1 = 0[/tex], [tex]t_1^{z}[/tex] is infinite. Thus it seems the integral [tex]\int_{0}^{\infty}{dt\, t^{z-1}e^{-t}}[/tex] has an infinite contribution right after t=0 for z<0.
 
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  • #2
properties of the integrand may be seen by expanding it into a Laurent series
 
  • #3
vcdfrexzaswq said:
The http://en.wikipedia.org/wiki/Gamma_function" is the integral [tex]\Gamma(z)=\int_{0}^{\infty}{dt\, t^{z-1}e^{-t}}[/tex] .

Only when z has positive real part. It is extended to the rest of the complex plane by analytic continuation or, equivalently, by enforcing that [itex]\Gamma(z+1)=z\Gamma(z)[/itex] holds everywhere.
 
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  • #4
gel said:
Only when z has positive real part. It is extended to the rest of the complex plane by analytic continuation or, equivalently, by enforcing that [itex]\Gamma(z+1)=z\Gamma(z)[/itex] holds everywhere.

Ok thanks
 

Related to Trouble understanding the Gamma function

1. What is the Gamma function?

The Gamma function is a mathematical function that is defined as an extension of the factorial function to complex numbers. It is denoted by the Greek letter Γ (gamma) and is used to solve problems in areas such as number theory, combinatorics, and statistics.

2. Why is it difficult to understand the Gamma function?

The Gamma function can be challenging to understand because it involves complex numbers and has many different properties and applications. It also has a complex relationship with other mathematical functions, making it a difficult concept for some to grasp.

3. How is the Gamma function calculated?

The Gamma function can be calculated using various methods, such as the Lanczos approximation or the Stirling's approximation. It can also be calculated using numerical integration or by using special functions in computer software such as MATLAB or Mathematica.

4. What are the main uses of the Gamma function?

The Gamma function has various applications in mathematics, including the calculation of definite integrals, solutions to differential equations, and the computation of probabilities in statistics. It is also used in the fields of physics, engineering, and economics.

5. Are there any real-life examples of the Gamma function?

Yes, the Gamma function has real-life applications in fields such as physics, where it is used in the Schrödinger equation to describe the behavior of subatomic particles. It is also used in the calculation of probabilities in the stock market and in the design of computer algorithms for data compression.

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