Is the Gamma Function Convergent for Re(z)<0 and Im(z)≠0?

In summary, the integral \Gamma(z) = \int\limits_0^{\infty} t^{z-1} e^{-t} dt converges when \textrm{Re}(z)>0 and diverges when \textrm{Re}(z)\leq 0 and \textrm{Im}(z)=0. However, the case when \textrm{Re}(z)\leq 0 and \textrm{Im}(z)\neq 0 is more difficult. It is possible that the rapid oscillation could make the integral convergent, but it is also possible that it diverges. The analytic continuation in this case is necessary.
  • #1
jostpuur
2,116
19
[tex]
\Gamma(z) = \int\limits_0^{\infty} t^{z-1} e^{-t} dt
[/tex]

I can see that if [itex]\textrm{Re}(z)>0[/itex], then the integral converges, and that if [itex]\textrm{Re}(z)\leq 0[/itex] and [itex]\textrm{Im}(z)=0[/itex], then it diverges. However, I found the case [itex]\textrm{Re}(z)\leq 0[/itex] and [itex]\textrm{Im}(z)\neq 0[/itex] more difficult.

[tex]
t^{z-1} = t^{x-1}\big(\cos(y\log(t)) + i\sin(y\log(t))\big),\quad\quad z=x+iy
[/tex]

Clearly the positive and negative parts (of real and imaginary parts) are not integrable alone, but it could be, assuming that we don't yet know the right answer, that the rapid oscillation would make the integral

[tex]
\lim_{\delta\to 0^+} \int\limits_{\delta}^{\infty} t^{z-1} e^{-t} dt
[/tex]

convergent. So, indeed, is it convergent or not? Anyone knowing proofs?

edit: Heuristically speaking, when [itex]t\to 0^+[/itex], then [itex]\log(t)\to-\infty[/itex] rather slowly, so my guess is that the oscillations are going to be too slow for convergence.

edit edit: hmhmh... or then I could approximate [itex]e^{-t}=1+O(t)[/itex], and actually integrate [itex]t^{z-1}[/itex] with formula [itex]t^{z-1} = D_t \frac{1}{z}t^z[/itex], and see that it diverges...

Thanks for the attention, and I hope you enjoyed the entertainment.
 
Last edited:
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  • #2
You must use the analytic continuation in re z < 0.
 

1. What is the gamma function and why is it important in mathematics?

The gamma function is a special function in mathematics denoted by the Greek letter gamma (Γ). It is an extension of the factorial function and is defined for all complex numbers except the negative integers. The gamma function is important because it has various applications in fields such as probability, statistics, and complex analysis. It also plays a crucial role in the evaluation of certain integrals and series.

2. What is meant by "gamma function convergence"?

Gamma function convergence refers to the behavior of the gamma function as its argument approaches a particular value. In other words, it is the study of how the gamma function behaves near its singularities, which are located at the negative integers. This concept is essential in understanding the properties of the gamma function and its applications in mathematics.

3. What are the conditions for the gamma function to converge?

The gamma function converges for all complex numbers except the negative integers. This means that the only requirement for the gamma function to converge is that its argument must not be a negative integer. However, for certain values of the argument, the gamma function may have poles or branch points, which affect its convergence properties.

4. How is the convergence of the gamma function related to the Riemann zeta function?

The Riemann zeta function is closely related to the gamma function through the functional equation ζ(s)Γ(s/2) = π−s/2Γ(1−s/2)ζ(1−s). This equation allows us to extend the definition of the Riemann zeta function to all complex numbers except the point s=1, where it has a simple pole. Therefore, the convergence of the gamma function is essential in understanding the behavior of the Riemann zeta function for complex arguments.

5. What are some applications of gamma function convergence in real-world problems?

The concept of gamma function convergence has various applications in fields such as physics, engineering, and economics. For example, it is used in statistical distributions to model the behavior of random variables. It also plays a crucial role in the evaluation of integrals and series, which arise in many real-world problems. Additionally, the gamma function convergence is essential in the study of special functions and their applications in different areas of mathematics and science.

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