Proving analyticity of gamma function

In summary, the conversation discusses the proof of the analytical property of the gamma function using complex analysis. The proof involves defining a function and showing its uniform convergence on compact subsets, which implies the analytical property of the gamma function. The conversation also mentions the possibility of proving the analytical property using the derivative of the function. The reason for the analytical property is attributed to Morera's theorem and the Cauchy integral formula. A recommended reference for this topic is the third chapter of Freitag and Busam's "Complex analysis."
  • #1
nonequilibrium
1,439
2
Hello,

In our course of complex analysis we proved that the gamma function,
[tex]\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \mathrm d t[/tex]
for 0 < Re(z), is analytical.

We did this by defining
[tex]f_{\epsilon,R}(z) = \int_\epsilon^R t^{z-1} e^{-t} \mathrm d t[/tex]
about which we can prove that it is analytical by rewriting t^(z-1) as a power series (by using the definition of the complex power in function of the exponential and ln) and then bringing the sum outside of the integral by using the uniform convergence of the power series (on compact subsets).

Then we proved that [tex]f_{\epsilon,R} \to \Gamma[/tex] uniformly on compact subsets (of the right half-plane).

And apparently analyticity of the gamma function follows, but I don't see why. Surely not because gamma is the uniform limit of analytical functions, because a uniform limit doesn't preserve analicity. So I'm guessing there is an implicit last step I'm overlooking.

Can anybody help?
Thank you.

EDIT: personally, I think I could prove it the following way: first take the derivative of [tex]f_{\epsilon,R}[/tex], what I got is the same function but with an extra ln(t) in the integrand. However, I think one can still prove that (also) converges uniformly, but now to [tex]\int_0^\infty t^{z-1} e^{-t} \ln t \mathrm d t[/tex], and then, by the uniform limit of derivatives, the limit of the original function is differentiable and this also gives an explicit form of the derivative of the gamma function. However, this doesn't seem like the thought process that the above proof was using, and I'm interested in understanding the above proof.
 
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  • #2
Hi mr. vodka! :smile:

If a sequence of analytic functions converges (locally) uniformly then the limit function is analytic. In particular, if [itex]f_n\rightarrow f[/itex], then we have that

[tex]f_n^\prime\rightarrow f^\prime[/tex]

this is not true in the real case, but it is true in complex analysis!
 
  • #3
Interesting!

Why is this so? Do you know where I can find a proof?
 
  • #4
The analycity is a consequence of Morera's theorem. Recall that Morera's theorem says that if a continuous function has the property that

[tex]\int_\gamma f(z)dz=0[/tex]

for every closed C1-curve gamma, then f is analytic.


The clue is that uniform convergence allows you to exchange limit and integral, thus

[tex]\int_\gamma \lim_{n\rightarrow +\infty}{f_n(z)}dz=\lim_{n\rightarrow +\infty}{\int_\gamma f_n(z)}=0[/tex]

thus the limit is analytic. The statement about the derivatives follows from the Cauchy integral formula.

As a reference, I can recommend the third chapter of Freitag and Busam's "Complex analysis"...
 
  • #5
Brilliant :)

<3 Complex Analysis

thank you
 

Related to Proving analyticity of gamma function

1. What is the gamma function?

The gamma function, denoted by Γ(z), is an extension of the factorial function to complex numbers. It is defined as the integral from 0 to infinity of t^(z-1)e^(-t)dt. It is a fundamental function in mathematics and has various applications in areas such as probability, number theory, and physics.

2. Why is it important to prove the analyticity of the gamma function?

Analyticity refers to a function being differentiable at every point in its domain. Proving the analyticity of the gamma function is important because it allows us to use complex analysis techniques to study and understand the properties of the function. It also enables us to extend the use of the function to complex numbers, which is necessary for many applications.

3. What is the significance of the Cauchy-Riemann equations in proving analyticity?

The Cauchy-Riemann equations are a set of necessary and sufficient conditions for a function to be analytic. In order to prove the analyticity of the gamma function, we need to show that it satisfies these equations. This involves using techniques such as integration by parts and contour integration.

4. Are there any specific techniques used to prove the analyticity of the gamma function?

Yes, there are several techniques that can be used to prove the analyticity of the gamma function. Some common ones include the use of Cauchy's integral theorem, the residue theorem, and the power series expansion of the gamma function. These techniques require a solid understanding of complex analysis and calculus.

5. What are some applications of the analyticity of the gamma function?

The analyticity of the gamma function allows us to apply complex analysis techniques to study its properties, such as the location of its zeros and poles. This has applications in number theory, as the distribution of the gamma function's zeros is related to the distribution of prime numbers. It also has applications in physics, particularly in the study of quantum mechanics and statistical mechanics.

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