Contribution of Semi-Circle in Complex Analysis Integration

In summary, the conversation discusses choosing a semi-circle in a halfplane to use Cauchy's integral theorem for a given integral. The question is asked about the behavior of the function f(s)exp(st) as s goes to infinity and if the contribution from the semi-circle being zero implies that the limit of the function also goes to zero. The expert suggests using the behavior of e^{at} to determine the correct halfplane for the semi-circle.
  • #1
Niles
1,866
0

Homework Statement


Hi all.

I have the following integral:

[tex]
I = \int_{2 - i\infty}^{2+i\infty}{f(s) \exp(st)ds},
[/tex]

where [itex]f(s)[/itex] is some function. In order to perform this integral, I will choose to close the vertical line with a semi-circle in some halfplane (in order to use Cauchy's integral theorem), but this requires that the contribution from the semi-circle is zero.

Question: Now, let us say that for the specific [itex]f(s)[/itex] in this case, then the contribution from the semi-circle in e.g. the right halfplane does go to zero: With this in mind, then will the limit [itex]f(s)\exp(st)[/itex] for [itex]s\rightarrow \infty[/itex] be zero?Thank you very much in advance.Niles.
 
Last edited:
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  • #2
I don't quite understand your question... the contribution from the semi-circle in the right halfplane going to zero is a statement of integrating the function over the whole semicircle. Are you asking about how the function f(s)exp(st) behaves as s goes to infinity for any constant t (is t real or complex?)? Because if you already know how the integral over the semi-circle behaves you don't really care about that in terms of answering the question as far as I can tell
 
  • #3
The variable t is real.

What I am asking is that if the following is true

[tex]
\int\limits_{semi - circle, right\,\, halfplane} {f(s)e^{st} ds = 0},
[/tex]

then does this imply that

[tex]
\mathop {\lim }\limits_{s \to \infty } f(s)e^{st} = 0
[/tex]
?

The reason why I am asking is because I am looking for a method to find the half plane, where the contribution from the semi-circle will be zero. This I can do by the above limit (if what I am asking is correct), because we are guaranteed that one of the half planes will be correct.
 
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  • #4
Well, you can't just take a limit as s goes to infinity, since obviously we don't expect the function to uniformly decrease in all directions (if it did, we wouldn't have to worry about which half circle to take). The easiest way to make this determination is to note that if s=a+bi then

[tex]|f(s)e^{st}| = |f(s)||e^{at+bti}| = |f(s)|e^{at}|[/tex] as [tex]|e^{tbi}|=1[/tex] always. For most functions f that you see in this context, f grows and shrinks slowly compared to exp. In these cases, we're interested in the behavior of [tex]e^{at}[/tex] which if t is positive, you want to have a be negative, and vice versa (in order to make it small). It's a pretty quick and dirty way of figuring out which way the circle should probably be pointing
 
  • #5
Thanks. It is kind of you to help me.
 

1. What is complex analysis integration?

Complex analysis integration is a branch of mathematics that deals with the integration of functions of complex variables. It combines the techniques of calculus and complex numbers to solve problems involving complex functions.

2. How is complex analysis integration different from real analysis integration?

Complex analysis integration differs from real analysis integration in that it deals with functions of complex variables, whereas real analysis integration deals with functions of real variables. Complex analysis integration also has additional techniques and theorems that are specific to complex functions.

3. What is the Cauchy Integral Theorem?

The Cauchy Integral Theorem states that if a function is analytic (i.e. differentiable) in a closed and simply connected region, then the integral of that function along any closed path within that region is equal to zero.

4. How is the Cauchy Integral Theorem used in complex analysis integration?

The Cauchy Integral Theorem is a powerful tool in complex analysis integration because it allows us to evaluate complex integrals without having to explicitly calculate the integral itself. Instead, we can use the theorem to relate the complex integral to the values of the function at points within the region.

5. What are some common applications of complex analysis integration?

Complex analysis integration has many applications in physics, engineering, and other areas of mathematics. Some examples include solving differential equations with complex coefficients, calculating electric fields and potential in electromagnetism, and studying fluid flow in fluid mechanics. It is also used in the study of number theory and in the development of algorithms for data processing and image processing.

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