Analitic function

  1. [tex]\oint dz\frac{e^{ikz}}{z}[/tex]

    How we know for [tex]k>0[/tex] is function analytic in upper or in lower half plane?
     
    Last edited: Dec 2, 2011
  2. jcsd
  3. jgens

    jgens 1,621
    Gold Member

    I am assuming you want to calculate:
    [tex]\int_{\gamma}\frac{exp(ikz)}{z} \, \mathrm{d}z[/tex]
    where [itex]\gamma[/itex] is some closed loop such that [itex]0 \in \mathrm{Int}(\gamma)[/itex].

    If this is the case, you can the integral using the Residue Theorem. That is, write [itex]exp(ikz)[/itex] as a power series. Divide each term of the power series by [itex]z[/itex] to obtain a meromorphic function. You can then perform the integration and the only term that contributes to the value of the integral is the residue.
     
  4. I asked what I want to know. I don't understand why if I have

    [tex]\oint \frac{e^{ikz}}{z}[/tex], [tex]k>0[/tex] function is analytic in upper half plane if I [tex]k<0[/tex] function is analytic in lower half plane? Why?
     
  5. jgens

    jgens 1,621
    Gold Member

    The contour integral is a number, not a function. So asking if the integral is analytic in the upper/lower half-plane doesn't seem to make much sense. If you want to consider the function all of whose values are equal to the contour integral, then this is just a constant function and is obviously analytic on the upper and lower half-plane.

    So, unless your question is about trivialities, I think you need to be more precise.
     
  6. My mistake. Why function [tex]\frac{e^{ikz}}{z}[/tex] is analytic in upper half plane for [tex]k>0[/tex]?
     
  7. jgens

    jgens 1,621
    Gold Member

    Well, exp(ikz) is entire and therefore analytic on the whole plane. If we divide the power series of exp(ikz) by z, we see that exp(ikz)/z has a simple pole at k = 0 but is well-defined everywhere else. Which means that our series expansion for exp(ikz)/z is valid on the upper half-plane.
     
  8. Why we close counture in upper half plane for k>0?
     
  9. jgens

    jgens 1,621
    Gold Member

    What do you mean by 'close counture'?
     
  10. When we calculate integral which I wrote we use contour in upper half plane for k>0. Why?
     
  11. jgens

    jgens 1,621
    Gold Member

    I do not know why you would choose to do this. Since the contour will be chosen over a closed curve and since exp(ikz)/z is analytic in the upper half-plane, this means if we integrate along any closed curve which lies entirely in the upper half-plane, the integral will necessarily be zero.
     
  12. I will also take a small conture around zero.
     
  13. Understand now?
     

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  14. I want to calculate integral [tex]\int^{\infty}_{-\infty}\frac{sinkx}{x}[/tex] use integration which I wrote. Why for k>0 in upper plane? Tnx.
     
  15. jgens

    jgens 1,621
    Gold Member

    Ah! That is not a contour in the upper half-plane; the upper half-plane excludes the real axis. And you take the integral that way because you know exp(ikz)/z is analytic everywhere except 0 and there is a theorem that involves evaluating real integrals using complex integrals like the one you have.
     
  16. Need to be more clear mate. We choose the upper half-contour for k>0 in that integral because we wish the integral over the large semi-circle to tend to zero as R goes to infinity. Consider the expression:

    [tex]e^{ikz}[/tex]

    for [itex]z=Re^{it}[/itex]

    thats:

    [tex]e^{ikR(\cos(t)+i\sin(t))}[/tex]

    Now consider it's absolute value:

    [tex]e^{-kR\sin(t)}[/tex]

    In the upper half-plane, sine is positive so that will tend to zero for k>0. And if k<0, they we'd have to divert the contour to the lower half-plane because then sin(t)<0.
     
  17. Thanks mate. Sorry again. Is there some easy way to see that that integral will go to zero? When you see

    [tex]\oint\frac{e^{ikz}}{z}[/tex]?
     
  18. Ok, that one "looks" like you're just going around the origin but really you mean the half-disc contour in either half-plane. Around the origin, the integral is 2pi i. Otherwise if it's around the discs. You could be more specific like:

    [tex]\mathop\oint\limits_{|z|=1} \frac{e^{ikz}}{z}dz[/tex]

    that's really clear or in the other case:

    [tex]\mathop\oint\limits_{D} \frac{e^{ikz}}{z}dz[/tex]

    then clearly specify in the text what D is. In regards to you question about the integral over the upper half-disc around the semi-circle, well, you just need to plug it all in and analyze it to see what happens as R goes to infinity.
     
  19. Thanks. What about

    [tex]\lim_{R\to\infty}e^{ikR\cos t}[/tex]?
     
  20. How about you answer that assuming k, R and t are real numbers. Use the Euler identity:

    [tex]e^{ix}=\cos(x)+i\sin(x)[/tex]

    What is the maximum in absolute value that expression attains? Does it ever settle down to a limit no matter how large x gets?
     
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