A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

Loser66

  • one year ago

Confuse!! Please, help Problem: Where is the function differentiable? \(x^2+y^2 +2ixy\) Where is it analytic?

  • This Question is Closed
  1. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \(U_x = 2x = V_y \\U_y = 2y = -V_x = -2y \iff y =0\) Hence the function is differentiable on x axis. My question: is it not that it is analytic on x axis also? why the answer is : "No where" ?

  2. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    hmm you sure this is right?

  3. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    What is your suspicion?

  4. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    checking the partials although it can't be that hard, i think it must be of by a minus sign somewhere

  5. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \(U(x,y) = x^2 +y^2 \\V(x,y) = 2xy\) \(U_x = 2x\\U_y= 2y\\V_x = 2y\\V_y = 2x\) I am sure about that.

  6. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    then you are off by a minus sign for cauchy riemman right?

  7. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \(U_x = V_y=2x\\U_y = -V_x\iff 2y = -2y \iff y =0\) I am sure about that also.

  8. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yes,

  9. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    for cauchy riemann \(U_y=-V_x\)

  10. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yes

  11. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    but is isn't

  12. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Hence, the conclusion is the function is differentiable when y =0

  13. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    That is it is differentiable on Real axis, right?

  14. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yeah on the real axis it is \(f(x)=x^2\)

  15. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    But for part b) Where is it analytic, From above it is differentiable on Real, it should be analytic on Real also, but the answer is "No where"

  16. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    It is not f (z)

  17. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    differentiable means the derivative exists from any direction

  18. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    f(z) = x^2 +y^2 +2ixy,

  19. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    and the whole thing differentiable at the point on Real axis only. We can't take off y from f, right?

  20. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    no just like on the real line differentiable means the limit exists from the left and the right, in complex plane the limit must exist from any direction

  21. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    btw if they didn't mention it, as i remember correctly "analytic" means differentiable, but also means you can write the function as a function of \(z\)

  22. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1443751466835:dw|

  23. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    which is probably not not the case here, \[f(z)=z^2=x^2-y^2+2xyi\]

  24. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i am pretty sure the answer is no the limit (difference quotient) has to exist from any direction

  25. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Another way of repeating what @satellite73 has already stated is that a function can't be analytic on a line - it can only be analytic on a disk. It can be differentiable on a line, but to be analytic the function has to be differentiable no matter how you approach the point in the complex plane.

  26. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I got it. Hence, from neighborhood of a point on Real axis, f is not differentiable,

  27. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the Cauchy-Riemann equations that govern complex differentiability of \(f(x+iy)=u(x,y)+iv(x,y)\) basically require that we have the following condition on a Wirtinger derivative: $$\frac{\partial f}{\partial\bar z}=0$$

  28. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    this is the connection to 'being written as a function of only \(z\), not \(\bar z\)' @satellite73

  29. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.