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Math Help - Complex analytic functions

  1. #1
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    Complex analytic functions

    Hey guys.
    I hope this is the right place to post this question.

    http://img641.imageshack.us/img641/70/97822806.jpg

    I have this "simple" complex function, and I need to decide where this function is analytic in the complex domain.
    So, I used the cauchy riemann equations as you can see, and I got to the condition x=y in order for this function to have a derivation. I know that this function is not analytic anywhere on the complex domain.
    Now, according to the solution we got, this function has derivation only in (0,0), is this true, shouldn't it be on all x=y?

    Sorry for the bad English.
    Thanks a lot.
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  2. #2
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    Quote Originally Posted by asi123 View Post
    Hey guys.
    I hope this is the right place to post this question.

    http://img641.imageshack.us/img641/70/97822806.jpg

    I have this "simple" complex function, and I need to decide where this function is analytic in the complex domain.
    So, I used the cauchy riemann equations as you can see, and I got to the condition x=y in order for this function to have a derivation. I know that this function is not analytic anywhere on the complex domain.
    Now, according to the solution we got, this function has derivation only in (0,0), is this true, shouldn't it be on all x=y?

    Sorry for the bad English.
    Thanks a lot.

    אם אתה בודק את משוואות קושי-רימן זה ברור שהמקום היחיד הן מתקייתות זה בראשית, אך הגדרה אנליטיות דורשת שהן יתקיימו באיזה שהוא תחום (קבוצה פתוחה) במישור המרוכב...ז"א, הפונקציה איננה אנליטית באף תחום

    If you check the Cauchy-Riemann equations then it's clear that the only place they're fulfilled is the origin, whereas the definition of analiticity requires that they exist in a domain (open subset of the complex plane), thus the function isn't analytic in any domain.

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    אם אתה בודק את משוואות קושי-רימן זה ברור שהמקום היחיד הן מתקייתות זה בראשית, אך הגדרה אנליטיות דורשת שהן יתקיימו באיזה שהוא תחום (קבוצה פתוחה) במישור המרוכב...ז"א, הפונקציה איננה אנליטית באף תחום

    If you check the Cauchy-Riemann equations then it's clear that the only place they're fulfilled is the origin, whereas the definition of analiticity requires that they exist in a domain (open subset of the complex plane), thus the function isn't analytic in any domain.

    Tonio
    אז זה מה שלא הבנתי, לפי משוואות קושי ריימן, המשוואות נכונות עבור
    x=-y
    או שאני מפספס משהו?

    תודה.
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  4. #4
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    Quote Originally Posted by asi123 View Post
    אז זה מה שלא הבנתי, לפי משוואות קושי ריימן, המשוואות נכונות עבור
    x=-y
    או שאני מפספס משהו?

    תודה.

    נדמה לי שאתה מפספס משהו: צריך להיות

    u_x=v_y\,,\,\,u_y=-v_x

    במקרה שלנו מקבלים

    u=x^2\,,\,v=y^2\Longrightarrow u_x=2x,\,v_y=2y\,,\,\,u_y=-v_x=0\Longrightarrow u_x=v_y\Longleftrightarrow x=y

    אבל אז מקבלים

    f(x,y)=x^2+ix^2\Longrightarrow u_x=2x=v_{y=x}\,,\,\,u_{y=x}=2x\,,\, -v_x=-2x\Longrightarrow  u_y=-v_x\Longleftrightarrow x=y=0

    Tonio
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