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Math Help - Analytic Function

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    Analytic Function

    Suppose f is an analytical function of the complex variables z=x+iy given by:
    f(z)=(2x+3y)+ig(x,y)
    where g(x,y) is a real valued function of real variables x and y. If g(2,3)=1, then g(7,3)=?

    From reading my book, I understand this going to be treated somewhat similar to a solving a differential equation; however, I am not sure what to do. I haven't made it to this section yet in Complex Analysis but in a week I have the GRE Math Subject Test and this is a practice question.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by dwsmith View Post
    Suppose f is an analytical function of the complex variables z=x+iy given by:
    f(z)=(2x+3y)+ig(x,y)
    where g(x,y) is a real valued function of real variables x and y. If g(2,3)=1, then g(7,3)=?

    From reading my book, I understand this going to be treated somewhat similar to a solving a differential equation; however, I am not sure what to do. I haven't made it to this section yet in Complex Analysis but in a week I have the GRE Math Subject Test and this is a practice question.
    Do you know about harmonic functions/the Cauchy-Riemann formulas?
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  3. #3
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    That is in the same section that I am not in yet.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by dwsmith View Post
    That is in the same section that I am not in yet.
    Ok, then I'm not sure how we can guide you.
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    Carefully.
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    Senior Member roninpro's Avatar
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    Maybe we can state the Cauchy-Riemann equations. They're easy to understand.

    Let f(x,y)=u(x,y)+i v(x,y). We have that f is analytic if and only if u_x=v_y and u_y=-v_x (simultaneously).

    Now, to apply these to your question, you know that u_x=2, u_y=3, v_x=g_x, and v_y=g_y. From the equations above, we have 2=g_y and 3=-g_x. You can integrate to recover g. Don't forget to use g(2,3)=1 to get the constant of integration.

    Give it a try.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by roninpro View Post
    Maybe we can state the Cauchy-Riemann equations. They're easy to understand.

    Let f(x,y)=u(x,y)+i v(x,y). We have that f is analytic if and only if u_x=v_y and u_y=-v_x (simultaneously).

    Now, to apply these to your question, you know that u_x=2, u_y=3, v_x=g_x, and v_y=g_y. From the equations above, we have 2=g_y and 3=-g_x. You can integrate to recover g. Don't forget to use g(2,3)=1 to get the constant of integration.

    Give it a try.
    Just a slight remark (payback for the simple connectedness thing ) we need to have that u,v are continuously differentiable.
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    Quote Originally Posted by roninpro View Post
    Maybe we can state the Cauchy-Riemann equations. They're easy to understand.

    Let f(x,y)=u(x,y)+i v(x,y). We have that f is analytic if and only if u_x=v_y and u_y=-v_x (simultaneously).

    Now, to apply these to your question, you know that u_x=2, u_y=3, v_x=g_x, and v_y=g_y. From the equations above, we have 2=g_y and 3=-g_x. You can integrate to recover g. Don't forget to use g(2,3)=1 to get the constant of integration.

    Give it a try.
    Integrating like this:
    \displaystyle\int 2dx=\int g_ydx\mbox{?}
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  9. #9
    Senior Member roninpro's Avatar
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    If you want to work with g_y, you have to integrate with respect to y to get g back.
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