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Thread: Analytic Function

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    Analytic Function

    Suppose f is an analytical function of the complex variables $\displaystyle z=x+iy$ given by:
    $\displaystyle f(z)=(2x+3y)+ig(x,y)$
    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 $\displaystyle z=x+iy$ given by:
    $\displaystyle f(z)=(2x+3y)+ig(x,y)$
    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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    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 $\displaystyle f(x,y)=u(x,y)+i v(x,y)$. We have that $\displaystyle f$ is analytic if and only if $\displaystyle u_x=v_y$ and $\displaystyle u_y=-v_x$ (simultaneously).

    Now, to apply these to your question, you know that $\displaystyle u_x=2$, $\displaystyle u_y=3$, $\displaystyle v_x=g_x$, and $\displaystyle v_y=g_y$. From the equations above, we have $\displaystyle 2=g_y$ and $\displaystyle 3=-g_x$. You can integrate to recover $\displaystyle g$. Don't forget to use $\displaystyle g(2,3)=1$ 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 $\displaystyle f(x,y)=u(x,y)+i v(x,y)$. We have that $\displaystyle f$ is analytic if and only if $\displaystyle u_x=v_y$ and $\displaystyle u_y=-v_x$ (simultaneously).

    Now, to apply these to your question, you know that $\displaystyle u_x=2$, $\displaystyle u_y=3$, $\displaystyle v_x=g_x$, and $\displaystyle v_y=g_y$. From the equations above, we have $\displaystyle 2=g_y$ and $\displaystyle 3=-g_x$. You can integrate to recover $\displaystyle g$. Don't forget to use $\displaystyle g(2,3)=1$ 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 $\displaystyle u,v$ 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 $\displaystyle f(x,y)=u(x,y)+i v(x,y)$. We have that $\displaystyle f$ is analytic if and only if $\displaystyle u_x=v_y$ and $\displaystyle u_y=-v_x$ (simultaneously).

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

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