1. ## 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.

2. Originally Posted by dwsmith
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?

3. That is in the same section that I am not in yet.

4. Originally Posted by dwsmith
That is in the same section that I am not in yet.
Ok, then I'm not sure how we can guide you.

5. Carefully.

6. 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.

7. Originally Posted by roninpro
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.

8. Originally Posted by roninpro
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:
$\displaystyle\int 2dx=\int g_ydx\mbox{?}$

9. If you want to work with $g_y$, you have to integrate with respect to $y$ to get $g$ back.