1. ## analytic function

$\displaystyle f:\mathbb{C}\longrightarrow\mathbb{C}$ is an analytic function and

$\displaystyle f(x+iy) = u(x,y) + iv(x,y)$

What geometric relationship exists between the level curves of $\displaystyle u$ and those of $\displaystyle v$?

Draw a rough sketch of these level curves for the case $\displaystyle f(z) = z^{2 }$to support your answer.

I haven't really any idea where to start this.

2. Originally Posted by Tekken
$\displaystyle f:\mathbb{C}\longrightarrow\mathbb{C}$ is an analytic function and

$\displaystyle f(x+iy) = u(x,y) + iv(x,y)$

What geometric relationship exists between the level curves of $\displaystyle u$ and those of $\displaystyle v$?

Draw a rough sketch of these level curves for the case $\displaystyle f(z) = z^{2 }$to support your answer.

I haven't really any idea where to start this.
Hint #1 :
Spoiler:
Cauchy-Riemann conditions

Hint #2 :
Spoiler:
Orthogonality of gradients of $\displaystyle u$ and $\displaystyle v$

Hint #3 :
Spoiler:
The gradient is perpendicular to level curves

Hint #4 : do the example first (it could also be hint #0 if you really don't know where to start).

3. well is this a start? Im clueless at picturing things in my head as regards graphing and plotting stuff

$\displaystyle f(z) = z^{2}$ and as $\displaystyle z = x+iy$

=> $\displaystyle f(x+iy) = (x^{2} - y^{2}) + 2xyi$

let $\displaystyle u = x^{2} - y^{2}$ and $\displaystyle v = 2xy$

therefore $\displaystyle U_{x} = 2x$ and $\displaystyle V_{y} = 2x$

so
$\displaystyle U_{x} = V_{y}$

and $\displaystyle U_{y} = -2y$ and $\displaystyle V_{x} = 2y$

so
$\displaystyle U_{y} = -V_{x}$

Therefore as the Cauchy Riemann equations are satisfied and the function $\displaystyle f(z) = z^{2}$ is analytic...

4. Originally Posted by Tekken
well is this a start? Im clueless at picturing things in my head as regards graphing and plotting stuff

$\displaystyle f(z) = z^{2}$ and as $\displaystyle z = x+iy$

=> $\displaystyle f(x+iy) = (x^{2} - y^{2}) + 2xyi$

let $\displaystyle u = x^{2} - y^{2}$ and $\displaystyle v = 2xy$

therefore $\displaystyle U_{x} = 2x$ and $\displaystyle V_{y} = 2x$

so
$\displaystyle U_{x} = V_{y}$

and $\displaystyle U_{y} = -2y$ and $\displaystyle V_{x} = 2y$

so
$\displaystyle U_{y} = -V_{x}$

Therefore as the Cauchy Riemann equations are satisfied and the function $\displaystyle f(z) = z^{2}$ is analytic...
What you need to do is plot a few level curves, like $\displaystyle u=-3,-2,-1,1,2,3,\ldots$ and same for $\displaystyle v$. It is easier for $\displaystyle v$: we have $\displaystyle y=\frac{v}{2x}$ (hyperbola). For $\displaystyle u$, if you don't know conics (these are also hyperbola), you can plot the level curves on your calculator or computer (equation $\displaystyle y=\pm\sqrt{x^2-u}$ (either + or -)), together with level curves for v and see what relation there is between both (you'll need an orthonormal basis to see it).

5. Originally Posted by Laurent
What you need to do is plot a few level curves, like $\displaystyle u=-3,-2,-1,1,2,3,\ldots$ and same for $\displaystyle v$. It is easier for $\displaystyle v$: we have $\displaystyle y=\frac{v}{2x}$ (hyperbola). For $\displaystyle u$, if you don't know conics (these are also hyperbola), you can plot the level curves on your calculator or computer (equation $\displaystyle y=\pm\sqrt{x^2-u}$ (either + or -)), together with level curves for v and see what relation there is between both (you'll need an orthonormal basis to see it).
Thanks i really appreciate your help. Unfortunately im not at the stage where i really understand your reply... Would you have any link to a website where i could read up on this sort of question, particularly "level curves"?

6. Originally Posted by Tekken
Thanks i really appreciate your help. Unfortunately im not at the stage where i really understand your reply... Would you have any link to a website where i could read up on this sort of question, particularly "level curves"?
I don't know if this help, but you can look here or rather there.

For a function $\displaystyle u(x,y)$, the level curves of $\displaystyle u$ are the sets of values $\displaystyle (x,y)$ that give the same value $\displaystyle u(x,y)$: for any $\displaystyle u_0\in\mathbb{R}$, the curve of level $\displaystyle u_0$ is $\displaystyle \{(x,y)\in\mathbb{R}^2|u(x,y)=u_0\}$.