Complex Analysis, Plotting basic stuff

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January 25th 2009, 05:15 AM
andreas

Complex Analysis, Plotting basic stuff

I have to plot these:

1) $\mid z \mid= arg Z$

2) $log \mid z \mid= -2 arg Z$

For the first one I got sort of heart shaped graph. I have no idea where to find systematic approach to such kind of problems.

When i try to substitute l Z l= $\sqrt{x^2+y^2}$ and $arg Z= \arctan{y/x}$ expressions get too complex...
January 25th 2009, 06:29 AM
Mush

Quote:

Originally Posted by andreas

I have to plot these:

1) $\mid z \mid= arg Z$

2) $log \mid z \mid= -2 arg Z$

For the first one I got sort of heart shaped graph. I have no idea where to find systematic approach to such kind of problems.

When i try to substitute l Z l= $\sqrt{x^2+y^2}$ and $arg Z= \arctan{y/x}$ expressions get too complex...

Consider what each expression stands for or represents. In the first, the modulus of Z is the length of the line that joins the point on the plane to the origin, and arg Z is the angle of Z. In other words, the length of the line is equal to its angle. So when the angle is zero, the length of the line is zero, when the line is at pi/2 radians, it has length pi/2. What you should get is an anticlockwise spiral whose radius grows continually.

What is the base of the logarithm in part 2?
January 25th 2009, 06:51 AM
Prove It

Quote:

Originally Posted by Mush

Consider what each expression stands for or represents. In the first, the modulus of Z is the length of the line that joins the point on the plane to the origin, and arg Z is the angle of Z. In other words, the length of the line is equal to its angle. So when the angle is zero, the length of the line is zero, when the line is at pi/2 radians, it has length pi/2. What you should get is an anticlockwise spiral whose radius grows continually.

What is the base of the logarithm in part 2?

It'd be a "natural" logarithm, as it is the inverse of the complex exponential. But I use that term loosely - logs don't really have a base in the complex number system because there's more to it than that...

If $Z \in \mathbf{C}$ then $\log{Z} = \ln{|Z|} + i\arg{Z}$ .

So $\log{|z|} = \ln{|(|z|)|} + i\arg{|z|}$ .

I think I just made things more confusing :P
January 25th 2009, 06:56 AM
andreas

Quote:

Originally Posted by Mush

Consider what each expression stands for or represents. In the first, the modulus of Z is the length of the line that joins the point on the plane to the origin, and arg Z is the angle of Z. In other words, the length of the line is equal to its angle. So when the angle is zero, the length of the line is zero, when the line is at pi/2 radians, it has length pi/2. What you should get is an anticlockwise spiral whose radius grows continually.

What is the base of the logarithm in part 2?

Yeah , I got the first one as you just said. So now I am pretty sure it is the correct solution. But what about the second one? Should it be done in the same fashion like first question or there exists more mature solution? I am currently thinking about it..
January 25th 2009, 07:13 AM
Opalg

If you are used to using polar coordinates, you may find that these problems look more familiar when you write $z = re^{i\theta}$ . Then the first equation has the polar form $r=\theta$ . This represents an archimidean spiral. The second one becomes $\log r = -2\theta$ , or $r = e^{-2\theta}$ , an equiangular spiral.