Author Topic: Does anyone have the definitions for a certain type of function in terms of z? (Read 867 times) Tweet Share

Linkage1992 · « **on:** October 31, 2010, 08:12:08 pm »

For instance,you know how |z+1| < 1 is the definition of a circle, could you please give me the definitions for hyperbolas, ellipses etc?
thankyou!

sajib_mostofa · « **Reply #1 on:** October 31, 2010, 08:17:22 pm »

$Arg(x+iy) = \theta$ gives a ray or straightline

$|z+x+iy| = |z-x+iy|$ gives a perpendicular bisector (another straightline)

Linkage1992 · « **Reply #2 on:** October 31, 2010, 08:23:06 pm »

thanks for that.

do you also know if there are any for hyperbolas and ellipses?

JinXi · « **Reply #3 on:** October 31, 2010, 08:24:31 pm »

|z+A|+ |z+B| = C where a,b,c are constants gives ellipse

Not sure about complex hyperbola's though, haven't come across them in Spec's course

m@tty · « **Reply #4 on:** October 31, 2010, 08:25:37 pm »

$\{z:|z-w||\bar{z}-\bar{w}|=r^2\} \text{ where } z,w \in \mathbb{C} \text{ and }r\in\mathbb{R} \text{ is a circle of radius r and center w.}$

EDIT: added modulus signs.

Linkage1992 · « **Reply #5 on:** October 31, 2010, 08:27:45 pm »

thanks. karma for everybody!

m@tty · « **Reply #6 on:** October 31, 2010, 08:35:21 pm »

Ah, for what I had earlier; there's meant to be modulus signs instead of brackets, sorry.

jasoN- · « **Reply #7 on:** October 31, 2010, 08:43:19 pm »

are you sure? I've got it down as:
$(z-(a+bi))(\bar{z}-(a-bi)) = R^2$ Center (a,b) Radius R
expanding:
$(x+yi-a-bi)(x-yi-a+bi) = R^2$
$((x-a)+(y-b)i)((x-a)-(y-b)i) = R^2$
$(x-a)^2 - (y-b)^2i^2 = R^2$
$(x-a)^2 + (y-b)^2 = R^2$

JinXi · « **Reply #8 on:** October 31, 2010, 08:50:35 pm »

What I got in my notes are |z-(a+bi)|=4,

Circle with center a,b with radius 4.

Sqrt((x-a)^2+(y-b)^2)) = 4, square both sides give

(x-a)^2+(y-b)^2 = 16, which is your typical circle

m@tty · « **Reply #9 on:** October 31, 2010, 08:52:21 pm »

Quote from: jasoN- on October 31, 2010, 08:43:19 pm

are you sure? I've got it down as:
$(z-(a+bi))(\bar{z}-(a-bi)) = R^2$ Center (a,b) Radius R
expanding:
$(x+yi-a-bi)(x-yi-a+bi) = R^2$
$((x-a)+(y-b)i)((x-a)-(y-b)i) = R^2$
$(x-a)^2 - (y-b)^2i^2 = R^2$
$(x-a)^2 + (y-b)^2 = R^2$

First I did it from memory, but looking at Derrick Ha's book, there are modulus signs.

...

Wait, they BOTH define the same region.

EDIT:

$|z-w||\bar{z}-\bar{w}|=r^2$

$z=x+yi \text{ and } w=a+bi$

$|x+yi-(a+bi)||x-yi-(a-bi)|=r^2$

$\sqrt{(x-a)^2+(y-b)^2}\cdot \sqrt{(x-a)^2+(-y+b)^2}=r^2$

$\implies (x-a)^2+(y-b)^2=r^2$

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Author Topic: Does anyone have the definitions for a certain type of function in terms of z? (Read 867 times) Tweet Share

Linkage1992

Does anyone have the definitions for a certain type of function in terms of z?

sajib_mostofa

Re: Does anyone have the definitions for a certain type of function in terms of z?

Linkage1992

Re: Does anyone have the definitions for a certain type of function in terms of z?

JinXi

Re: Does anyone have the definitions for a certain type of function in terms of z?

m@tty

Re: Does anyone have the definitions for a certain type of function in terms of z?

Linkage1992

Re: Does anyone have the definitions for a certain type of function in terms of z?

m@tty

Re: Does anyone have the definitions for a certain type of function in terms of z?

jasoN-

Re: Does anyone have the definitions for a certain type of function in terms of z?

JinXi

Re: Does anyone have the definitions for a certain type of function in terms of z?

m@tty

Re: Does anyone have the definitions for a certain type of function in terms of z?

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