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Complex Numbers (FP2 Edexcel)

Hey :smile:
Basically I'm struggling with complex number transformations so I was wondering if anyone knew of any documents/resources available for me to practice them? (Preferably a few pages of just transformation questions with answers if available). I know I could go through past papers but that would take a while and it would just be easier if they were all in one place.
Thanks for any help :smile:
Reply 1
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NicCx
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6
Anyone? :colondollar:
Reply 2
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Hasufel
16
do you mean linear mappings?
Reply 3
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NicCx
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Original post by Hasufel
do you mean linear mappings?


I've never heard them called that before, so I'm not sure. Questions like 5c on this paper. https://googledrive.com/host/0B1ZiqBksUHNYdHIxUkJmdndfMlE/June%202011%20QP%20-%20FP2%20Edexcel.pdf


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Reply 4
Avatar for Hasufel
Hasufel
16
Its called a Mobuis Transformation.

if we call it T, then

T(z)=az+ccz+d\displaystyle T(z)= \frac{az+c}{cz+d}

and there is a "circle preserving property" Theorem which says:

" if C is a circle in the z plane, and T is a Mobuis Transformation, then the image of C is either a circle or a line in the extended complex plane. The image is a line if and only if c is not zero, AND the pole (where the denominator of the transformation is equal to zero) z= -d/c is on the circle.

In this case, the pole z=3i is ON the circle (0^2)+(3-1)^2=2^2=radius ^2.

(using the point 0+3i) in the cartesian eqn for the circle)

the question is really saying " if the point after the transformation has a v co-ordinate of zero, show that it`s "pre-image lies on the circle"

in other words, that any point on the circle has to be mapped to a certain line.

To find the equation of the line in the w = or (u,v) plane, we can pick, just for convenience, the 2 points in the cartesian eqn of the circle where y=0, ie where

x=+root 3, x= - root 3

put these into the eqn of the transformation, and you end up with an equation:

T(z) = w = u+iv = k+0i

in other words, a horizontal line v=0, -infinity < u < infinity

so, this means that every point on the circle is transformed to a point on the horiz line v = 0 , meaning that the point we started with (on the circle) is mapped somewhere onto the horizontal line v = 0. (interestingly, the point z = 3i maps to infinity)
(edited 9 years ago)

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