i got this question i cant seem to figure how to work out after knowing the mapping
#complex analysis question
dim kraken
tribal basin
parameterise for y=x+1 and get y(t)=t+1, x(t)=t t∈ℝ
z=x+iy
(x+iy+1)/(x+iy-1)
(t+i(t+1)+1)/(t+i(t+1)-1)
rationalise the denominator
-(t+1+i+it)(it+i+1-x)/(2t²+2)
((2t²+2t)/(2t²+2))-i((2+2t)/(2t²+2))
now I'll graph this and post it here:
which is (x-0.5)²+(y+0.5)²=0.5
or |z-0.5+0.5i|=√0.5
@dim kraken
dim kraken
parameterise for y=x+1 and get y(t)=t+1, x(t)=t t∈ℝ
z=x+iy
(x+iy+1)/(x+iy-1)
(t+...
Thank youuuu
late mapleBOT
@dim kraken has given 1 rep to @tribal basin
tribal basin
@dim kraken I made a minor error in my modulus form, the radius is not 0.5², it's √0.5
if you don't mind me asking, what exactly were you struggling with?
I have a habit of just giving answers and not really helping the surrounding understanding
dim kraken
if you don't mind me asking, what exactly were you struggling with?
i just saw this now ooop, so in this module i need to find the decomposition of the f(z) graph in terms of translation, magnification, inversion and then calculate the mapping of the line. i can figure out the decomposition but when calculating the mapping using it i never get the right answer
i can show you the answer my lecturer gave if that would help
tribal basin
ok
tribal basin
ok that makes sense, he just solved it with complex coordinates rather than parametrics
if you find parametrics easier then do those
dim kraken
ok that makes sense, he just solved it with complex coordinates rather than para...
but i need to do it with complex coordinates because thats what the module is on xDD
like i dont understand how he got the answer for the first translation
tribal basin
parametrics and complex coordinates are very similar
dim kraken
i know its -1 but then het got (-1+i)+ sqrt(2)e etc
tribal basin
(f(t),g(t))→f(t)+ig(t)
dim kraken
and i didnt get it T_T
dim kraken
(f(t),g(t))→f(t)+ig(t)
hmmm
tribal basin
so z~(t)=(t-1)+i(t+1) because it was translated by -1, right?
then you expand the brackets and get z~(t)=t-1+ti+i→(-1+i)+t+ti where t+ti=t(1+i)→(e^(iπ/4))√2
so now you have (-1+i)+√2(e^iπ/4)t
-1 on the real part
@dim kraken
dim kraken
so z~(t)=(t-1)+i(t+1) because it was translated by -1, right?
then you expand th...
ohhhhh wait i see it now xDD that makes sm more sense
i then got confused when he did the magnification
where we did w=2w
tribal basin
notice on the step before when the 2(u²+v²)+u+v is converted into a circle equation
where did the 2 go?
because it's equal to 0 the 2 was divided out
which makes it necessary to do the magnification on a later step
dim kraken
but wouldnt dividing w by 2 give you 1/4 at the end and not 1/2
tribal basin
(u/2 + ¼)² when you get u to be just u you get (u+½)² which because that multiplication in the brackets is 2, it is 2 everywhere within the bracket. but it's effect is 2² so 4
dim kraken
OHHH wait so u only divide the u and v components for the magnifcation rather than the whole thing?
tribal basin
pretty much
tribal basin
sure
tribal basin
yeah
tribal basin
ok
dim kraken
x(t)=t y(t)=3t-⅓
yeah i got that far xDD
but idk why he did the magnification first rather than the translation
i thought u was supposed to do it in order?
tribal basin
doesn't really matter which order as long as you account for it, but I agree he should have done it in order for ease of understanding
dim kraken
doesn't really matter which order as long as you account for it, but I agree he ...
can i add you so i can ask you more questions if i need it xDD
tribal basin
sure, I won't always be able to answer them in a timely manner though
so might be a better idea to post them here
dim kraken
sure, I won't always be able to answer them in a timely manner though
thank youuuuuuu uve helped a lot so far hehe
late mapleBOT
@dim kraken has given 1 rep to @tribal basin
tribal basin
no problem
dim kraken
no problem
wait no i dont get how he got y=3x 😭
tribal basin
wait no i dont get how he got y=3x 😭
he just applied a translation of +1
+1
I can't read
dim kraken
but why
wasnt the translation supposed to be +3
this module is gonna be the death of me i swear
tribal basin
makes it easier to deal with, constants won't change whether it's a straight line or a circle
tribal basin
wasnt the translation supposed to be +3
nah, +1 only shifts the function, but it makes y=3x so much easier to work with
then you have x(t)=t and y(t)=3t
so much better
dim kraken
but why wasnt the magnification eq y=9x-1 instead of y=3x-1
because isnt magnification supposed to be for the whole eq
tribal basin
because it's applied only to the constant
dim kraken
all the time?
tribal basin
with analysing these functions and just finding whether it's a straight line or a circle or whatever you can almost do whatever you want with a constant
dim kraken
so he just applied the magnification to the constant alone and left the t component alone?
dim kraken
would it be wrong for me to continue on with 9t or
tribal basin
well, it would get you the same answer in terms of straight line or circle, but not the desired answer
treat it as though he just did the translation ⅓
dim kraken
why 1/3 😭 this is so confusing
tribal basin
not for this method of solution, your lecturer is a very strange man with his solutions
dim kraken
not for this method of solution, your lecturer is a very strange man with his so...
but if i was to use the decomposition from part a would i get the same answr
tribal basin
same overall answer yes
dim kraken
same overall answer yes
okay let me try that and then get back to you
dim kraken
same overall answer yes
NVM ITS SO LONG but from my understanding, what your saying is when it comes to finding out whether its a line or circle the value of the translation is insignificant and same with the magnification of it?
dim kraken
okay yeah i get that
dim kraken
like no scalar multiple of 2x will make it 2x²
OHHHHHH I GET why you said he couldve just translated it by 1/3 now, so then the magnification step wouldve been unnecessary when doing that
tribal basin
lmao
tribal basin
no?
dim kraken
whaaaa
tribal basin
what's the general form for a complex number
x+iy
what's the parametric for y and x
t=x(t) 3t=y(t)
t-3it
dim kraken
ohhhhhh yeah i did it the wrong way oops
how would you apply the rotation ?
@tribal basin
tribal basin
not entirely sure without seeing what you have and how you got it
dim kraken
i got the same w(t) =t-3it as him and in part a it rotates by -pi/2
but idk how to apply that to the w(t)
tribal basin
rotation by -pi/2 means you find the line orthogonal with the same constants
well
yeah pretty much, negative 90° and 90° with a line is the same
dim kraken
yeah pretty much, negative 90° and 90° with a line is the same
ahhh okayy, thank you smm
