f (z) = lamdba (a - z)/(1 - abar*z) |lambda| = 1; |a| <1
where f is any one-to-one analytic function mapping delta = {z:|z| < 1} onto itself
set g(z) = (a - f(z))/(1-abar*f(z))
now they say that g is one-to-one analytic function mapping delta onto delta (why?) and g(0) = 0
and then they...
so the way book states it, for the sc. lemma to work, |f(z)| has to be less than or equal to 1 and and z has to be less than 1. However, the book seems to use the lemma in some problems even if one of the conditions is not satisfied ... any help with gretaly appreciated
also ...if you could...
thats what i thought .... i wrote down "open" as my answer and the prof circles it and I dont think I got any points for it ... yes, i didnt write connected, but I should at least get half the points or something. oh well maybe he didnt gimme any credit, because I didnt explain why I think it's...
in that case ... wouldnt it be an open set?
and it will be above real axis? (meaning the boundary is upper plane or lower plane? getting confused with terminology a little)
Let C be a simple closed curve. Show that the area enclosed by C is given by 1/2i * integral of conjugate of z over the curve C with respect to z.
the hint says: use polar coordinates
i can prove it for a circle, but i am not sure how to extend it to prove it for any given closed curve
{z^2: z = x+iy, x>0, y>0}
i am a lil confused about the notation to represent the set ...
i'm used to seeing {z: z = x+iy, x>0, y>0}
but what effect does squaring z have?
i thought the set was open simply because x>0 and y>0 ... but apprently i was wrong ... (or maybe not?) ... i...
no we havent covered cross ratio in class yet.
so i *sorta* get how to send a circle to a line ... ...
the circle |z| = 1 and the line Re((1+i)w) = 0 ... so i sent 1 to infinity, -1 to 0 and i = -1 and i came up with i*(z+1)/(z-1) ...
the book has the answer as (1-i)*(z+1)/(z-1)
i...
woopsie ... let's restart then ...
I send i to infinity, -i to zero ... what about the third point?
it has to do something with fixing orientation, right? how does that work?
(like i know the third point can not be totally arbitrary like the first two points were, right?)
if i apply max/min modulus to both f and 1/f, then it means both f and 1/f dont have a local max/min ... so what does it tell me?
i'm sorry ... i dunno if it's lack of confidence or what .. but i still dont see it
thats what i thought ....
but i still have no insight about f having any zero inside lambda ....
like i kinda see it visually ... like i know it makes sense ... but i got no clue how to "prove" it
the way prof showed it in class ... he said pick three points on the circle and send it to three points on the line (or vice versa if we are trying to map a line to a cirle) ... but i am not sure how do i know what those three points are going to map to ....
on a circle |z| = 1, we could...
uhh .... sorry ...
now i am getting the answer quite fine .... just wanna make sure what i am doing is right ....
so it travels 9pi/2 counting 0 to R and along the radius of R centered at origin... to iR ... but when it reaches iR, it travel clockwise to land on the positive real axis ... so...
what effect does multiplying by i have?
i am still lost ...
can you direct me to a list of those "meanings of geometric addition" etc ... maybe i am taking more complicated approach than i should be ... but things dont make much sense
Suppose that f is analytic on a domain D, which contains a simple closed curve lambda and the inside of lambda. If |f| is constant on lambda, then either f is constant or f has a zero inside lambda ...
i am supposed to use maximum/modulus principle to prove it ...
here is my take:
if f...
okay ... so i believe i am doing something wrong then ...
f(z) = z^9 +5z^2 + 3
(i have to determine number of zeroes in the first quadrant)
i came up with the 9pi/2 ... i am sure i am doing something wrong ... i should be getting 8 pi/2, right?
Verify that the linear fractional transformation
T(z) = (z2 - z1) / (z - z1)
maps z1 to infinity, z2 to 1 and infinity to zero.
^^^ so for problems like these, do I just plug in z1, z2 and infinity in the eqn given for T(z) and see what value they give?
in this case, do i assume 1/ 0 is...
(changes in arg h (z) as z traverses lambda)/(2pi) =
# of zeroes of h inside lambda +
# of holes of h inside lambda
now the doubt i have is what happens when the change i get in h (z) is say 9 pi/2 .... because then i would have a 2.5 on left side of the eqn ... so do i round it up and...
can someone here go over " translation, angle expansion, and how to map strips to the upper half plane. "?
and can someone go over how to put them together?
lol you see my problem? i have no clue how to even have a start
find a one-to-one analytic function that maps the domain {} to upper half plane etc ...
for questions like these, do we just have to be blessed with good intuition or there are actually sound mathematical ways to come up with one-to-one analytic functions that satisfy the given requirement...
nope it wasnt zero... thats the problem!
btw it's kinda hard to post my work, did you get a zero for it?
i assure you it's not a hw problem ... lol
if i had time, i would scan my work, but i need to know how to do it by tomorrow.
see .. .what i was tryina to do ... is ... prove that second partial derivative of log |f (z)| with respect to x + second partial derivative of log |f (z)| with respect to y = 0 ... of course it wasnt giving me happy result lol ...although it is technically right ...
i mean in order to show log |f (z)| is harmonic on the domain, dont I need to prove second partial derivative of log |f (z)| with respect to x + second partial derivative of log |f (z)| with respect to y = 0?
I hope my question makes sense ...
so .. if f (z) = u + iv is analytic on D, then u and v are harmonic on D...
now ...
if f (z) never vanishes on the domain ...
then show log |f (z)| is harmonic on the domain ...
Recall: harmonic means second partial derivative of f with respect to x + second partial derivative of f with...