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Mar
29
revised If $f(z)$ maps the unit disk onto itself $k$ times, prove that f(z) must be a rational function and show that the degree of its denominator $\leq k$.
added 296 characters in body
Mar
29
comment If $f(z)$ maps the unit disk onto itself $k$ times, prove that f(z) must be a rational function and show that the degree of its denominator $\leq k$.
@Chilango. Good point about the origin. Can you explain your suggestion and why that should work for $f(z)$? I've seen something similar called the Blaschke product, but not in relation to my question.
Mar
29
asked If $f(z)$ maps the unit disk onto itself $k$ times, prove that f(z) must be a rational function and show that the degree of its denominator $\leq k$.
Mar
26
accepted What does it mean for a function to tend uniformly to $\infty$ on every compact set?
Mar
26
comment What does it mean for a function to tend uniformly to $\infty$ on every compact set?
Thank you. However, how/where did you get your definition of uniform convergence to infinity in your second paragraph?
Mar
25
asked What does it mean for a function to tend uniformly to $\infty$ on every compact set?
Mar
23
asked What is the form of an analytic function that maps $|z|<1$ onto the $n$-times covered disk $|w|<1$?
Mar
19
asked Let $f(z)=\sum_{-\infty}^\infty e^{2\pi inz}e^{-\pi n^2}$. Show there is a unique $z_0$ in the unit square such that $f(z_0)=0$.
Mar
18
asked Find all entire functions in the complex plane such that $|f(z)| \leq 2 |z|^{3/2} - 1$ for $|z| \geq 2$ and $f(0)=1$.
Mar
14
asked If $A_1, A_2, A_3$ are disjoint circles in the complex plane, prove there is a fourth circle or line that is perpendicular to all.
Mar
9
awarded  Yearling
Mar
7
comment For $T: V \to V$, suppose $A = A^*$ where $A = [T]_\mathcal{X}$. Find another basis/matrix where $B \neq B^*$ for $B = [T]_\mathcal{Y}$.
@FriedrichPhilipp. Thank you. How do you know to choose such a matrix $S$? As in, how do you know to choose a transition matrix that is not unitary? The intuition escapes me.
Mar
7
asked For $T: V \to V$, suppose $A = A^*$ where $A = [T]_\mathcal{X}$. Find another basis/matrix where $B \neq B^*$ for $B = [T]_\mathcal{Y}$.
Feb
24
comment Let $f(z) = z^4 - 2z^3 + z^2$. Evaluate $\frac{1}{2\pi i} \int \frac{f'}{f} dz$ and $\int \frac{zf'}{f} dz$
Is there a theorem that states the result you are using for the sum of the zeros of a monic polynomial?
Feb
23
accepted Let $f(z) = z^4 - 2z^3 + z^2$. Evaluate $\frac{1}{2\pi i} \int \frac{f'}{f} dz$ and $\int \frac{zf'}{f} dz$
Feb
23
asked Let $f(z) = z^4 - 2z^3 + z^2$. Evaluate $\frac{1}{2\pi i} \int \frac{f'}{f} dz$ and $\int \frac{zf'}{f} dz$
Feb
17
accepted What is the form of the general mobius transformations that map the line $\Re(z)=2$ and the unit circle into concentric circles?
Feb
16
comment What is the form of the general mobius transformations that map the line $\Re(z)=2$ and the unit circle into concentric circles?
Also, how did you come up with the map $\varphi(\zeta)$? It seems to come out of nowhere for me.
Feb
16
comment What is the form of the general mobius transformations that map the line $\Re(z)=2$ and the unit circle into concentric circles?
Thanks. Can you explain your symmetry argument for finding $r$? Why should $\varphi(1/2)=r$ and $\varphi(0)=-r$?
Feb
15
asked What is the form of the general mobius transformations that map the line $\Re(z)=2$ and the unit circle into concentric circles?