16,440 reputation
542109
bio website
location
age
visits member for 3 years, 2 months
seen 12 hours ago

2d
awarded  Constituent
Dec
18
comment Suppose $f$ is entire and one-to-one. Show that $f(z)=az+b$.
@LebronJames My pleasure. Good luck with your studies.
Dec
18
comment Suppose $f$ is entire and one-to-one. Show that $f(z)=az+b$.
@LebronJames If it has a root with multiplicity $2$, then locally at that root it looks like $w=z^2$, which is clearly not injective. Ahlfors provides a rigorous explanation of this somewhere his book. Essentially, WLOG we may assume the root is at zero, so $f(z)=z^2(1+\dots)$, and then you can take a square root locally since the second factor is nonzero around $0$.
Dec
14
comment Prove or disprove If f is continuous and g is dicontinuous then f + g is discontinuous
Do you know the theorem that the sum of two continuous functions is continuous?
Dec
14
answered Problem 11 Section 2.6 in Erwine Kryszeg's Introductory Functional Analysis With Applications
Dec
14
comment Describe the Riemann surface for $w=z^2-1$.
@mathjacks My pleasure!
Dec
14
revised Describe the Riemann surface for $w=z^2-1$.
added 515 characters in body
Dec
14
comment Describe the Riemann surface for $w=z^2-1$.
@mathjacks You want to choose a branch of $z=\sqrt{w+1}$. The branch points here are $\pm i$. Recall that at a branch cut, the argument "jumps" by $i \pi$. So make two cuts, one for each branch point, going down the imaginary axis. It looks like you must cut out everything from $i$ downwards. But below $-i$, you have two "jumps" of $i \pi$, or a single jump of $2i \pi$, which doesn't change anything, so you can continue the function across that infinite ray. So only $i$ to $-i$ needs to be cut out.
Dec
12
comment Describe the Riemann surface for $w=z^2-1$.
@mathjacks I'm not sure which function you're referring too, but in general we know that most numbers have two square square roots, so two sheets are necessary. Also, I misread the question and gave an incorrect answer. See my edit.
Dec
12
revised Describe the Riemann surface for $w=z^2-1$.
edited body
Dec
11
answered Describe the Riemann surface for $w=z^2-1$.
Dec
10
awarded  Caucus
Dec
8
comment Which conformal maps should one have memorized?
@LebronJames It sounds like you would benefit from reading a good textbook. I recommend Stein and Shakarchi's Complex Analysis.
Dec
8
comment Which conformal maps should one have memorized?
@LebronJames That's a lot of questions! The maps I indicated are conformal when restricted to appropriate domains. A reflection is not even holomorphic. If by inversion with respect to a circle, you mean something like $1/z$, that's conformal when properly restricted. Generally, if you're trying to solve PDE problems (I have in mind Laplace's equation), you need the mapping to be conformal (look at the details in whatever examples you have and you'll see why).
Dec
7
awarded  Popular Question
Nov
25
awarded  Popular Question
Nov
23
comment Proving $f(z)$ entire function in complex analysis
@mint Ok, so you test on triangles instead. You are right that it won't make a difference. The problem reduces to showing that the two curves you get in the first paragraph, when integrated over, give zero. Consider just the one that lies in the closed disk. Think about shrinking it a little bit so it lies entirely in the open disk. Then Cauchy's integral theorem tells you the integral over this shrunken curve is zero. Does this make sense so far?
Nov
23
answered Proving $f(z)$ entire function in complex analysis
Nov
23
comment Proving $f(z)$ entire function in complex analysis
I believe $\partial B_1(0)$ is more appropriate notation. And really, you should use $\mathbb D$, not $B_1(0)$.
Nov
9
comment Show that $\sqrt [3]{2}-\sqrt [3]{4}$ is algebraic
What prevents you from just plugging it into the given cubic and simplifying?