All Questions
42
questions
0
votes
1
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62
views
To justify a complex-valued function is continuous
A complex-valued function is defined on the unit disk as $f(z) = \int_{0}^{1} \frac{1}{1-tz} dt$. How can we show that the function is continuous ?
My Approach: As the integrand is analytic in $z$, it ...
- 379
3
votes
1
answer
70
views
Complex analysis, Ian Stewart Exercise 4.7.5: Proving $\sqrt{z}$ is continuous on $\mathbb{C}\setminus\{x\leq0\}$
This is exercise 4.7.5 in Ian Stewart's "Complex Analysis
(The Hitch Hiker’s Guide to the Plane)":
Let $C_{\pi} =\{z\in\mathbb{C}:z\neq x\in\mathbb{R},x\leq0\}$ be the 'cut plane' with the ...
- 451
0
votes
0
answers
78
views
Why does this show Log can't be extended to whole $\mathbb{C}^*$
Why does the following show Log can't be extended to whole $\mathbb{C}^*$?
Here's another proof which I think I understand, though I'm not sure what's the connection between the two proofs:
I ...
- 451
2
votes
1
answer
84
views
Weak hypothesis for Morera's theorem?
Morera's theorem states that : Let $f(z)$ is a continous in a domain $D$. If $\int_Cf(z)dz = 0$ for every simple closed contour lying in $D,$ then $f$ is analytic in $D$.
Does there exist a function $...
- 751
0
votes
1
answer
34
views
Why $g$ is continuous on A?
I am trying to understand the proof of Schwarz Lemma below:
But I do not understand why $g$ is continuous on A? we do not know the formula for $f$ and so we do not know $f'(0),$ could someone explain ...
- 2,087
0
votes
1
answer
57
views
How can this function be written without cases?
The function,
$$
f(x) = \begin{cases}
(e^x-1)/x & x\neq 0 \\
1 & x = 0
\end{cases}
$$
is continuous and differentiable at $x=0$. By composition, the $x\neq 0$ case is analytic everywhere ...
- 531
2
votes
0
answers
42
views
A function satisfying a condition is a polynomial of degree $\leq 1$
Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function s.t $f(x)=\frac12(f(x+r)+f(x-r))$ for every $r>0, x\in\mathbb{R}$. Prove that $f$ is a polynomial of degree $\leq 1$.
This is a question ...
- 4,653
0
votes
0
answers
115
views
Reason(s) why $f(z)=1/(z-z_0)$ is not analytic at $z=z_0$
I am studying complex analysis and I am confused about why $f(z)=1/(z-z_0)$ is not analytic at $z=z_0$. For a function to be analytic, it must be differentiable and single-valued. Obviously, $f'(z)=-1/...
- 39
0
votes
0
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32
views
A question regarding differentiability and the boundary of analytic functions
Consider a Jordan curve $\Gamma\subset \mathbb{C}$ and let $\Omega$ be the interior of $\Gamma$ (which is well-defined by the Jordan curve theorem). Let $f:\Gamma\cup\Omega\rightarrow\mathbb{C}$ be ...
- 852
1
vote
1
answer
56
views
Limit of integral with gamma function
By some relations of Whittaker functions and a comment I read in a paper I came up with the following identity
$$\lim_{s\to 0+} \frac{1}{\Gamma(s)} \int_{0}^\infty e^{-vt} t^{s-1} (1+t)^{s+1}dt =1,$$
...
- 6,184
0
votes
1
answer
27
views
Use of uniformity of a limit for proof of continuity when right continuity is known
Consider a function $f: [0,\infty) \rightarrow \mathbb{R}$. Suppose that, for any $t \in (0, \infty)$
$$
\lim_{h \downarrow 0 } \, \, ( f(t+h) - f(t) )= O(h),
$$
namely that the function $f$ is ...
- 200
0
votes
1
answer
81
views
Unable to think about argument involving integratibility of a function in a research paper
I am self studying a research paper in analytic number theory and I am unable to think about how G(x,z) is integrable with z belonging to Complex Number and |z|$\geq$1.
Adding relevant image
My ...
user775699
0
votes
1
answer
301
views
Prove that the as Zn approached infinity, the Chordal Metric of Zn and zero approached zero.
We've been asked this problem to prove that as $z_n\to \infty$, the chordal metric $\rho(z_n,\infty)\to 0$. Where $\rho(z_1,z_2)= d(z_1',z_2')$ where $z_1'$ and $z_2'$ are points on the Riemann Sphere,...
- 41
1
vote
0
answers
25
views
Prove that complex function is uniformly continous [duplicate]
$f: \mathbb{C} \rightarrow \mathbb{C}, \; z \mapsto f(z) := \frac{z^2}{1+|z|}$
How would I go about proving this is uniformly continuous? Currently I have only practically dealt with analyzing ...
- 33
1
vote
1
answer
64
views
Location of complex roots of a polynomial
Let $f(x,y)$ be a polynomial function in $x\in\mathbb{C}$ and $y\in\mathbb{R}$. Suppose for a fixed $y=y_0$, the complex roots of the equation
\begin{align}
f(x,y_0)=0 \tag{1}
\end{align}
are located ...
- 1,398