All Questions
15
questions
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62
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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
0
votes
0
answers
56
views
Prove that if $f(z)$ is continuous on closed region then it is bounded in that region
While reading text on complex analysis, I found a following question:
Question: Prove that if $f(z)$ is continuous on closed region then it is bounded in that region.
My attempt: Isn't the boundedness ...
0
votes
0
answers
36
views
Identity theorem for (real) analytic functions on lower dimensional subsets
For simplicity, we will deal with $\mathbb{R}^2$. Let's assume we have an one-dimensional submanifold $M_1 \subset \mathbb{R}^2$ and two analytic function $F,G: M_1 \rightarrow \mathbb{R}$.
If I know $...
- 661
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
1
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113
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Show that if $M ≥ 0$ and $|f(z)| ≤ M$ for all $z ∈ ∂V$ , then $|f(z)| ≤ M$ for all $z ∈ V $
Suppose that V is a bounded open subset of the plane and $f ∈ C(\overline V)
∩ H(V)$ i.e. $f$ is continuous on $\overline V$ and $f\restriction_V$ is holomorphic on $V.$
Show that if $M ≥ 0$ and $|f(z)...
- 61
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
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69
views
Let $f(z) = \textrm{Log}(\textrm{Log} (z + 2i))$. Where is $f$ continuous?
Let $f(z) = \textrm{Log}(\textrm{Log} (z + 2i))$. Where is $f$ continuous?
I have been having issues on determining the continuity of this double logarithmic complex function.
Do I approach this ...
user741289
2
votes
1
answer
475
views
For complex function : $f'(z)$ exists $\implies$ $f$ continous in $z$?
Consider a complex function $f(z): A\subset\mathbb C \to\mathbb C$.
If the derivative of $f$ exists then $f$ must necessarily be a continuous function?
Is the following true? $f'(z)$ exists $\...
- 205
0
votes
1
answer
93
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What does a function look like if its derivative is not continuous? [closed]
In a previous question about the Cauchy-Riemann condition in complex analysis, I learned that a function can have derivatives in a region, but its derivatives might not be continuous.
My question is:
...
- 1,201
1
vote
0
answers
49
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Let an analytic function $f$ have domain-set the punctured disk $\Delta^*(z_0,r)$. Assume the existence of a complex number $w_0$ and a sequence
Let an analytic function $f$ have domain-set the punctured disk $\Delta^*(z_0,r)$. Assume the existence of a complex number $w_0$ and a sequence $\{r_n\}$ of positive real numbers satisfying $r>r_1&...
- 485
-2
votes
1
answer
605
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Continuity of a function with complex variables [closed]
How could I show if or not the following piece-wise defined function is continuous at the point $z=-i$?
$$f(z)=\left\{ \begin{matrix} \frac{z^2+2iz-1}{2z^2+iz+1}, & z \neq -i \\ 0, & z=-i \...
- 31
2
votes
2
answers
114
views
Is $f(x)=\frac{|x|^2}{x}$ continuous?
$$f(x) =
\begin{cases}
\frac{|x|^2}{x}, & \text{if $x \neq 0$} \\
0, & \text{if $x=0$}
\end{cases}$$
Can someone please explain if f is continuous? Assume $x$ is a complex number
Hints ...
- 535
1
vote
1
answer
2k
views
Continuity of $f(z) = Log z$ , for $z$ complex, non-real $-\ln|z|$, for $z$ real
At what points in the complex plane is this function continuous (If there is any)?
Would it be correct to conclude that $f$ is then continuous for all $z$ in the complex plane less the real numbers?...
- 176
0
votes
1
answer
1k
views
Determining whether a function is uniformly continuous
Determine whether $(4x-3)/(x-2)$ is uniformly continuous on the open interval $(1,2)$.
I'm not sure how to start this as I have only answered these questions with closed intervals?
- 21
1
vote
2
answers
264
views
Find a function that satisfies the condition.
Let $\epsilon > 0$ be fixed and $t$ a variable that takes values in the universal covering space of ${\mathbb{C} \setminus \{0\}}$.
Find a continuous function $f(s$) such that
$$|t \log t| = |t| \...
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