All Questions
Tagged with complex-analysis continuity
88
questions with no upvoted or accepted answers
4
votes
0
answers
106
views
Do both real and imaginary roots of a cubic equation need to continuous?
I have a cubic equation:
$X^3-UX^2-KX-L=0$ (1)
with $X=1-E+U$, $K=4(1-\gamma^2-\lambda^2)$, $L=4\gamma^2U$.
I solve Eq. (1) for the variable $E$ numerically for $U=2$ and different sets of parameter $\...
- 141
4
votes
0
answers
1k
views
Why is $\log z = \ln r + i\theta$ ($r>0, \alpha <\theta < \alpha + 2\pi$) discontinuous at $\alpha$?
In one book on complex variables it is written that, given the function $\log z = \ln r + i\theta$ (for proper citation, let's call it function (2), as in the book) ($r>0, \alpha <\theta < \...
- 9,708
3
votes
0
answers
116
views
Show continuity of Euler gamma function using DCT
The Euler Gamma function is defined by
$$\begin{align}
&\Gamma : \{z\in \mathbb{C} : \Re(z) > 0\} \rightarrow \mathbb{C} \\
&\Gamma(z) = \int_{0}^{\infty} x^{z-1}e^{-x}dx
\end{align}$$
My ...
- 31
2
votes
0
answers
33
views
Continuity of a function defined by an improper integral
Let $c > 0$ and let the function $f : (0, \infty) \to \mathbb{C}$ be defined as
$$
f(y) = \int_{c - i\infty}^{c+i\infty} \frac{y^s}{s(s+1)} \, ds.
$$
I want to show that $f$ is continuous.
My ...
- 163
2
votes
0
answers
50
views
Surface integral of a complex Log function
I am trying to calculate the surface integral of a complex Log function i.e.
$$ \int\int_{|z|<1}{Log(x+i y-(x_0+iy_0)))dxdy}$$
where $z=x+iy$ and $x_0,y_0 \in \mathbb{R}$ .
I know that for analytic ...
- 21
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
2
votes
0
answers
37
views
Showing Differentiability/Continuity at endpoints of closed interval?
I am given the function
$\gamma:[-1,\frac{\pi}{2}] \rightarrow \mathbb{C}$
$\gamma(t) =
\begin{cases}
t+1 & \text{for $-1 \leq t \leq0$} \\
e^{it} & \text{for $0 \leq t \leq\frac{\pi}{2}$} ...
- 296
2
votes
0
answers
146
views
How do we define fractional derivatives for complex argument functions?
NOTE: This question is not about fractional derivatives of complex order. That topic has already been discussed on this site, here, for example. No - this question is more simple.
How do we define $\...
- 12.5k
2
votes
0
answers
417
views
Continuous function F holomorphic everywhere in a region D except on line L.
I am trying to prove that if a continuous function $F$ is holomorphic everywhere in a region $D$ except on line $L$ that passes through $D$, then $F$ is holomorphic on $D$.
However, I am having ...
- 2,922
2
votes
0
answers
441
views
Example of Looman-Menchoff theorem
I would like to know of one example of a function that strictly satisfies the Loomann-Menchoff theorem.
Namely, I want a complex function $f=u+iv$, defined on some $A \subseteq \mathbb C$, such that
...
- 2,588
2
votes
0
answers
56
views
Multiplicative maps on C*
Suppose $\mathbb{C}$ is the complex plane and $\mathbb{C}^{*} = \mathbb{C} \setminus \lbrace 0 \rbrace$.
Let $ \varphi : \mathbb{C}^{*} \rightarrow \mathbb{C}^{*} $ be a continuous multiplicative ...
2
votes
1
answer
238
views
Continuous function can be written as an exponential
Suppose a function $f: \mathbb{R} \to \mathbb{C}$ is continuous, never becomes $0$ and $f(0) = 1$. I want to proof that there exists a unique continuous function $g:\mathbb{R} \to \mathbb{C}$ such ...
- 163
2
votes
0
answers
443
views
Separate continuity and analyticity in one variable implies joint continuity?
A theorem of Hartog states that if $U \subseteq \mathbb C$ is open and $f : U \times U \to \mathbb C$ is analytic when we fix any variable ('separately analytic'), then it is continuous.
In general, ...
- 26.5k
2
votes
0
answers
257
views
Limit Point 2D Bolzano-Weierstrass
I'm struggling with the following problem:
a) Let A be the set of complex numbers. A number z is called, as in the real case, a limit point, the limit point of the set A if for every $\epsilon>0$, ...
- 499
2
votes
0
answers
49
views
Continuity of a complex function
Let $$f\left(z\right)=\left(\frac{1}{z}-\frac{1}{2}\right)\left(\frac{1}{z-2}\right).$$
Define the function at $z=2$ so it will be continuous there.
So we have to find the limit of the function at ...
- 13.1k