All Questions
10
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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
0
votes
0
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86
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Continuity on boundary of convergence power series
I’m stuck trying to prove the following: Given $f(z) = \log(2+z^2)$, I consider its power series representation around $z=0$, which is $i(z) = \log(2)+\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n2^n}z^{2n}$...
- 1
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1
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69
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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
0
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1
answer
506
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How do I directly show that Log is discontinuous on the interval $\{-\infty<\Re z\le 0\}?$
Given the knowledge that the exponential function maps bijectively each half-open horizontal strip of width $2\pi$ onto $\mathbb C\setminus \{0\}.$ So, in particular, if we restrict it to $-\pi<\...
- 2,338
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40
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Uniform convergence of logarithm on a compact set.
Given that $\phi$ and $\phi_n,n \geq 1$ are continuous functions from $\mathbb{R}$ into $\mathbb{C}$ such that $ \phi(0) = \phi_n(0) = 1$, $\phi(x) \neq 0$ and $\phi_n(x) \neq 0$ for any $x$. Suppose ...
- 163
2
votes
1
answer
238
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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
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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
-1
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1
answer
27
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Prove that Log is defined on D [closed]
$D=D(0,R)$ is the disk of center $0$ and radius $R$.
Given that $a>R$ and $\Phi(z)=\frac{a-z}{a+z}$, I have proved that $\forall z\in D$, $\operatorname*{Re}(\Phi(z))>0$.
Prove that $f = \...
- 984
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68
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Non zero continuous path $[0,1]\to \mathbb C$ has continuous logarithm
Let $\gamma:[0,1]\to \mathbb C$ be continuous, and not passing through $0$. How can we prove that, using complex analysis, there is a continuous $G:[0,1]\to \mathbb C$ so that $\gamma=e^G$ ?
This can ...
- 67
1
vote
2
answers
264
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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| \...
- 942