All Questions
6
questions
3
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70
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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
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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
2
votes
1
answer
166
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How misleading is it to regard $i$ as *the* square root of $-1$?
Many mathematicians regard writing $i=\sqrt{-1}$ as at best an abuse of notation, and at worst simply incorrect. This is because every non-zero complex number has two square roots, and there is no ...
user883638
1
vote
1
answer
1k
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on the (double) discontinuity of dilogarithm along a branch cut
Define the function
$$Li_s(z)=\sum_{k=1}^\infty \frac{z^k}{k^s}$$
for |z|<1. Let's focus on $s=2$.
It can be extended to a holomorphic function on $\mathbb C \setminus [1,\infty)$
$$Li_2(z)= -\...
- 2,390
1
vote
1
answer
279
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How to define Square Root
I'm trying to understand how to define the square root of a complex function "globally". Let's say we have some function from some set $X$ onto $\mathbb{C} - \{0\}$: $$ f:X\to\mathbb{C}-\{0\} $$ and ...
- 1,116
1
vote
2
answers
476
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Is discontinuity along a line equivalent to branch cut?
Suppose I claim the analytic function $f(z)$ has a branch cut along the positive real line, how would one go on to prove this?
Is it sufficient to prove that $f(z)$ is discontinuous across this line?
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