Tagged Questions
3
votes
2answers
44 views
Product rule for logarithms works on any non-zero value?
The product rule for logarithms states that:
$$\log_b M + \log_b N = \log_b (M\cdot N)$$
Most sources that I've found* state that this rule only applies when $M$ and $N$ are positive. It's true that ...
1
vote
0answers
79 views
Complex Logarithm
For what values of $p$ is the following valid?
$$\log(z^p) = p\log(z)$$
where
$$\log(z) = \ln(|z|) + i[\arg(z)+2\pi n]$$
where $n$ is an integer.
I heard the expression above should not be valid for ...
2
votes
1answer
74 views
Complex logarithm and injectivity
Please forgive the trivial nature of this question: let U be a connected domain inside the punctured unit disk so that every curve inside it has winding number zero around the origin. Is the complex ...
2
votes
2answers
200 views
Determination of complex logarithm
Good day everyone. I was reading the more advanced lectures on complex analysis and encountered a lot of questions, concerning the determination of complex logarithm. As far I don't even understand ...
2
votes
2answers
102 views
About the logarithm of the negative unit
$${e}^{iz} = \cos(z)+i\sin(z)$$
and
$$e^{i\pi}=-1$$
But then
$$\ln(-1)$$
can be infinite many numbers (positive and negative), as $z$ is the natural logarithm of that number and the solution to the ...
2
votes
1answer
93 views
How to determine periodicity of complex log in different bases?
How do you determine the "period" of a complex logarithm as a multivalued function in an arbitrary (real or complex) base?
I apologize in advance if my terminology is incorrect, but let me illustrate ...
2
votes
1answer
202 views
How to find logarithms of negative numbers?
Logarithms of negative numbers must be complex.
But how do you find $\ln{(-2)}$ expressed in something like $x \cdot i$ where $x \in \mathbb{R}$?
40
votes
4answers
1k views
A new imaginary number? $x^c = -x$
Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...
6
votes
1answer
119 views
Complex Logs and Roots of Unity
I need to find all the solutions to the following using logarithms:
$(e^z-1)^3=1$ where z is a complex number.
I am told that using roots of unity I can break this equation down but I must be missing ...
7
votes
4answers
273 views
Has anyone talked themselves into understanding Euler's identity a bit?
What does the ratio of the circumference of a circle to its diameter have to do with the base of the natural logarithm and $\sqrt{-1}$?
3
votes
2answers
214 views
Is it standard to say $-i \log(-1)$ is $\pi$?
I typed $\pi$ into Wolfram Alpha and in the short list of definitions there appeared
$$ \pi = -i \log(-1)$$
which really bothered me. Multiplying on both sides by $2i$:
$$ 2\pi i = 2 \log(-1) = ...
6
votes
5answers
655 views
Understanding imaginary exponents
Greetings!
I am trying to understand what it means to have an imaginary number in an exponent. What does $x^{i}$ where $x$ is real mean?
I've read a few pages on this issue, and they all seem to ...
