# Tagged Questions

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### Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
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### Why $\ln(1)\neq 2\pi ik$

Given that $e^{2\pi ik}=1$ for all $k \in \mathbb{Z}$, why isn't $\ln{e^{2\pi ik}}=2\pi ik$? On the other hand $\ln(1)=0$. What am I missing here?
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### Complex Logarithm Derivation

I don't understand how the definition of the complex logarithm was derived. It is $log(z) = ln|z| + i Arg (z)$, where $z = x + iy$. I've tried all sorts of method to find this definition but ...
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### A uniformly convergent series

How does one show that the series $$\sum_{k = 1}^\infty \left\{\frac{s}{k} - \log\left(1 + \frac{s}{k}\right)\right\}, \quad s \in \mathbb{C} \setminus \{0, -1, -2, \ldots\}$$ is uniformly convergent? ...
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### What is the value of $\ln \left(e^{2 \pi i}\right)$

I know that $$e^{2 \pi i} = 1$$ so by taking the natural logarithm on both sides $$\ln \left(e^{2 \pi i}\right)=\ln (1)=0$$ however, why isn't this $2 \pi i$ as expected? Is it beacuse logarithms ...
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### Conditions required for $(z_{1}z_{2})^{\omega}=z_{1}^{\omega}z_{2}^{\omega}$, where $z_{1},z_{2},\omega\in\mathbb{C}$

I am having trouble finding the conditions on $z_{1}$ and $z_{2}$ in order for: $$(z_{1}z_{2})^{\omega}\equiv z_{1}^{\omega}z_{2}^{\omega}\qquad \forall\omega\in\mathbb{C}$$ My first step was to ...
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### What calculate $\ln i$

I would like to know how to calculate $\ln i$. I found a formula on the internet that says $$\ln z=\ln|z|+i\text{Arg}(z)$$Then $|i|=1$ and $\text{Arg}(i)$ is?
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### Find the value of K. Use of l Hospital's rule and expansion is not allowed.

Let $f(x) =\log_{cos3x} (\cos2ix)$ if $x \ne 0$ and $f(0) = k$ where ($i$= iota) is continuous at $x = 0$, then find the value of $K$. Use of l Hospital's rule and expansion is not allowed.
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### Can we *ever* use certain log/exp identities in the complex case?

This article on Wikipedia points out that certain identities for the log and exponential functions which are familiar from the real case require care when used in the complex case. Failures in the ...
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### Does $i = -\frac{(2\;W({\pi\over2}))}{\pi}$

Let $x = -\frac{(2\;W({\pi\over2}))}{\pi}$, where $W$ denotes the Lambert W-function. As $${\log(i^2)\over i} = \pi$$ and $${\log(x^2)\over x}=\pi$$ Does $x = i$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 $x \in \mathbb{R}$?
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### 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 ...
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### 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 ...
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### 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 $\sqrt{-1}$?
### 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) = ...
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 ...