1
vote
1answer
21 views

Principal branch of the complex logarithm does not always satisfy the product formula

My book asks to prove: $\text{Ln}[i \cdot (-1+i)]$ does not equal to $\text{Ln}(i) + \text{Ln}(-1+i)$ where $\text{Ln}$ gives the principal log of the complex number. I don't see why this is true ...
0
votes
1answer
40 views

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 ...
3
votes
1answer
77 views

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?
1
vote
1answer
26 views

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 ...
0
votes
0answers
128 views

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? ...
6
votes
3answers
102 views

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 ...
1
vote
1answer
65 views

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 ...
4
votes
2answers
82 views

Integral of $\frac{1}{x^2+1}$ using complex partial fractions.

Is there any way to evaluate the following integral via a complex partial fraction decomposition? $$ \int \dfrac{1}{x^2 + 1} \text{ d}x $$ So far I have: $$ \begin{aligned} \int \dfrac{1}{x^2 + 1} ...
1
vote
0answers
59 views

Is this correct: $\ln({(-1)}^{2x-1})=(2x-1)\ln(-1)$?

I would expect the answer to be positive, but it appears otherwise for some values of $x \geq 1$. Here is a simple C++ code that I have used in order to test this: ...
1
vote
1answer
94 views

Logarithm rules for complex numbers

Are the logarithm rules true for complex numbers? We know that for positive real numbers $a$, $b$, $c$ and real number $d$ that: $$\log_b\left(a^d\right)=d\log_b(a)$$ $$\log_b(a) = ...
0
votes
4answers
88 views

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?
1
vote
1answer
62 views

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.
0
votes
2answers
110 views

How do I solve this integral?

As stated the title, I get to a point which I can't do anything, and I'm sure I've made a mistake some where, here is my full working out: $$ \int e^{ix}\cos(x)dx \\ u = e^{ix} \text{ | } u'= ie^{ie} ...
1
vote
1answer
68 views

logarithm of a complex number?

I have a task to study a function like this one: $$F(z) = \frac{\ln(e^{iz^4})}{z^3}$$ I'm trying to simplify this: since the exponential is the inverse function of $\ln()$ can we simplify it to ...
4
votes
3answers
103 views

Does $\operatorname{Log}(1+i)^2 =2\operatorname{Log}(1+i)$

And similarly, does $\operatorname{Log}(1-i)^2=2\operatorname{Log}(1-i)$? If we were dealing with real numbers, it would hold. But I'm guessing that the fact that there are imaginary numbers involved ...
0
votes
1answer
89 views

complex logarithm

I have $e^{jk}$ and I want to take a logarithm from it, $\log(e^{jk})$ must be $jk$, right? here some example I have tried to do with matlab. $$\log(e^{j2})=j2$$ $$\log(e^{j3})=j3$$ but for $e^{j4}$ ...
1
vote
1answer
85 views

How to prove the following inequality of logarithm?

Let $x,y,z\in\mathbb{C}.$ Suppose $$z=\frac{1}{2}(xy\pm\sqrt{x^2y^2-4(x^2+y^2)} ).$$ Show that $$log^+|z|\leq log^+|x|+log^+|y|+log 2.$$ Where $log^+\phi=max\{0,log\phi\}.$ Here we are also ...
3
votes
1answer
66 views

A double sum and its relation to a simple sum, is this an identity for any complex number $S=a+i b$ and any integers n and t?

Does: $$\sum _{m=1}^t \lim_{s\to \text{S}} \, \zeta (s) \sum _{k=(m-1) n+1}^{m n} \frac{1-\text{If}[k \bmod n=0,n,0]}{k^{s-1}}$$ equal: $$\lim_{s\to \text{S}} \, \zeta (s) \sum _{k=1}^{n t} ...
3
votes
3answers
177 views

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 ...
0
votes
1answer
55 views

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 $x = i$?
3
votes
2answers
168 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
116 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
100 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
420 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
127 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
161 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
412 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 $x \in \mathbb{R}$?
45
votes
4answers
2k 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
229 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 ...
8
votes
6answers
430 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
353 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) = ...
10
votes
5answers
2k 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 ...