All Questions
Tagged with logarithms complex-numbers
258
questions
4
votes
2
answers
287
views
Showing $\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right) \, dx=\frac{\pi^2 + 4}{2}$
While exploring possible applications for exponential substitution, I stumbled upon the following integral identity:
$$\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right)...
- 1,011
-2
votes
0
answers
141
views
Solving $\sqrt{x+1}=-2$ and looking for complex solutions [duplicate]
So the question was
$$\sqrt{x+1}=-2$$
And obviously there is no value for it,
However,
If you do the thing with $e$ and $\ln{}$
$$e^{\ln{\sqrt{x+1}}}$$
and
$$e^{\frac{1}{2}\cdot (\ln{x+1})}$$
Then ...
- 19
-4
votes
0
answers
108
views
Find $\arcsin c$, $\, c\in\Bbb C$ [duplicate]
A math fan sent me a solution of the weird equation $\sin z=2$ posted in Quora. It is Weird because in real calculus, we experienced that $-1\leq \sin x\leq 1$. I saw this question in so many places ...
- 11.9k
2
votes
0
answers
87
views
Does complex log(1) have other values than 0?
I want to determine all values of $$ \left[\log \left(3+2 i^{2}\right)\right]^{1-i} $$
First I simplify to $$\left[\log \left(3-2\right)\right]^{1-i}$$
resulting in $$\left[\log \left(1\right)\right]^{...
- 21
2
votes
0
answers
74
views
Principal branch of $z^{1-i}$
I am solving a problem
Find the principal branch of $z^{1-i}$.
I wanted to verify my solution.
I know we can write $z^{1-i} = e^{(1-i)\cdot \text{Log}_e(z)}$
Since the principal branch of $\text{Log}...
- 148
0
votes
1
answer
92
views
Conjugate of complex number raised to complex power
Why is $\overline{z^{n}} = (\bar{z})^{n}$ true only for $n \in \mathbb{Z}$? What about a real or complex exponent in general?
Let $$\log z=\{\ln \rho + i (\theta + 2k\pi)\mid k\in\Bbb Z\}$$
then
$$
\...
- 1
0
votes
1
answer
125
views
Non-power series proof that $|\log (1 + z)| \le -\log (1 - |z|)$
Given $z \in \mathbb{C}$, $|z| < 1$, we have the bound
\begin{align*}
|\log (1 + z)|
&= \left| \sum_{n \ge 1} \frac{(-1)^{n+1}z^n}{n} \right| \\
&\le \sum_{n \ge 1} \frac{|z|^n}{n} \\
...
- 868
0
votes
1
answer
183
views
The polynomial P(x) is equal to $(i + x)^{2024}$. Let $S$ be the sum of all the real coefficients of $P(x)$. Find $\log_{2} S$. [closed]
I think the answer to this problem is $1012$, but I'm not sure.
Any ideas? Any and all help is appreciated.
- 119
-1
votes
1
answer
144
views
Calculate $i^{\log(−𝑖)}$ (principal value) [closed]
I am working on a complex number problem and need some guidance on how to calculate the principal value of the expression "$i^{\log(-i)}$". I understand that the principal value of a complex ...
0
votes
1
answer
51
views
Using the main branch of $z^\lambda$, calculate $1^i$
I do believe this is a simple question, but I am still unsure if my solution correct.
We have that:
$$z^\lambda=e^{\ln(z)\cdot \lambda}$$
And also that:
$$\ln(z)=\ln\left(\sqrt{a^2+b^2}\right) + i\...
- 57
-2
votes
2
answers
119
views
An integral Wolfram Alpha cannot take: $\displaystyle\int_{-1}^{+1}(-1)^{i\pi e^{-x\pi}}dx$
So I was bored yet again, and decided to evaluate some integrals. After a while, I came up with this:$$\int_{-1}^{+1}(-1)^{i\pi e^{-x\pi}}dx$$which is based off of $\color{red}{\text{this}}$ integral ...
- 2,652
0
votes
0
answers
37
views
Fast addition of logarithmic values
Given two values $\log(a)$ and $\log(b)$ of complex values $a$ and $b$. Is there a numerically fast way to compute $\log(a + b)$ (on a CPU)?
I'm aware that, $\log(a + b) = \log(a) + \log(1 + \exp(\log(...
- 577
1
vote
0
answers
46
views
An estimate for a logarithm occuring in analytic number theory
Let $s$ range over the complex numbers and write such numbers as $s=\sigma+iT$ with $\sigma,T$ real. In textbooks on analytic number theory, I have found the following estimate:
$$\frac{\log|s|}{|s|} \...
- 2,436
0
votes
2
answers
99
views
Why is $0≠Log(e^{i2n\pi})=i(2n\pi)$
$\\Log$ denotes the multi-valued natural log defined over the complex numbers
I know that $Log(z)$ is multivalued and that we can choose a particular branch if we want to, but why isn't it the case ...
- 887
0
votes
0
answers
32
views
Existence and understanding holomorphic root functions
I have to prove that
a) for all $|z|<\frac{\pi}{2}$ exists a root function of the function $f(z)=\cos(z)$, which means there is a holomorphic function $g:\mathbb{D}\to \mathbb{C}$, such as $g(z)^2=...
- 177