All Questions
Tagged with logarithms complex-analysis
552
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:
$$\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right)...
- 1,011
4
votes
1
answer
178
views
Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.
The standard branch of logarithm, $\log:\mathbb{C}\setminus (-\infty,0]\to\mathbb{C}$ is defined as
$$
\log(z):=\ln|z|+i\operatorname{Arg}(z)\tag{2}
$$
where $\arg(z)$ is the standard branch of the ...
- 936
0
votes
1
answer
41
views
Integral of Complex logarithm makes sense?
For example, I know that the principal branch of logarithm is not defined over negative real axis.
I think the integral of this logarithm along a circle doesn’t make sense.
Moreover, I know that the ...
- 11
-2
votes
0
answers
141
views
Solving $\sqrt{x+1}=-2$ and looking for complex solutions [duplicate]
So the question was
$$\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})}$$
Then ...
- 19
-4
votes
0
answers
108
views
Find $\arcsin c$, $\, c\in\Bbb C$ [duplicate]
A math fan sent me a solution of the weird equation $\sin z=2$ posted in Quora. It is Weird because in real calculus, we experienced that $-1\leq \sin x\leq 1$. I saw this question in so many places ...
- 11.9k
1
vote
0
answers
66
views
Why the log function is so important on the plane?
I am studying right now some Complex Analysis and I have seen the importance of the (complex) logarithm function in almost every subject in it. Now I'm intrigued with that (possible) relation between $...
- 141
2
votes
0
answers
40
views
How is the dilogarithm defined?
I am pretty happy with the definition of the "maximal analytic contiuation of the logarithm" as
$$
\operatorname{Log}_{\gamma}\left(z\right) =
\int_\gamma ...
- 424
0
votes
1
answer
77
views
How to show $\log(z) = \log(r) + i \theta$ without implicitly assuming $z = r \exp (i \theta)$ - from Penrose Road to Reality
In Roger Penrose's book Road to Reality - Chapter 5 - he goes to great lengths to arrive at the standard polar expression for a complex number $w = r e^{i \theta}$ via a discussion of complex ...
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 $$\left[\log \left(1\right)\right]^{...
- 21
2
votes
0
answers
74
views
Principal branch of $z^{1-i}$
I am solving a problem
Find the principal branch of $z^{1-i}$.
I wanted to verify my solution.
I know we can write $z^{1-i} = e^{(1-i)\cdot \text{Log}_e(z)}$
Since the principal branch of $\text{Log}...
- 148
1
vote
0
answers
42
views
Finding the imaginary part of $\ln\left(\ln\left(\frac{1+e^{2ix}}{2}\right)\right)$, where $x\in[0,\frac\pi2]$
I am looking for the imaginary part of the following expression
$$\ln\left(\ln\left(\frac{1+e^{2ix}}{2}\right)\right)$$
where $x\in[0,\frac\pi2]$.
The attempt I made was the following
$$\begin{align}
\...
- 203
6
votes
0
answers
156
views
Finding a closed form for $ \int_0^1 \frac1x \ln\left(\frac{\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x $
I want a closed form for the following integral
$$
\int_0^1 \frac1x\;\ln\left(\frac {\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x
$$
An integration by parts ...
- 203
0
votes
0
answers
48
views
$\log \Gamma(z + 1) = \log \Gamma (z) + \log z$ issue
I was reading proof of Binet's first expression for $\log \Gamma(z)$; that is for $\Re z > 0$,
$$
\log \Gamma(z) = \left(z - \frac 1 2\right)\log z - z + \frac 1 2 \log (2\pi) + \int_0^\infty \left(...
- 2,376
2
votes
1
answer
53
views
Question about three lines theorem
I'm trying to prove that the function defined as $$F_\epsilon(z)=F(z)M_0^{z-1}M_1^{-z}e^{\epsilon z(z-1)},$$ where $F$ is an holomorphic function on $0<Re z<1$ and continuous and bounded on the ...
0
votes
0
answers
21
views
Determine the domain-sets of $ Log(e^z-e^{-z})$
Determine the domain-sets of $$ Log(e^z-e^{-z})$$
I only know that $e^z-e^{-z} > 0$ and so z must be positive. Also, $e^z = e^{x+iy} $
The solution is $\Bbb{C}$ \ { $k \pi i | k \in \Bbb{Z}$ }
I ...
- 19