2
votes
0answers
16 views

Moving the branch cut of the complex logarithm

The complex logarithm is defined as $\log z:=\operatorname{Log} |z|+i\arg z$ , with the branch cut on the non-negative real axis. Determine a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$ ...
0
votes
0answers
16 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 ...
2
votes
0answers
42 views

Integral of Difference of Logs

I get the expansion of $h$ to be $$ h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r $$ $$ \Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}} $$ ...
1
vote
2answers
19 views

Conformal Mapping Between Two Domains (log)

Does anyone have a recommendation as how to go about solving this problem? I want a conformal from G to H where $$ G = \{ z \in \Bbb C \ | \ |z|<1, |z+i|>\sqrt{2} \}, S = \{ z \in \Bbb C \ | \ ...
1
vote
0answers
35 views

Why doesn't Logz/z have zeros?

Our book claims that $\frac {Logz}{z}$ has no zeros, where Logz is the principle branch of the complex natural logarithm. However, $Logz=log|z|+iArg(z)$, correct? So $Log1=log|1|+iArg(1)=0+i0=0.$ ...
0
votes
1answer
23 views

Limits involving logarithm and argument in the complex plane

$\operatorname{Log}((2/n) + 2i)$ as $n \to \infty$ $\operatorname{Log}(2 + (2i/n))$ as $n \to \infty$ $\operatorname{Arg}((1+i)/n)$ as $n \to \infty$ $(\operatorname{Arg}(1+i))/(n)$ as $n \to \infty$ ...
0
votes
1answer
17 views

For which $z$ is the following true: $Log(iz^2) = \frac{i\pi}{2} + 2Log(z)$.

The question on our worksheet is: For which $z$ is the following true: $Log(iz^2) = \frac{i\pi}{2} + 2Log(z)$. We raised both sides by $e$ and concluded this equation holds for all $z\neq 0$, but are ...
0
votes
0answers
55 views

Complex exponentiation

So I've got this question that is a bit difficult to ask, since it uses a term in my language that I can't properly translate into English. For $z\in\mathbb{C}^*$ and $a\in\mathbb{C}$ it would be ...
1
vote
1answer
64 views

Derivative of $\operatorname{Log}(\operatorname{Log}(z^2))$

Please help me with this question: (i don't know how to start) Suppose that $f(z)$ = $\operatorname{Log}(\operatorname{Log}(z^2))$. Find $f'(z)$ where it exists, and determine the set of points at ...
1
vote
2answers
41 views

Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
1
vote
3answers
107 views

Solve the equation $\log(z^2-1)=i\pi/2$

I set $z=x+yi$, so: $$ \log[(x+yi)^2-1]=\log(x^2+2xyi-y^2-1)=\log (r+iθ)=i\pi/2$$ than I get $x^2-y^2=1$ and I have no idea how to continue.
17
votes
2answers
479 views

Is this function a constant?

I am a french guest and I hope that my english isn't too bad... So here is my issue : I consider an entire function $f$ which satisfies the following property for all complex number $z\in \mathbb{C}$ ...
1
vote
3answers
38 views

Value of $2^{-1-i}$ in the complex

I am trying to find the value of $2^{-1-i}$. I rewrite it like this, $2^{-1-i}=e^{\ln(2)(-1-i)}={1\over{e^{\ln2}e^{i\ln2}}}=1/2$ Since $e^{i\ln2}=e^{Re(i\ln2)}=e^0=1$. This looks way nicer than ...
1
vote
0answers
22 views

Mapping a deleting ray to a horizontal strip

So, this is my question: D is a domain obtained by deleting the ray $x\leq 0$. And $G(z)$ is a branch of $log(z)$ on $D$. I want to show that G maps D onto a horizontal strip of width $2\pi$. Show ...
0
votes
1answer
37 views

Exponential of $\bar{z} $

I am currently reading the book Complex Variables by Stephen Fisher, there is one paragraph that was written like this: Establishing the following relation, and they write ...
1
vote
1answer
64 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
91 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 ...
1
vote
0answers
24 views

Can an entire $f$ satisfy $x>k | f(x+yi)=\ln(x+yi+z)+o(1) $?

Let $z$ be a complex number. Let $i$ be the imaginary unit. Let $x,y,k$ be positive real numbers. Consider $$x>k | f(x+yi)=\ln(x+yi+z)+o(1) $$ true for all $x>k,y$ and some $k,z$. Is there ...
3
votes
1answer
72 views

Keyhole integral and version of $\log$ in $\frac{\log t}{(t^2+1)^2}$

I want to calculate $$\int_0^\infty \dfrac{\log t}{(t^2+1)^2}dt$$ It certainly looks like a contour integral. I'm thinking about the keyhole contour where the "hole" is around the origin and along ...
3
votes
3answers
77 views

Regarding Cauchy - Goursat theorem with Log function

If $f(z) = \operatorname{Log}{z+2} $ and $C$ is $|z|=1$ , can Cauchy-Goursat theorem be applied at all? I was having the impression that log function resemble a ray $$\ln{r}+i\theta$$ therefore there ...
3
votes
1answer
77 views

Complex Analysis: Log Function

I want to approach this problem with maximum understanding of everything that is going on. I have the function $F(z)=\log(z^2+4)$, and I want to give a region in which it is analytic. I guess I ...
3
votes
1answer
81 views

How to prove $\operatorname{Log}(z) = \log(|z|)+i\arg(z)$.

The value of the principal branch of the logarithm can be evaluated by the formula \begin{align*} \operatorname{Log}(z) = \log(|z|)+i\arg(z), \end{align*} where $\arg(z) \in (-\pi,\pi)$ and ...
6
votes
1answer
89 views

Complex exponent properties?

Here is a line in a proof in a complex analysis text: $\sqrt{1-z^2}=\sqrt{1-z}\sqrt{1+z}$ I know you can't do this in general, but when can you do it? Here is what I tried: ...
1
vote
1answer
65 views

Show whether $\log r$ has a conjugate harmonic function on $\mathbb{C} \setminus \{0\}$

Can someone help me understand this passage in a student-written wiki article? The question is whether $u(x+iy) = \log \sqrt{x^2+y^2}$ has a conjugate harmonic function on $\mathbb{C}\setminus \{0\}$. ...
2
votes
1answer
84 views

Why isn't $\log(-1)$ defined?

Why isn't $\log(-1)$ defined. It can be defined as being equal to $i\pi$. Why don't we define the $log$ function over Complex Numbers as well.
0
votes
1answer
26 views

Compute log(z) for the following

Compute $\log{z}$ for the branch of log determined by $\pi < \theta \leq 3 \pi$ (a) $2 + 2i$ (b) $4 e^{i\theta\over 7}$ (c) $-i$
0
votes
1answer
41 views

Principal Logarithmic Question

Here is a question that is driving me insane: Show that $p.v \sqrt{z-1}\times p.v\sqrt{z+1}=-p.v.\sqrt{z^2 -1}$ for $Re(z)<-1.$(p.v. stands for the principal singular valued logarithmic ...
2
votes
1answer
50 views

Choosing a branch for $\log$ when comparing $\prod_{n=1}^\infty(1+a_n)$ and $\sum_{n=1}^\infty \log{(1+a_n)}$

On Ahlfors on p. 191 he is talking about the relation between $\prod_{n=1}^\infty (1+a_n)$ and $\sum_{n=1}^\infty \log(1+a_n)$. He says: Since the $a_n$ are complex, we must agree on a definite ...
0
votes
0answers
42 views

Why is the Chern class of a line bundle well defined?

Let $M$ be a compact Riemann surface with a finite open covering $$ M = \bigcup_{i=1}^{n} U_i $$ which has the property that every intersection is contractible (i.e. it is a good cover). To each two ...
0
votes
1answer
102 views

Branch cut for $\log (iz)$ in the region $\{z:\mathrm{Im}(z)>0\}$

If someone could explain branch cuts and branch points to me that would be fantastic. I understand that a branch cut is a curve that we remove from the domain to make a function (usually a logarithm) ...
5
votes
2answers
269 views

Definition of a logarithm

My question is as follows: Is the below a useful elementary way of dealing with negative arguments? If not, what is a better (elementary or not) way of dealing with negative arguments of the ...
2
votes
1answer
51 views

nth Root of a Rational Function

Suppose I have two polynomials $p(z)$ and $q(z)$ and a positive integer $n$. Suppose I wanted to define $r(z)=(\frac{p(z)}{q(z)})^{1/n}$ on $\Omega$ such that r(z) was analytic and single valued. On ...
2
votes
2answers
94 views

Evaluating $\int _{-1}^{e} \frac{1}{x}dx$

Here very easily by the Fundamental Theorem of Calculus $$\int _{-1}^{e} \frac{1}{x}dx=\ln(e)-\ln(-1)$$ From Euler's identity $e^{i \pi}$=-1 we can easily deduce that $\ln(-1)=i \pi$. Thus the ...
2
votes
1answer
87 views

Questions about $\ln(z)$ recurrence and fixed points.

Define property $A_R$ for an analytic function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ for $|z|<R$ where $R$ is a radius. And $f(z)$ is analytic within the radius $R$ ...
2
votes
1answer
42 views

Question about fixpoints and zero's on the complex plane.

Define property $A$ for an entire function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ $2)$ $f(z)=z$ has exactly one solution $=>z=0$ (follows from $1)$ ) $3)$ $f(z)$ is not a ...
2
votes
1answer
54 views

Logarithms and the Identity Theorem

My teacher today asked us how can we use the Identity Theorem to show why can't we define the complex logarithm on $\mathbb{C}-\{0\}$. Can someone give me a hint on how can I do this?
4
votes
3answers
810 views

Find a branch of $f(z)= \log(z^3-2)$ that is analytic at $z=0$.

Find a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$. Can anyone help me on this question? I have no idea how to find a branch. The definition of branch given in lecture is $F$ is a branch ...
4
votes
1answer
95 views

Does a logarithmic branch point imply logarithmic behavior?

The complex logarithm $L(z)$ is given by $$L(z)=\ln(r)+i\theta$$ where $z=re^{i\theta}$ and $\ln(x)$ is the real natural logarithm. It is well known that $L(z)$ then sends each $z$ to infinitely many ...
2
votes
1answer
51 views

Growth rate of Taylor convergents near pole

For any fixed $z_0\in\mathbb{C}\setminus \{0\}$ and $\beta\in\mathbb{R}^{+}$, prove that $$\left.T_n\left(\log^{\beta}z;z_0\right)\right|_{z=0}\sim\log^{\beta} n$$ Note: I observed that this holds ...
3
votes
1answer
207 views

contour integration of logarithm function

I'm new to contour integral involving branch point and stuck on this particular integration. Here is the problem: $$\int_{\mathcal{C}}\log z\,\mathrm{d}z,$$ where $\mathcal{C}$ is a closed square ...
0
votes
1answer
233 views

Integral of $z^{n} \log z $ on the unit circle under two assumptions

I'm asked to calculate $\int_{|z| = 1} z^{n} \log z dz$ in two ways: (1) if $\log 1 = 0$; (2) if $\log (-1) = i \pi$. I understand it means that in case (1) I have to work with the principal ...
3
votes
1answer
89 views

Show that map is conformal

I want to show that the map $\phi(r,\theta) = r^\lambda (\cos(\lambda \theta), \sin(\lambda \theta))$, where $\lambda \in \mathbb{C}$, is conformal on the slit plane $\{(r,\theta)| r > 0, -\pi < ...
1
vote
1answer
68 views

The bound of a log function

It looks like we can control $\log\frac{1+z}{1-z}$ by $\log\frac{1+r}{1-r}$ if $|z|=r<1$ where the logarithm is defined on the branch obtained by deleting the negative imaginary axis. I tried to ...
2
votes
3answers
163 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 ...
3
votes
1answer
132 views

How is $(\arg F)(z)$ of complex function calculated?

My book on complex analysis introduces $\arg(z)$ and $\arg(z-a)$ after the complex logarithm is introduced. It shows that the two are just the oriented angles between $z$ and the point $z_0$ of the ...
4
votes
1answer
176 views

Solving a transcendental equation consisting of a quadratic part and a part involving inverse Lambert W functions

Question statement I would like to solve the following equation in the two variables $x$ and $y$: \begin{gather} 0 = x^2 - a y^2 + i b [x y - W^{-1}(x)W^{-1}(y)] , \end{gather} where $a$ and $b$ are ...
5
votes
2answers
137 views

How to formally show that $f(z)$ is analytic at $z=0$?

Let $z$ be a complex number. Let $$f(z)=\dfrac{1}{\frac{1}{z}+\ln(\frac{1}{z})}.$$ How to formally show that $f(z)$ is analytic at $z=0$? I know that for small $z$ we have ...
7
votes
4answers
820 views

Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$

How would I go about evaluating this integral? $$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$ What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
17
votes
5answers
1k views

Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$

Evaluate: $ \displaystyle \int_0^{\pi} \ln \left( \sin \theta \right) d\theta$ using Gauss Mean Value theorem. Given hint: consider $f(z) = \ln ( 1 +z)$. EDIT:: I know how to evaluate it, but I am ...
2
votes
1answer
621 views

Evaluating $f(z)=\sqrt{z^2-1}$, given the branch I am on.

I'm working on a problem in Gamelin's Complex Analysis (Chapter IV, Section 2, page 109, exercise #4). I'm asked to consider the branch of $f(z)=\sqrt{z^2-1}$ on $D=C\setminus (-\infty,1]$ that is ...