0
votes
1answer
38 views

True or false logarithmic branches

Say whether the following are true or false. Give a short proof. 1) $log(-z)+i{\pi}$ is a branch of the logarithmic function whose branch cut is the non-negative real axis 2)If $g(z)$ is a branch of ...
3
votes
0answers
117 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting two rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
0
votes
1answer
38 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 ...
1
vote
2answers
49 views

Definition of $a^b$ for complex numbers

Problem statement Let $\Omega \subset C^*$ open and let $f:\Omega \to \mathbb C$ be a branch of logarithm, $b \in \mathbb C$, $a \in \Omega$. We define $a^b=e^{bf(a)}.$ $(i)$ Verify that if $b \in ...
2
votes
0answers
116 views

What is ${\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$?

This is a new integral that I propose to evaluate in closed form: $$ {\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$$ where $\log (z)$ denotes the principal value of ...
6
votes
4answers
162 views

What did Johann Bernoulli wrong in his proof of $\ln z=\ln (-z)$?

Some people say, Johann Bernoulli has proven $\ln z=\ln (-z)$ in the following way $$\ln ((-z)^2 )=\ln(z^2)\;\;\;\Rightarrow\;\;\;2\ln(-z)=2\ln z\;\;\;\Rightarrow\;\;\;\ln (-z)=\ln z$$ While the ...
1
vote
2answers
222 views

Find a function that satisfies the condition.

Let $\epsilon > 0$ be fixed and $t$ a variable that takes values in the universal covering space of ${\mathbb{C} \setminus \{0\}}$. Find a continuous function $f(s$) such that $$|t \log t| = |t| ...
1
vote
1answer
23 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
126 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? ...
0
votes
2answers
44 views

Does the logarithm inequality extend to the complex plane?

For estimates, the inequality $\log(y)\le y-1,$ $y>0$ is often helpful. Is there any sort of upper bound for the logarithm function in the complex plane? Specifically, $|\log(z)|\le$ something for ...
0
votes
2answers
21 views

Branch of logarithm which is real when z>0

I am familiar with the complex logarithm and its branches, but still this confuses me. I read this in a textbook: "For complex $z\neq 0, log(z)$ denotes that branch of the logarithm which is real ...
2
votes
0answers
38 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 $z=0$ ...
1
vote
1answer
64 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
53 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
41 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
39 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
27 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
25 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
63 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
66 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
44 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
154 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.
25
votes
3answers
807 views

If there are entire $G_k$s such that $f=\exp\circ\exp\circ\cdots \circ\exp\circ G_k$ ($k$ times), must $f$ be 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 each complex number $z\in ...
1
vote
3answers
39 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
23 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
43 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
67 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
101 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
86 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
78 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
93 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
95 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
96 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
84 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
90 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
42 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
53 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
45 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
126 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
288 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
60 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
96 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
57 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
894 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
102 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
242 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 ...