1
vote
2answers
28 views

Branch of $n$th root of $f$ is holomorphic

The problem states to prove that if $h$ is a branch of $f^{1/n}$ for integer $n > 0$ (i.e. $h(z)^n = f(z)$ for $z \in G$, $h$ continuous), then $h$ is holomorphic, where $f$ is a holomorphic ...
1
vote
1answer
36 views

Principal square root of a product of complex numbers with positive real part

Given $n$ complex numbers $z_i$ with $\Re z_i>0$, why is it that $$\prod_i\sqrt{z_i}=\sqrt{\prod_i z_i}?$$Numerically, this appears to be the case, however, I don't see an easy way to prove it.
1
vote
1answer
26 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 ...
7
votes
2answers
102 views

Why is this function a really good asymptotic for $\exp(x)\sqrt{x}$

$$f(x)=\sum_{n=0}^{\infty} a_n x^n\;\;\;\;\; a_n = \frac{1}{\Gamma(n+0.5)}$$ Why is this entire function a really good asymptotic for $\exp(x)\sqrt{x}$, where for large positive numbers, ...
3
votes
3answers
247 views

Find a real entire function $f(z)$ asymptotic to $\ln(x^2+1)$ for real $x$.

Find a real entire function $f(z)$ asymptotic to $\ln(x^2 +1)$ for real $x$. More specific I want $f(0)=0$ and $\frac{1}{2} \ln(x^2+1) < f(x) < 2 \ln(x^2+1)$. Or prove it does not exist.
2
votes
0answers
109 views

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration $$ \int_{0}^{1}{\ln\left(\,x\,\right)\over \,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}}\,{\rm d}x $$ I am not sure as to how to work with the branch ...
2
votes
2answers
56 views

Is it possible to use complex logarithm to integrate $1/(z+i)$ along a path?

Evaluate the following on the path $\gamma_1$ with endpoints $[-1,1+i]$ $$ \begin{align} I_1=\frac{i}{2}\int_{\gamma_1} \frac{1}{z+i}dz -\frac{i}{2}\int_{\gamma_1}\frac{1}{z-i}dz \end{align} $$ Am I ...
0
votes
2answers
56 views

Show that $\log[(1+i)^2]\neq 2\log(1+i)$

The problem is as stated in the title. I found that the $\mathrm{Log}[(1+i)^2] = 2\mathrm{Log}(1+i)$. We know that $$\mathrm{Log}(z)=\ln(r)+i\theta$$ Now, without defining a branch, doesn't that mean ...
2
votes
1answer
57 views

Proof of the analyticity of complex logarithm

Let $a\in(-\pi,\pi]$ and $f:G\to\mathbb C$, $G = \{ z\in\mathbb C\setminus\{0\},\operatorname{Arg}z\neq a \}$ $$f(z)=\ln|z|+\imath \arg_a z,\quad a<\arg_az<a+2\pi$$ Prove that $f$ is ...
0
votes
1answer
115 views

Finding the transfinite diameter of the level sets of complex logarithm

Given a simply-connected domain $|g(z)|\ge C$ how can I find the analytic conformal mapping guaranteed by the Riemann mapping theorem? In particular I'm interested in finding the transfinite diameter ...
0
votes
1answer
40 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 ...
4
votes
0answers
145 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
41 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
53 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
1answer
140 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
169 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
226 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
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? ...
0
votes
2answers
47 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
43 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
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 ...
2
votes
0answers
55 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
64 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
43 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
72 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
68 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
187 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
820 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
27 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
45 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
69 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 ...
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
89 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
82 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
103 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
103 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
108 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
95 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
43 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
51 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
142 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) ...