# Tagged Questions

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### 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 ...
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### 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.
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### 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 ...
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### 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, ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### ${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 $\mathbb{C}\setminus \{0\}$. ...
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### 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.
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### 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 ...
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### 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 ...
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 ...