# Tagged Questions

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### At what point is this piecewise function continuous?

let $$f(z) = \begin{cases} z & |z|\leq 1 \\ |z|^{2} & |z|> 1 \end{cases}\ \text{where}\ z\in \mathbb{C}$$Does anyone could help me ?Thanks!
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### Problem on Complex numbers involving a point on a Circle

Question: The Complex number $z$ is represented by the point $T$ in the Argand Diagram.Given that $$z =\frac{1}{3+it}$$ where $t$ is a variable, show that i) as $t$ varies, $T$ lies on a circle, and ...
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### Finding the set of analytic functions whose image is a subset of a given set

Let $A=${$z\in\mathbb{C}||z|=1$} and $B=${$z\in\mathbb{C}||z|<2$}. I want to find the the set of analytic functions such that $f(B)\subset A$. Is there a way to solve this? Hope someone could help ...
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### Calculating the real and imaginary parts of a holomorphic function

Calculate the real and imaginary parts of the holomorphic function $f(z)=z^2\cos(z)-e^{z^3-z}$ and verify directly that each of these functions is harmonic. I believe I know how to the question, ...
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### The differentiability of the complex valued function $(Rez)(Imz)z\over|z|^2$

$$f(z) = \left\{ \begin{array}{ll} \Re(z)\Im(z)z\over|z|^2 & \quad z \neq 0 \\ 0 & \quad z = 0 \end{array} \right.$$ I want to prove that this ...
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### Find similar estimate for complex numbers

According to Borwein, page 356 Prop. 2, $\left|\ln\left(\frac{4}{k}\right)-\operatorname{I}(1,k)\right|\leq 4k^2\left(8+\left|\ln k\right|\right)$ holds for $k\in\left(0,1\right]$. ...
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### Does $\lim_{x \to 0}({z^2\over \overline z})$ exist? $(z\in \mathbb{C})$

I am trying to figure out if $\lim_{x \to 0}({z^2\over \overline z})$ exists or not. This is a way I though to show that this does not exist but I am not entirely sure. Let $a_n={1\over n}$ and ...
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### $f:U \rightarrow \mathbb{C}$ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$

I want to prove that if $f:U \rightarrow \mathbb{C}$ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$. This is my rather incomplete approach to ...
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### Is this log identity true?

I'm wondering if the exponent property carries forward to the complex log. In other words, for some complex numbers $z$ and $w$ does $\ln(z^w) = w\ln(z)$?
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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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### Is $\{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\}$ open or closed?

I am trying to figure out if the set $\{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\}$ is open or closed or maybe none of that. I hope someone could provide a hint to solve this. Can this set be ...
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### All Values of a Complex expression

I am asked to find all values to $$\left(\frac{1-i}{\sqrt2}\right)^{1+i}$$ I do not know how to approach a power with complex part. Any help would be appreciated.
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### The annulus $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open

I want to prove that the set $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open. This is my attempt. Let $z \in A_{r,s}(z_0)$. Then $|z-z_0|-r>0$. Let $r'=[|z-z_0|-r]/2$. Then ...
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### Quadratic formula with complex coefficients

Let $a,b$ and $c$ be complex numbers. I'm trying to prove that this version of the usual quadratic formula: $$z=\frac{-b+(b^2-4ac)^{\frac{1}{2}}} {2a}$$ solves the quadratic equation ...
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### If the sum of absolute values of complex numbers is at least $1$, then some subset of these numbers has absolute value at least $C$

There is a challenging problem in a book of mine on complex analysis, and I seriously do not even know where to start. I'm more than sure I don't properly understand the problem. Prove that there ...
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### Find solution of equation $(z+1)^5=z^5$

I attempt to solve the equation $(z+1)^5=z^5$. My first approach is to expand the left hand side but ı get more complicated equation. So I couldn't go further. Secondly, I write equation as, since ...
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### Why is Euler's formula defined for non-integer values?

Say that for some complex number $w$ $$e^{wi} = a$$ Now raise both sides to $1/4$. $$e^{wi/4} = a^{1/4}$$ Now $e^{wi/4}$ has a single defined value. Yet $a^{1/4}$ can have multiple values. So why ...
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### Why is Euler's formula valid for all $n$ but not De Moivre's formula?

The Wikipedia page on De Moivre's Formula says the formula doesn't hold for non-integer $n$, since non-integer powers of a complex number can have multiple values. It then goes on to say that this ...
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### Guidance for complex numbers/analysis problem needed [duplicate]

I'm looking at this one problem in a book of mine, but I can't even seem to start it. Let $z_1,z_2,...$ be a countable set of distinct complex numbers. If $|z_j-z_k|$ is an integer for every ...
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### Proving $\arg(zw)=\arg(z)+\arg(w)$

This is my attempt I know this is incomplete or may even be wrong. Let $θ_1 \in \arg(z)$ and $θ_2 \in \arg(w)$. Then, $θ_1+θ_2 \in \arg(z)+\arg(w)$. Also, $θ_1+θ_2 \in \arg(zw)$. Is this sufficient ...
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### Find loci of the points in complex plane such that $\mathrm{Im}(\frac{z-z_1}{z-z_2})^n=0$

Find loci of the points in complex plane such that $$\mathrm{Im} \left (\frac{z-z_1}{z-z_2}\right )^n=0,$$ where $n\in\mathbb{N}$, $z_1, z_2$ are the given points in $\mathbb{C}$. When $n=1$, it ...
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### Every line or circle in $\mathbb{C}$ are solution sets to the equation…

Here is a complex analysis homework problem I can't quite figure out: Prove that every line or circle in $\mathbb{C}$ is the solution set of an equation of the form $a|z|^2+\bar{w}z+w\bar{z}+b=0$, ...
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### complex limits, how to show they go to 0?

In complex integration my book uses that some limits go to zero as R goes to infinity. However I do not now how to show this, these two limits are: $e^{-\pi(R^2+2iRy-y^2)}$, where y is a real number ...
### Complex analysis: Rewrite $\cos^{-1}{i}$ in algebraic form
I'm stuck in this problem (complex analysis), my answer is not the one reported in the book: Rewrite $\cos^{-1}{i}$ in the algebraic form. A: $k\pi + i \frac{\ln{2}}{2}\ \forall\ k \in \mathbb{Z}$ ...
### Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$?
Is there anything special with the form: $$\left|\frac{z-a}{1-\bar{a}{z}}\right|$$ ? With $a$ and $z$ are complex numbers. In fact, I saw it in a problem: If $|z| = 1$, prove that ...