Tagged Questions

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Definition of $0^z$, $z$ complex

Usually we define the function $a^b$ over the complexes using the exponential function, as $e^{b\log a}$. This function has some issues with multivalued-ness, but it still more or less satisfies ...
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Alternative definition of complex number, showing it is equivalent to the tradidional one.

The author of a book makes an alternative definition of the complex numbers, later he shows that this definition is equivalent to the ordinary definition where we define $i^2=-1$. Here is his ...
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$f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set $f(A)$

Let $f:\mathbb R^2\to\mathbb R^2$ be defined by the equation $$f(x,y)=(x^2-y^2,2xy).$$ Show that $f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set ...
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Prove that the functions $g_k(z) = f_k \circ h_k(z)$ form a normal family.

I am having a bit of trouble with the following complex analysis question which originates from a qual. Some help would be awesome. Let $f_k :\mathbb{D} \rightarrow \mathbb{C}$ be a normal family of ...
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Is this $\epsilon-\delta-$proof correct?

I have to Show that $$\mathbb{C} \rightarrow\mathbb{R}; z \rightarrow \Re z$$ is a continuous function using the $\epsilon-\delta-$criteria. So what I did is the following: I have to Show that ...
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How come complex numbers represent coordinates?

I'm wondering why complex numbers represent coordinates without being on the form of a tuple (a,b). The complex numbers come in the form: $a+bi$ where $a$ denotes the real part and $bi$ denotes the ...
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How to prove $x_{n+1}y_n-x_ny_{n+1}=2^{3n}\sqrt{7}$

For $n \in \mathbb{N}$ let $z_n=(1-i\sqrt {7})^n, ~x_n = Re~z_n,~y_n = Im~z_n$. I want to show the following: $x_{n+1}y_n-x_ny_{n+1}=2^{3n}\sqrt{7} ~~~(n \in \mathbb{N})$ My only idea was to show ...
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Why do imaginary numbers work (somewhat philosophical question)?

Asking as a layman, I've always puzzled over imaginary numbers and how they can be used to solve problems involving real numbers or quantities only (e.g. contour integration methods or Fourier ...
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How does $\mathrm {e}^z$ and $\log z$ look like as complex functions.

I want to visualize complex functions $\mathrm e^z$ and $\log z$ in $C$, here $z\in\Bbb C$. I want to know their behavior and zeros and singularities. Can anyone explain me in an easy way. Thank you ...
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Is there a single valued definition for $\sqrt[n]{z}$ on $\Bbb{C}$?

Since the $n$th root of a complex number has $n$ possibilities, which one do you choose so that its restriction to $\Bbb{R}$ is the positive square root if $r$ is positive and $\sqrt{|r|}i$ if $r$ is ...
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Find Limit of Complex integration

How should we prove that $\int_{\mathbb{C}}\frac{|G(w)|}{|z-w|}dA(w)\longrightarrow0$ as $|z|\longrightarrow\infty$where $G(z)\in L^{1}(\mathbb{C})$ and $|G(z)|\leq\frac{C_{0}}{|z|^{1+\varepsilon}}$ ...
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Triangle Inequality with complex numbers: Prove that ||x|âˆ’|y||â‰¤|x|-|y|.

Prove that $||x| - |y|| \le |x| - |y|$ for all $x,y \in \mathbb{C}$. I fully understand the other inequality: $|x+y| \le |x|+|y|$ for all $x,y \in \mathbb{C}$. But I have no clue how to start ...
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Convergence of the Zeta and Phi functions

I want to show that the following functions (in the picture) are absolutely and locally uniformly convergent if real part of complex number $s$ is bigger than 1. Absolute part for zeta function is ...
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If $f: \mathbb{C}\to\mathbb{C}$ is bounded, then is it a constant? [closed]

If a function $f: \mathbb{C}\to\mathbb{C}$ is bounded, then it is a constant. Is it true or false?
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Fourier Analysis

I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function? Can someone provide me simple information ...
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Accumulation Points of a Complex Sequence

Let $z$ be a complex number of absolute value 1: $z=e^{i\theta}, 0 \le \theta \lt 2\pi$. What are the accumulation points of the sequence $\lbrace z^n \rbrace$? Distinguish between the case where ...
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$x^2+1=0$ uncountable many solutions [duplicate]

Possible Duplicate: Why are the solutions of polynomial equations so unconstrained over the quaternions? Coudl someone explain me the following: Why should $x^2+1=0$ have uncountable ...
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find all the solutions of $e^{a+ib}=e^{a-ib}+2i$ $Re(a+ib) \le 0$

find all the solutions in the complex field of the system $$e^{a+ib}=e^{a-ib}+2i$$ $$Re(a+ib) \le 0$$ where $a+ib$=z and a-ib=conj(z)
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Prove $i\notin \mathbb R$

Prove that $\imath$ (defined by $\imath^2=-1$) does not have a position on the Real number line. That is, show that there does not exist two real numbers $a$ and $b$ such that $a<\imath<b$. ...
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Complex Analysis and Limit point help

So S is a complex sequence (an from n=1 to infinity) has limit points which form a set E of limit points. How do I prove that every limit point of E are also members of the set E. I think epsilons ...