4
votes
0answers
64 views

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 ...
2
votes
1answer
71 views

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 ...
2
votes
4answers
71 views

$f$ 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 ...
0
votes
2answers
69 views

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 ...
1
vote
1answer
51 views

Construct a non-constant analytic function $f : \Omega_1 \to \Omega_2$ or show that this is impossible.

I am having a lot of difficulty with the following past qualifying exam problem. Any help would be awesome. Thanks. Let $\Omega_1 = \mathbb{C}\setminus \left \{\{0\} \cup \{\dfrac{1}{n}:n\in \Bbb ...
5
votes
1answer
93 views

$\lim_{x\to 2} \, \sqrt{x-2}$

$$\lim_{x\to 2} \, \sqrt{x-2}$$ When you take the right hand limit for this expression, you get $0$. However, if you take the left hand side it gives an imaginary number. However, do you consider ...
0
votes
2answers
18 views

$\big| |z|-|w|\big| \leq |z-w| \implies \big|c_{1} |z|- |w|c_{2}\big| \leq |c_{1}z- c_{2} w| ? $ ($z, w \in \mathbb C, C_{1}, C_{2} >0$)

By triangle inequality, we get, $$\big| |z|-|w|\big| \leq |z-w|; (z, w\in \mathbb C.)$$ Take any $C_{1}, C_{2} > 0$ and fix it. My Question is: Can we expect: $$\big|C_{1} |z|- |w|C_{2}\big| ...
1
vote
1answer
19 views

$|g(t_{1}) e^{-(t_{1}-x)^{2}}- g(t_{2})e^{-(t_{2}-x)^{2}}|\leq |f(t_{1}) e^{-(t_{1}-x)^{2}}- f(t_{2})e^{-(t_{2}-x)^{2}}| $?

Suppose $f, g: \mathbb R \to \mathbb C$ such that $|g(t_{1}) -g(t_{2})| \leq |f(t_{1})- f(t_{2})| $ for every $t_{1}, t_{2} \in \mathbb R.$ Take any $x\in \mathbb R$ and fix it. Edit: We also assume ...
0
votes
1answer
32 views

what are contraction(Lipschitz) maps on $\mathbb C$?

We say a map $f:\mathbb C \to \mathbb C$ is contraction(Lipschitz) if $|f(z_{1})- f(z_{2})| \leq C |z_{1}- z_{2}|$ for every $z_{1}, z_{2} \in \mathbb C$ and $C$ is some constant. Trivial Examples: ...
3
votes
1answer
69 views

Why is $\int |e^{ix}|^2 dx = x + C$?

Quick question: Wolfram Alpha tells me that $$\int |e^{ix}|^2 dx = x + C$$ Why is that?
0
votes
1answer
30 views

Behaviour of Hankel function $H_s^{(1)}(x)$ near $x=0$

I am looking for a reference to the fact $H_s^{(1)}(x) \approx i (\frac{2}{x})^s \frac{\Gamma(s)}{\pi}$ for small $x$,and $s\in \mathbb{C}$. I think it is obtained from some integral representation of ...
1
vote
1answer
44 views

Continuity and other properties of complex exponential

So I think I can do the others, but part (i) about showing the continuity of $a^z$ has me stumped. I always get really stuck when it comes to proving continuity (I am using the metric spaces ...
3
votes
1answer
32 views

Effect of sigmas inequality on sequences

We have two nets of complex numbers $\{z_\alpha\}_{\alpha\in I},\{w_\alpha\}_{\alpha\in I}$ for some set $I$ which might be uncountable, and we have $$\sum_{\alpha\in I}|w_\alpha|\leq ...
0
votes
1answer
47 views

Complex number in polar coordinates

I have to get $\Im$, $\Re$, the absolut value as well as the argument $\phi$ of the complex number $$z = \left(-\frac{1}{\sqrt2}+\sqrt\frac{3}{2}i\right)^8$$ I do this by transforming $z' = ...
0
votes
2answers
82 views

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 ...
2
votes
1answer
56 views

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 ...
0
votes
1answer
49 views

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 ...
4
votes
2answers
202 views

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 ...
3
votes
3answers
140 views

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 ...
0
votes
1answer
27 views

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 ...
1
vote
0answers
28 views

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 $|G(z)|\leq\frac{C_{0}}{|z|^{1+\varepsilon}}$ ...
1
vote
2answers
674 views

Triangle Inequality with complex numbers: Prove that ||x|−|y||≤|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 ...
3
votes
1answer
131 views

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 ...
-1
votes
2answers
94 views

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?
0
votes
1answer
72 views

A rather ugly limit [duplicate]

Evaluate $$\lim_{n \rightarrow \infty} n \sin (2\pi e n!).$$ I wanna ask what's wrong with my method: Define $C_n= n \cos (2\pi e n!)$ and $S_n=n \sin (2\pi e n!)$, then $C_n+iS_n=ne^{i2\pi ...
0
votes
0answers
37 views

Fractional linear transformations with given properties

I need a function of the form $\displaystyle f(z):= \frac{az+b}{cz+d}, \qquad z\in\mathbb{C}-\{-\frac{d}{c}\}, \qquad ad-bc\neq0$ which carries the half-plane $\{z\in\mathbb{C}\ |\; ...
2
votes
5answers
296 views

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 ...
3
votes
3answers
478 views

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 ...
6
votes
2answers
205 views

$x^2+1=0$ 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 ...
1
vote
2answers
178 views

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)
1
vote
2answers
224 views

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 $a<\imath<b$. ...
1
vote
1answer
257 views

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 ...
2
votes
1answer
109 views

Real numbers mapped onto a sphere

We can compare real values if they were greater, lesser but we cannot do same for complex numbers. What if we map real values(within some small range) onto a sphere and declare each one of them as ...
2
votes
1answer
118 views

Definition of a real without using complex

I have a set of arithmetic functions from $D\subset\mathbb C$ to $\mathbb C$ (addition, division, trigonometric functions, ...). Each of those functions can also be restricted from $E\subset \mathbb ...
3
votes
2answers
90 views

question about binomial expansion's coefficients

I am trying to show that if $$\left( 1+x\right) ^{n}=p_{0}+p_{1}x+p_{2}x^{2}+\ldots $$ and n being a positive integer, then $$p_{0}-p_{2}+p_{4}+\ldots = 2^{\frac {n} {2} }\cos \dfrac {n\pi } {4}$$ and ...
5
votes
2answers
267 views

Primitive roots of unity

I am trying to show that, If $$f\left( x\right) =a_{0}+a_{1}x+\ldots +a_{k}x^{k}$$ then $$\dfrac {1} {n}\left\{ f\left( x\right) +f\left( wx\right) +\ldots +f\left( w^{n-1}x\right) \right\} ...
0
votes
1answer
93 views

Is there a forumla for number of primes preceding a natural number?

I am guessing there is no known analytical function which gives such a formula. This question came to mind while attempting a rather basic proof. I am trying to show that the number of primitive ...
2
votes
1answer
124 views

Rudin Question (Integration of Complex Functions) [pg.325]

I was reading Rudin and I stumbled upon a proof that I do not seem to understand. It is on page 325 of Baby Rudin $3^{rd}$ edition. In case you do not have a copy I shall write some background ...