0
votes
0answers
17 views

Derivatives, Cauchy-Riemann Equations [on hold]

Given the function, $w=z^4$ and I want to find the following solutions for this equation, Find real functions u and v such that w=u+iv Show that Cauchy-Riemann equation holds at all points in the ...
0
votes
2answers
40 views

Uniform convergence of the series

Test the uniform convergence of the series $$ \sum_{n=1}^\infty \frac{1}{z^2 - n^2 \pi^2}$$ $$ \forall z \not= \pm n\pi,\;\; where n \in\mathbb N$$ Can I find $M_n$ such that $$ ...
2
votes
2answers
32 views

Express each function in the form $u(x,y) + iv (x,y)$

I was doing some homework with complex numbers and I'm stuck with these two, I hope that someone can solve these and clear it up for me. Thank you. ln(1+z) z/(3+z) Samples,
0
votes
0answers
16 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 ...
1
vote
1answer
24 views

Finding the locus represented by complex variable equations?

I'm trying to solve these two problems related to complex number but hardly found a solution. I hope that someone can solve these and clear it up for me. Thank you. |z+2|=2|z-1| |z+5|-|z-5|=6
0
votes
1answer
45 views

Number of zeros of $ z^7+4z^4+z^3+1$

How many zeros does $z^7+4z^4+z^3+1$ have in each of the regions |z|<1 and |z|<2? I know I should use Rouche's Theorem but I can't find a $|f(z)| > |p(z)-f(z)|$ and $f(z)$ have equal number ...
0
votes
0answers
14 views

How to deal with x* when solving complex-variable linear equation(s) of x?

The theory of linear algebra can be directly applied to linear equation(s) of complex variables with the form \begin{equation} \sum_i a_i x_i=c\ldots\ldots(1) \end{equation} with $a_i,c\in ...
0
votes
3answers
49 views

Algebraic Equation?

$$Ve^{i\theta} = We^{i\phi}$$ where, $V$ and $W$ are some real constants. From this my book concludes: $\theta = \phi$. How does it conclude this? I don't see why its valid to just equate the ...
1
vote
2answers
64 views

Complex numbers confused!!

If you give me a complex number say $z=2+3i$, then I can easily find $\text{Im}(z)=3$ and $\text{Re}(z)=2$ but when this polar coordinates stuff came, I lost my head! So say ...
0
votes
1answer
25 views

Plot of a domain in the complex plane

I am trying to plot the following domain in the complex plane: $\lbrace x\in\mathbb{C}|\: |x^{2}-1|<r\rbrace$ for some $r>1$. I know that in general to take a square root of a complex number ...
0
votes
1answer
12 views

Residues more than one singularity at 0

Having trouble calculating the residue at 0 for this integral within the unit circle I understand that its a pole of order 3 because both the z^2 and the sinz have singularities at 0. Is there an ...
1
vote
0answers
33 views

Test the uniform convergence of the series in indicated region

Test the uniform convergence of the series I tried to find $M_n$ such that $|\sum_{n=1}^ \infty(-1)^n\frac{z^{2n-1}}{1-z^{2n-1}}|\le M_n $ by using Abel's Theorem This is the question : Test the ...
1
vote
2answers
33 views

Why does $| a_n + \frac {a_{n-1}} {z} + \ldots + \frac {a_{0}} {z^n} | \ge | a_n | - |\frac {a_{n-1}} {z} + \ldots + \frac {a_{0}} {z^n} |$ hold?

Why does $$| a_n + \frac {a_{n-1}} {z} + \ldots + \frac {a_{0}} {z^n} | \ge | a_n | - |\frac {a_{n-1}} {z} + \ldots + \frac {a_{0}} {z^n} |$$ hold by the trinagle inequality for $z, a_i \in \mathbb C$ ...
1
vote
1answer
57 views

Complex number with 3 dimensions [duplicate]

I was looking back on complex analysis and asked myself: ''Why is there no complex number in 3 dimensions ?''. To place this question let me define with what I mean with 3 dimensions in the following. ...
0
votes
1answer
69 views

Why isn't $i$ affected by powers?

When finding roots of complex functions we can write for example: $$z=2-2i$$ Let's find complex numbers $w$ such that $$w^4 = 2-2i$$ $$\large z = \sqrt{8} e^{ \frac{- \pi }{4} i}$$ This reads: ...
-1
votes
0answers
46 views

Does any series leads to $\pi i $ in complex plane [closed]

Does any series leads to $\pi * i$ in complex plane , is their any series like we have in real plane.
-2
votes
0answers
38 views

Determine the number of zeros of the upper half-plane [closed]

$$z^4 + 3iz^2 + z - 2 + i$$ Can anyone please help me????
0
votes
1answer
21 views

Finding isolated singularities

I am having trouble categorizing the singularities of the following complex valued function: $$f(z) = \frac{z^2}{sin(z)}$$ It seems like the isolated singularities are $2n\pi$ where ...
1
vote
1answer
47 views

Showing $\operatorname{Log}(z-i)$ is not analytic

Show that the function $\operatorname{Log}(z-i)$ is analytic everywhere except on the half line $y=1$ $(x\leq 0)$. I know that ...
1
vote
2answers
41 views

Limit concept under Complex analysis

Prove that $$ \lim_{z\to i} \dfrac{3z^4-2z^3+8z^2-2z+5}{z-i} = 4+4i $$
1
vote
1answer
25 views

Making $-{{\pi i}\over n} e^{\alpha i}({{1 - e^{2 n \alpha i}\over{1-e^{2 \alpha i}}}})={\pi \over {n sin(\alpha)}}$; $\alpha={{2m+1}\over{2n}} \pi$

As part of a (much) longer problem in complex analysis, I need to show that the equality mentioned in the title makes sense, but I can't seem to find the right algebra tricks to get from point A to ...
0
votes
0answers
29 views

Showing where complex function is analytic and differentiable.

I've been asked to show where the following function is analytic and differentiable; $$f(z) = x^4 + i(1-y)^4$$ for $z = x + iy$ First, I noted that $u(x,y) = x^4$ and $v(x,y) = (1-y)^4$. Then, I ...
-1
votes
1answer
35 views

Determine the real number a so that… [closed]

Determine the real number "a" so that $$ Z= \frac{1+3i}{2-ai} $$ has: $$\arg z = 0 $$
0
votes
2answers
67 views

Complex Number Question - $|z^{z}|$

Find all possible values of $$\mid z^{z} \mid$$ using the polar for of $z$. I have tried putting it into polar form but nothing comes out that seems easy to work with/looks like a reasonable simple ...
1
vote
1answer
43 views

Using Liouvilles theorem to show that f is identically constant on all of $\mathbb C$

Use Liouvilles theorem and the fundamental theorem of calculus to prove that for an entire function $f$, if there exists $M \in \mathbb R: Re(f(z)) \leq M$ $ \forall z \in \mathbb C $, then $f$ is ...
0
votes
1answer
29 views

Determining the number of complex roots (including multiplicities) of a polynomial

Could someone please explain/show me how to determine the number of complex roots including multiplicities of a polynomial such as $P(z):= 5i z^{37} - (6 +2i)z^{4} + 4z^2 - i$ Would i need to ...
4
votes
6answers
157 views

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$. My attempt: As the polynomial is a cubic, it must have atleast one real ...
2
votes
2answers
87 views

Paradox with function representation

Let assume the function $\eta(E)$ has the following representation: $$\eta(E) = \sqrt{\frac{a}{E}}$$ where $a$ is the known positive constant, and $E \in [-\infty, +\infty]$. I know that $\sqrt{a} = ...
0
votes
1answer
52 views

proving a limit of a function by definition

Consider $f: \Bbb{C} \to \Bbb{C}$ defined by $$ f(z) = \begin{cases} z^3 + 2z &\text{if } z \ne i \\ 3 + 2i &\text{if } z = i \end{cases} $$ Prove that $$ \lim_{z \to i} f(z) = i $$ using the ...
0
votes
2answers
32 views

Convergence of complex power series question

I need some help to solve this problem and find the domain of convergence of the following power series: $$\displaystyle\sum_{n=0}^\infty(2^n+i^n)(z-2i)^n$$ Thank you!
0
votes
0answers
29 views

Example of holomorphic function from unit disc to itself

let $f:\mathbb{D} \to \mathbb{D}$ be analytic function with $f(0)=0$,where $\mathbb{D}$ is the open disc $\{z \in \mathbb{C}:|z|<1 \}$ then $1.|f'(0)|=1$ $2.|f(\frac{1}{2})|\leq \frac{1}{2}$ ...
2
votes
1answer
68 views

Cauchy's Theorem and Cauchy's formula

I came across the following problem in our last midterm exam. I am completely stuck as to how to begin the solution: If $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$), then $|f|$ has no ...
2
votes
2answers
54 views

What happens if to postulate that complex numbers whose argument differs by $2 \pi$ are not equal?

What happens if to postulate that complex numbers whose argument differs by $2\pi$ are not equal? What properties such system will have? Will all analytic functions be entire?
0
votes
0answers
32 views

Why are the real part and imaginary part of normal distribution function independent?

As I said in title, why are the real part and imaginary part of normal distribution function independent? I need a detail derivation to proof it. Thank you.
0
votes
1answer
56 views

If $|z|<1$ , show that $\left|\frac{1}{2}\arg (\frac{1+z}{1-z}) \right| < \frac{\pi}{2}$

If $|z|<1$ , show that $\left|\frac{1}{2}\arg (\frac{1+z}{1-z}) \right| < \frac{\pi}{2}$
1
vote
1answer
41 views

Proof complex series

I have to prove this: $\displaystyle\sum_{n=1}^\infty n\alpha^n = \displaystyle\frac{\alpha}{(1-\alpha)^2}$ if $|\alpha | < 1$ I think this is a geometric series, and i have to solve it with a ...
2
votes
2answers
90 views

Sum of $\sum\limits_{n=1}^\infty q^n \sin(nx)$

How to find $\sum\limits_{n=1}^\infty q^n \sin(nx)$, where $|q|<1$ and $x \in \mathbb{R}$? I was thinking about rewriting it as $\sum\limits_{n=1}^\infty (q(\Im(\cos x+i\sin x)))^n$. It is a ...
1
vote
2answers
93 views

prove that there are no reals such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{x+y}$

I'm going through an introduction to complex analysis and there are two problems that I'm having problems with. A) Prove that there are no reals $x$ and $y$ such that $\displaystyle\frac{1}{x} + ...
1
vote
2answers
54 views

Need help solving this equation with complex numbers

$$(z^3 + 1)^3 = 1$$ where $z$ is an element of the complex number system. Can someone show me the most efficient way of finding all the solutions for $z$ here and also if possible please demonstrate ...
0
votes
4answers
40 views

Induction of logarithmic derivatives of complex functions?

I am trying to use induction to prove the logarithmic derivative of a complex function (called $P(Z)$ here). I define a function $P(z) = (z-z_1)(z-z_2)...(z-z_n)$ and then I want to use induction on ...
0
votes
2answers
61 views

How could I prove this trigonometric identity?

Show that: $$\left(\frac{1+\tan \theta}{1 - \tan \theta}\right)^n = \frac{1+i\tan n\theta}{1-i\tan n\theta}$$ Original image: http://i.stack.imgur.com/q8Yxj.jpg
0
votes
1answer
31 views

composition of complex functions

I'm given that $g(z)= \ln r+i\theta$ where $(r>0,0<\theta<2\pi)$ . I've already shown this function is analytic and that its derivative is $g'(z)=\frac{1}{z}$. Now it wants me to show ...
0
votes
2answers
46 views

Complex function mapping the unit circle onto an interval

Show that the function $f(z) = z^2 + z^{-2}$ maps the unit circle onto the interval $[-2, 2]$. Okay so far, doing previous questions I firstly try and find the inverse mapping. Here I considered the ...
1
vote
2answers
34 views

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined?

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined ? Okay. It is easily shown that something goes wrong if you define equality in ...
0
votes
1answer
33 views

how to solve this problem on complex analysis

in a probelm set I found $(x + \sqrt2j)(x − \sqrt2ij) =(x^2 + 2x + 2)$ where $j=√i$ but i can't understand how this happened. I have done this $(x + \sqrt2j)(x − ...
2
votes
1answer
33 views

How to find out the real part of this expression in term of k.

$$A=\frac{e^{i \pi k}-1}{e^{i \pi k h}-1}$$ Where $k=p+q$, $p$ and $q$ natural numbers $h\in]0,1[$ real number We can consider that the denominator is never $0$. The result may be someway like ...
0
votes
0answers
22 views

Providing an upper bound for a sum of complex numbers

Let $(\alpha_l)_{l=0}^{k-1}$ and $(\beta_l)_{l=0}^{m-1}$ be two sequences of complex numbers where $m>k$. It is known that $$ 0<\frac{1}{m}\sum_{l=0}^{m-1}|\beta_l|^2\leq A $$ where $A>0$. ...
2
votes
1answer
26 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 ...
1
vote
3answers
57 views

How to find out the real part of this.

I have to sum this: $$S:=\cos\left(\frac{\pi}{M+1}\right)+\dots+\cos\left(\frac{M\pi}{M+1}\right)$$ Where $M$ is a given natural number. I tried with this: Since $$e^{i a}=\cos a+i\sin a$$ and ...
0
votes
1answer
81 views

Finding where the Cauchy-Riemann Equations are satisfied

I have to find where the Cauchy-Reimann equations $\left( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \text{ and } \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial ...