The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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10 views

Use differential form to prove meromorphic function on compact riemann surface has same zeros and poles

I am reading mine's modular form note, proposition 1.12 states that the sum of residues of a differential form on compact Riemman surface is 0. Then he states that applies this to $df/f$, then we can ...
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2answers
34 views

Why are the zeros of $f$ isolated?

I'm reading Conway's complex analysis book and on page 79 he proved the following theorem: I think the theorem he is mentioning is a corollary which says that each zero of $f$ has a finite ...
2
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1answer
29 views

Proving that a complex-valued function has limit infinity at a finite point

Question: Show using the $\epsilon -\delta$ definition that $$\lim \limits_{z \to i} \frac{z-1}{z^2+1} = \infty$$ My attempt: We have to prove that for any $M \in \Bbb R^+$, $\vert ...
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0answers
29 views

Find a Mobius Transformation that carries the points $ -1, i, 1+i$ to the following:

My goal is to find a Mobius transformation that transforms $-1, i, 1+i$ onto the points a) $0, 2i, 1-i$ b) $i, \infty, 1$ For part a, I know that the Mobius transformation $M$ will be such that ...
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4answers
62 views

How to prove $\lim_{s \rightarrow \infty} \zeta(s) = 1$?

$\lim_{s \rightarrow \infty} \zeta(s) = 1$ I have seen a proof using the fact $1 \leq \zeta(s) \leq \frac{1}{1-2^{1-s}}$ but this relies on proving the inequality first which is quite cumbersome. I ...
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1answer
21 views

Calculate the vertical asymptote of the absolute of a complex rational function

I have a function with the following shape: $$ f(x) = \left|\frac{a_0 + a_1x+a_2x^2 + ... + a_nx^n}{b_0 + b_1x+b_2x^2 + ... + b_nx^n} \right| $$ The constant $b_0=1$ (I don know if it matters) When ...
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2answers
29 views

given $-\pi < \theta \leq \pi$ prove $f(z) = z^{1/3}$ is not entire.

I don't want the solution at all, but I'm incredibly stuck, and I really need some (hopefully not much) help. What I've considered: Liouville's Theorem Not applicable because f is not bounded. ...
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1answer
16 views

Show that $p = u \cdot (\zeta -1)^{p-1}$, where $u$ is an invertible element of $Z[\zeta]$ [on hold]

Show that $p = u \cdot (\zeta -1)^{p-1}$, where $u$ is an invertible element of $Z[\zeta]$. This outcome is the result of this link. So I think I have to use the previous result and the ...
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0answers
41 views

Why is that if $z^n = |z|^2$, then $|z| = 1$?

We have $z^{n-1} = \bar{z}\ \forall\ n > 2$ which gives us $z^n = |z|^2$, but I dont see why that means $|z| = 1$?
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1answer
28 views

Examples of holomorphic, complex differentiable, always positive functions

I am looking for classes of functions which are: 1) holomorphic 2) |f(z)|>0 for all z 3) complex differentiable (i.e. f(z)=mod(z) is not valid) ...
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1answer
23 views

solve and skecth $\log{|z|}=-2\arg(z)$

Ive asked this question a week ago, but nobody managed to answer but it is doing my heading from then. I know usually You demand some initial work done on the question but I just dont know how to ...
1
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1answer
32 views

Proving that $h(z)=\overline{h(\overline{z})}$ for all $z \in \mathbb{C}$ assuming that $h$ is holomorphic and the real line maps itself.

I have trouble proving that $h(z)=\overline{h(\overline{z})}$ for all $z \in \mathbb{C}$ under the assumption that $h$ is holomorphic and the real line maps itself or in other words: $h(z) \in ...
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3answers
18 views

Finding disc of convergence

Find the disc of convergence $$\sum_{n=0}^\infty z^{n^{3}}$$ I have applied the ratio test but I can not seem to come up with a conclusion.
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2answers
23 views

A function that satisfies $|f(z)-\bar z|<0.9$ is not analytic in the unit circle.

I've came accros this excersize: Suppose that $D=\{z:|z| \le 1\}\subset \mathbb C$ and $$f:D\rightarrow\mathbb C$$ suppose that for every $z\in D$ such that $|z|<1$ $$|f(z)-\bar z|<0.9$$ where ...
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4answers
39 views

Prove this integral is analytic

Let $\phi$ be a continuous (complex valued) function on the real interval [−1, 1] inside C, and define $$f(z)=\int_{-1}^1\frac{\phi(t)}{t-z}dt$$ Show that f is analytic on C less the interval [−1, ...
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1answer
16 views

Find a power series centered at the origin that satisfies the Bessel

Find a power series centered at the origin that satisfies the Bessel differential equation $$zf''(z)+f'(z)+zf(z)=0$$ with initial condition $f(0)=1$. Show that this series converges for all z in C. I ...
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1answer
20 views

Using the Maximum Modulus Principle to prove that every holomorphic function on a compact Riemann surface is constant

I have read in a number of sources (including here) that a holomorphic function on a compact Riemann surface must be constant. The reason given has always been the Maximum Modulus principle, but ...
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2answers
87 views

Integral of $\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$

I'm having trouble integrating $$\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$$ where $\alpha$ is a real number and $i = \sqrt{-1}$. I'm guessing that I ...
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1answer
38 views

Complex Continuity [on hold]

Is the function $f$, defined by $$ f(z) = \begin{cases} \frac{z^2+2iz-1}{2z^2+iz+1} & \text{ if } z \not \in \{-i\}\\ 0 & \text{ if } z = -i \end{cases}$$ continuous at $−i$? Explain your ...
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1answer
38 views

Prove Complex Limits from first principle definition [on hold]

Show from first principles, that is using the definition of limit, that $$\lim_{z\to i}\frac{z-1}{z^2+1} = \infty$$ Please can someone actually show me the procedure, struggling to understand it ...
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0answers
22 views

Prove equivalence between two Bessel functions relations

Given the following equation $$\frac{J_{n - 1} (u)}{uJ_n (u)} - \frac{K_{n-1}(w)}{wK_n(w)} = 0$$ (where $J$ is the Bessel function of the first kind, $K$ is the modified Bessel function of the ...
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2answers
59 views

Does the function $z+\frac{1}{z}$, $z\in \mathbb{C}$, have real-world applications?

The function $$z+\frac{1}{z}$$ seems to play a role in complex analysis. However, does it have any applications besides what can be deduced in pure mathematics?
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0answers
26 views

Complex exponentials [duplicate]

How do I solve: $$ e^{4z}+e^{3z}+e^{2z}+e^z+1=0 $$ I'm getting lost on where to start. I tried using the definition $$ e^z=e^x(\cos(y) +i\sin(y)) $$ But that doesn't seem to do me any good. I also ...
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0answers
28 views

Show that $\lim_{z\to z_0} cf(z)=ac$ (where $c$ is a complex number)

How do I prove this? Suppose that $a, b$ and $c$ belong to $\mathbb C$ and that $$\lim_{z\to z_0} f(z)=a$$ and $$\lim_{z\to z_0} g(z)=b.$$ a - $\lim_{z\to z_0} cf(z)=ac$ (where $c$ is a ...
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1answer
20 views

Winding number, Conway text

I have a question about this statement in the Cauchy's Integral Formula in Conway text. In the Integral formula, it states that " Let $G$ be an open subset of the plane... If $\gamma$ is a closed ...
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1answer
82 views

Solve $e^{4z} +e^{3z} + e^{2z} + e^z + 1 = 0$.

Solve $$e^{4z} +e^{3z} + e^{2z} + e^z + 1 = 0.$$ I have attempted this problem with the usual definition by writing $z=x+iy$ and using $e^z = e^x(\cos y + i \sin y)$ but got a large unsolvable mess. ...
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1answer
36 views

Are there complex numbers whose sines are zero?

I recently learned that $\sin(z)$ has an extension into the complex plane, namely: $$\frac{e^{iz}-e^{-iz}}{2i}$$ Is there any complex number $z=a+bi$, with $b≠0$ such that $\sin(z)=0$ ? I am ...
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1answer
13 views

Prove that: $n^2+3n^3 + 6^{lgn} is $ $\theta(n^3)$

I'm asked to prove that: $n^2+3n^3 + 6^{lgn} is $ $\theta(n^3)$ I know that for Big O, I need to show: $f(n) <= c*g(n)$ But I'm not sure how to show this, since it involves theta. Any help would ...
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1answer
35 views

How to bound this complex number from below?

I am doing an $\epsilon-\delta$ proof ($z \rightarrow i, f(z) \rightarrow \infty$) and currently have the absolute value $$|f(z)|=\left|\frac{z-1}{z^2+1}\right|$$ and I wish to make a statement about ...
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0answers
25 views

how to prove this function has zeros interlacing and including those of Riemann zeta

Let $\chi (t) = \dfrac{4 i \pi \zeta (t) \left( \left( \ddot{\Psi} \left( \frac{t}{2} \right) - \ddot{\Psi} \left( \frac{1}{2} - \frac{t}{2} \right) \right) \zeta (t)^3 - 48 \zeta (t) \dot{\zeta} (t) ...
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1answer
38 views

Zeroes of Complex Cosine

Find the zeroes of $\cos z=2$. Attempt: $\cos z=\cos(x+iy)=\cos(x)\cos(iy)-\sin(x)\sin(iy)=\cos(x)\cosh(y)+\sin(x)\sinh(y)=2$ I don't know how to proceed form here...
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0answers
23 views

what is $\int_{\gamma}\frac{2}{(z+2)^2}dz$ with $\gamma(t)=t+it\sin(\frac{\pi}{t})$ for $t>0$?

Again a question about integration. Consider the integral $$\int_{\gamma}\frac{2}{(z+2)^2}dz,$$where $\gamma:[0,1]\to\mathbb{R}$ such that $\gamma(0)=0$ and $\gamma(t)=t+it\sin(\frac{\pi}{t})$ if ...
2
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0answers
58 views

Help with finding the numerical average of $x^x$ from $(-4,-2)$.

I wanted to find the approximate average of all defined points in $(x)^{x}$ from $[-4,-2]$ To first solve this I found the following defined sets when $x<0$. ...
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2answers
25 views

Conformal mapping $z+\frac{1}{z}$, how to see the mapping to hyperbolas?

http://www.webassign.net/zillengmath4/20.2.pdf p.2. The conformal map $z+\frac{1}{z}$ maps circles $|z|=r$ to ellipses and $arg(z)=\theta$ to hyperbolas. I believe one can display both using the ...
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1answer
54 views

Is $\int_{|z|=2}\frac{z}{(z-3)^2}dz=0?$

I have a question. What is $$\int_{|z|=2}\frac{z}{(z-3)^2}dz?$$ In my optinion it must be zero, because the singularity $3$ is outside $\{z\in\mathbb{C}:|z|<2\}$, is it correct? Regards
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2answers
39 views

Complex differentiability and differentiability in R2

In $\mathbb R$ for a derivative to exist (or a limit generally) it is necessary that the limit be the same in both directions (from below and above) and this is the same in $\mathbb C$ where for a ...
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2answers
30 views

How do I evaluate this sum :$\sum_{n=1}^{\infty}\frac{{(-1)}^{n²}}{{(i\pi)}^{n}}$?

I'm interesting to know how do i evaluate this sum :$$\sum_{n=1}^{\infty}\frac{{(-1)}^{n²}}{{(i\pi)}^{n}}$$, I have tried to evaluate it using two partial sum for odd integer $n$ and even integer $n$ ...
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1answer
36 views

page $102$ from Ahlfors.

He talks about a function $f(a)$ for which all the derivatives vanish. He shows inside a circle within our domain $\Omega$, for any circle $C$ we take, there $f$ is identicaly zero. Then he shows ...
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1answer
32 views

Complex numbers as linear operators?

If it is valid to interpret multiplication by a complex number as a dilative rotation, does that mean that it can be viewed as a function $$f: R^2 \rightarrow R^2$$ making it a linear operator?
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22 views

What general mobius transformation maps $|z-1|=1$ to itself and $|z+1|=1$ to $|w-3|=3$.

What is the most general mobius transformation that maps $|z-1|=1$ to itself and $|z+1|=1$ to $|w-3|=3$. I want to find the most general form of such a linear transformation, I'll denote it $T$. ...
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0answers
27 views

Find maximum value or upper bound of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$ [duplicate]

If $|z_{1}|=2,|z_{2}|=3,|z_{3}|=4$,then find maximum value of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$. My attempt:I considered 3 circles having centre origin and radii as $2,3,4$. Then I ...
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4answers
139 views

How many values does $\sqrt{\sqrt{i}}$ have?

Wolfram says, there are only two roots, but $\sqrt{i}$ already gives two roots. So if we express them in Cartesian form we can take square roots of them separately and end up with four roots. ...
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0answers
37 views

Is there any generalization of Riemann Mapping theorem?

Given any two regions in complex plane when can we say they are conformally equivalent? I mean does there exists some "complex-geometric" invariant which determines whether two regions are conformally ...
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1answer
30 views

Disc of convergence involving logs

Find the disc of convergence: $$\sum_{n=2}^\infty \frac{z^{n}}{n(log(n))^p};(p>0)$$ I have tried geometic series, ratio test, root test... What would be your thought on the best test to use?
6
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2answers
130 views

Computation of an iterated integral

I want to prove $$\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\frac{\sin(x^2+y^2)}{x^2+y^2}dxdy=\frac{\pi^2}{2}.$$ Since the function $(x,y)\mapsto\sin(x^2+y^2)/(x^2+y^2)$ is not ...
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1answer
51 views

What function can turn $z=x+iy$ into something involving $xy$?

What function can turn $z=x+iy$ into something involving $xy$? What function takes the real parts of $z$ and then multiplies them? Or would I perhaps need to consider the point $(x,iy)$, rather than ...
0
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1answer
51 views

Why is it valid to set $r=e^t$ in $f(r)=\frac{r+r^{-1}}{2}$?

$f(r)=\frac{r+r^{-1}}{2}$ $f(re^{i \theta})=\frac{re^{i\theta}+r^{-1}e^{-i\theta}}{2}=\frac{r+r^{-1}}{2}\cos\theta+i\frac{r-r^{-1}}{2}\sin\theta$ Why is it then valid to set $r=e^t$, $-\infty≤t≤0$ ...
0
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2answers
39 views

Continuity of a function with complex variables

How could I show if or not the following piece-wise defined function is continuous at the point $z=-i$? $$f(z)=\left\{ \begin{matrix} \frac{z^2+2iz-1}{2z^2+iz+1}, & z \neq -i \\ 0, & z=-i ...
3
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2answers
46 views

Sketch the set of points satysfing an inequality $|z+1|+|z-1|\leq2$

The inequality is $$|z-1|+|z+1|\leq2$$ I used a triangle inequality to show that Since triangle inequality states: $$|z+w|\leq|z|+|w|$$ Then $$|z-1+z+1|\leq|z-1|+|z+1|\leq2$$ So $$|2z|\leq2$$ From ...
0
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1answer
22 views

Show that $\Sigma_{j=0}^n z^j=\frac{1-z^{n+1}}{1-z}$ [duplicate]

As in the question I have to show that $$\sum_{j=0}^n z^j=\frac{1-z^{n+1}}{1-z}$$ So if we suppose that the above is true then clearly ...