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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38 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
38 views
+50

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} ...
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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
24 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
66 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 of $x^x$ when $x<0$. ...
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2answers
26 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
55 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
0
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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 ...
2
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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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1answer
33 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 ...
6
votes
4answers
142 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. ...
2
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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
31 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
votes
2answers
132 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 ...
0
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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 ...
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1answer
55 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 $r=e^t$, $-\infty≤t≤0$ ...
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2answers
40 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 ...
0
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1answer
39 views

Give the power series expansion of $\log z$ about $i$

I'm reading Conway's complex analysis book and I'm trying to solve the exercise 5 from page 74. In this exercise the author asks for the radius of convergence and power series expansion of $\log z$ ...
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0answers
29 views

Inequality on the unit circle-part 2

This is the follow up question to my earlier one Inequality on the unit circle . It seems to be $$\left|np(z)+(\alpha-1)zp'(z)\right| \\\leq\left|np(z)+(\alpha-z)p'(z)\right|$$ on $|z|=1,$ for ...
1
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1answer
26 views

Inverse of a Linear transformation in complex analysis

I am given $T z= \frac{z+2}{z+3}$ , where $T_1$ is a linear transformation on the complex number $z$. I need to find its inverse $T^{-1}z$. I considered a complex number $w =\frac{z+2}{z+3} $. This ...
2
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3answers
35 views

Compute the integrals using the residue theorem

Compute the following integrals: $I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$ $J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$ I do not know where to begin. I know I am supposed to use the substitution ...
0
votes
1answer
34 views

Closed sets and accumulation points

In complex analysis how to prove that if $S$ is closed in $\mathbb{C}$ then it contain all of its accumulation points. If $S$ is closed then $S$ contain all its boundary points.(If $z_{0} $ is a ...
0
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1answer
30 views

Finding an Entire function with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$

I am really stuck on a homework problem, which boils down to the following: We need to exhibit an entire function $f$ with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$. The only sorts of functions ...
1
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0answers
30 views

How to approach solving this Fourier series [on hold]

$$f(x):=\frac{1}{e^{2+\cos x}-1}$$ Source. Hi. I need to find Fourier series for this function. This is even function so Fourier coefficient $b_n$ is 0. Basically I need to solve this integral ...
0
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0answers
21 views

Showing a function has a global primitive on the unit disk minus the origin

I've reached a dead end for a problem with proving there is a global primitive for a continuous function on the unit disk D minus the origin with the condition that $\lim_{z\rightarrow 0}{zf(z)}=0$. ...
0
votes
1answer
34 views

Finding $\sqrt{(1 + \sqrt{3i})}$

Find $\sqrt{(1 + \sqrt{3i})}$. I am trying to use the fact that $\sqrt{(1 + \sqrt{3i})} = re^{i \theta} = r(cos \theta + i sin\theta)$ but I am having trouble figuring out where to go from here. Any ...
0
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0answers
9 views

Regarding Smirnov domains

Suppose $G$ is a Smirnov domain that contains infinity in the plane (we can think of it as the exterior to a closed Jordan curve) and $\phi$ is a conformal mapping from $\mathbb{D}$ onto $G$. Can we ...
1
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2answers
46 views

If $f$ is analytic in $D$ and $|f(z)|<M$ everywhere on $|z|=1$, show for all $z:|z|<1$, $|f(z)| \leq M |\frac{z-a}{\bar a z - 1}|$

Suppose $f(a)=0$ for some $|a|<1$. If $f$ is analytic in $D$ containing the unit disk and $|f(z)|<M$ finite for all $z:|z|=1$, show for all $z:|z|<1$, that $$ |f(z)| \leq M ...
1
vote
0answers
27 views

Prove for $z$ in the unit disc (real/complex analysis)

Prove for $z$ in the unit disc $$1+\binom{k+1}{1}z+\binom{k+2}{2}z^2+\cdots+\binom{k+n}{n}z^n+\cdots=\frac{1}{(1-z)^{k+1}}\quad(k=0,1,2,\ldots)$$where the coefficients on the left are the binomial ...
0
votes
2answers
26 views

Show that the series is not absolutely convergent but is uniformly convergent in the whole complex plane

Show that the series is not absolutely convergent but is uniformly convergent in the whole complex plane. ...
0
votes
1answer
17 views

Help finding a second homogeneous polynomial of degree 5 that are also harmonic

Essentially I have to find 2 homogeneous polynomial of degree 5 that are also harmonic. Knowing z=(x+iy) is analytic I found my first polynomial to be f(z)=z^5 and that multiples of this would ...
0
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2answers
42 views

What's the relationship between hyperbola, hyperbolic functions and the exponential function?

The hyperbolic functions can be expressed using the exponential function. However how are these related to "hyperbolas"?
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0answers
17 views

If $b \in (-\infty, \infty)$ in $z=a+bi$, then how to mark the range of $z$?

Let $a$ be fixed. If $b \in (-\infty, \infty)$ in $z=a+bi$, then how to mark the range of $z$?
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0answers
26 views

How to classify harmonic functions on the punctured disk without the Schwartz reflection principle?

I am working through old qual problems at the University of Minnesota and am trying to find an alternate solution to the following problem. Determine all continuous functions on $\{ z : 0 < \left| ...
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0answers
19 views

Find the disc of convergence of the power series (real/complex analysis)

Find the disc of convergence: $$\sum_{n=1}^\infty \frac{(z-3)^{n}}{n(n+1}$$ I have tried geometic series, ratio test, root test... but I seem to get stuck each time. What would be your thought on ...
2
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0answers
63 views
+50

Complex Root of Unity Analogue of Forward Difference Operator

In my studies I have come across a couple of operators; in particular; $$\Delta[f(x)]=f(x+1)-f(x)$$ $$\Delta^*[f(x)]=f(x+1)+f(x)$$ $\Delta$ has been called the Forward Difference Operator. I was ...
2
votes
5answers
77 views

How do I solve the following equation: $z^4+z^3+z+1=0$

Is there an existing method to solve the following equation: $z^4+z^3+z+1=0$?
1
vote
1answer
22 views

$\frac{1}{2}(z+\frac{1}{z})$ range is $[-1,1]$, when $|z|=1$?

The range of $$\frac{1}{2} \left(z+\frac{1}{z}\right), \quad z \in \mathbb{C}$$ should be $[-1,1]$, when $|z|=1$? Any idea how to see it? I tried de Moivre (since it has the $|z|$ term), but it ...
0
votes
2answers
32 views

Cantor Intersection Theorem extension [duplicate]

Task at hand: Show that in the Cantor Intersection Theorem, "compact" cannot be replased by "closed"; that is, find a nested sequence $\{F_n\}_{n=1}^\infty$ of nonempty closed sets in C such that ...
0
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0answers
30 views

Find the set of all accumulation points for the following set

Find the set of all accumulation points for the following set. $$\left\{\frac{1}{m}+\frac{i}{n}; m,n\in N\right\}$$ I am trying to partition the set in a way to find the points but I can not seem to ...
2
votes
3answers
44 views

Image of a family of circles under $w = 1/z$

Given the family of circles $x^{2}+y^{2} = ax$, where $a \in \mathbb{R}$, I need to find the image under the transformation $w = 1/z$. I was given the hint to rewrite the equation first in terms of ...
0
votes
0answers
29 views

Existence of a curve with index 1 around a compact set

Let $K \subset \mathbb{C}$ be compact. If $U$ is an open set containing $K$, I want to show that there exists a collection of (piecewise $C^1$) curves $\gamma_1...\gamma_n$ such that 1) For $ x \in ...
1
vote
1answer
31 views

If $(z_{n}) \in \mathbb{C}$, $z_{n} \to \infty$ as $n \to \infty$, what happens to $|z_{n}|$, $Re(z_{n})$, $Im(z_{n})$, $Arg(z_{n})$?

Suppose the sequence $(z_{n}) \in \mathbb{C}$ converges to infinity as $n \to \infty$. I need to determine what this implies about $|z_{n}|$, $Re(z_{n})$, $Im(z_{n})$, $Arg(z_{n})$. I know that a ...
1
vote
0answers
28 views

$(2+z^2) f(z) + 3 z + 4 = f(z+1)$?

Consider the equation $(2+z^2) f(z) + 3 z + 4 = f(z+1)$ Such that $f(z)$ is analytic near the positive real axis and the functional equation holds for real $z>0$. Can we express Some solutions ...
0
votes
0answers
15 views

Fourier transform of windowed complex exponential

I have a function on the form $$f(x) = g'(x)*e^{i\pi g(x)}.$$ Where $g'(x)$ is a window function with support in the range $-R \ldots R$. I want to find the fourier transform $\mathcal F(\omega)$ ...