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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1answer
88 views

Showing $f=0$ almost everywhere

Let $\psi_n(x)=e^{-x^2/2}P_n(x)$ where $P_n$ is a degree $n$ polynomial with real coefficients. Assume that $$\int_{\mathbb{R}}e^{-x^2/2}P_n=0.$$ Suppose that for any $f\in L^2$, such that ...
4
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1answer
66 views

$f$ has simple pole at origin. Compare $\rm Res (f,0)$ with $\rm Res (f(1/f), 0 )$

I'm preparing for a qualifying exam (instead of working assignments I have due now) and I have a small question about a problem that was on a past qualifying exam: Suppose $f$ has a simple pole at ...
2
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2answers
95 views

Uniform convergence of complex series with $|z|=1$ but $z\neq 1$.

Prove that $\sum\limits_{k=0}^\infty\frac{z^k}{k+1}$ converges where $|z|=1$ but $z\neq 1$. This gives an example of a power series with radius of convergence 1 that converges at every point of the ...
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0answers
42 views

Show that $\frac{f(z)}{z }$ does not have a limit as z goes to 0

let $f(z)=\frac{z^5}{|z^4|}$ if $z \neq 0$ and $f(0)=0$. Show that $\frac{f(z)}{z}$ does not have a limit as $z\to 0$. I'm not sure how to start on this one.
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2answers
141 views

There aren't non-holomorphic polynomials, right?

Full disclosure: I'm taking my first complex analysis course as a graduate student and the title of my question looks like a dumb question to me. In any case, there's a problem in my book that deals ...
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2answers
40 views

Finding a function $u$

Let $u$ satisfies $u_{xx}+u_{yy}=0$ in the region $x^2+y^2<9$ and the boundary condition $u=\cos^2\theta$ at $r=3$. Find $u$. Can someone show me how to proceed with this?
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1answer
52 views

Can a complex function be holomorphic at a point whose limit approaches infinity?

Let $f: \Omega \rightarrow \mathbb{C}$ be a complex function, $z_0 \in \Omega$. If $\lim_{h \rightarrow 0}\frac{f(z_0+h)-f(z_0)}{h} = \infty$, is $f$ considered holomorphic at $z_0$? I would think ...
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2answers
132 views

Meaning of principal value integrals

Consider the integral $I = [PV]\int_{-\infty}^{\infty} \frac{exp(iax)}{x} dx$ where $a$ real and positive, and $[PV] $ denotes 'the principal value of'. Using a semicirle contour in the upper half ...
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2answers
68 views

Find an analytic function that maps the disk $\{|z|<1\}$ onto the disk $\{|w-1|<1\}$ so that $w(0)=1/2$ and $w(1)=0$

Find an analytic function that maps the disk $\{|z|<1\}$ onto the disk $\{|w-1|<1\}$ so that $w(0)=1/2$ and $w(1)=0$ The 3 points theorem: Given 3 point $z_1, z_2, z_3 $ always map ...
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2answers
89 views

Evaluating, $\sum_{n=1}^{\infty} n$ [duplicate]

Evaluating: $$S = \sum_{n=1}^{\infty} n$$ Apparently, $S = \zeta(-1) = -\frac{1}{12}$ Which is crazy. How can $1 + 2 < 1$ ? Anyway, How do you evaluate $S = \zeta(-1)$ Hint please? Thanks
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1answer
21 views

What is $f(D)$ if $f(z)=\frac{z+1}{z-1}$ and $D=\{z\in\mathbb C\mid \bar z=-z \}$

I have $$\mathcal D=\{z\in\mathbb C\mid \bar z=-z \}$$ and \begin{align*}f:\mathbb C\setminus \{1\}&\longrightarrow \mathbb C\setminus \{1\}\\ z&\longmapsto \frac{z+1}{z-1}.\end{align*} I ...
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1answer
74 views

Show that $\log(\sin(x)) = -\log(2) + …$

Show that: $$\log(\sin(x)) = -\log(2) - \sum_{n \ge 1} \frac{\cos(2nx)}{n}$$ $$\sin(x) = \frac{e^{ix} - e^{-ix}}{2i}$$ $$\log(\sin(x)) = \log(e^{ix} - e^{-ix}) - \log(2i)$$ $$= \log(e^{ix} - ...
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1answer
63 views

Notation (Complex Analysis) function

Does this notation $${\Large \Im f(z)}$$ mean the imaginary part of a complex function?
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1answer
27 views

Proving that $g(z)$ is bounded in a neighborhood $N(c,\delta)$

I need help understanding a proof that appears in "Complex Analysis" by John M. Howie. The theorem Suppose that $f$ has a simple pole at $c$, with residue $\rho$, and let $\gamma^*$ be a circular ...
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2answers
75 views

Convergent power series R=1

Give an example of a power series with $R=1$ that converges uniformly for $|z|\leq 1$, but such that its derived series converges nowhere for $|z|=1$. I have tried many different series but can't ...
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0answers
54 views

Finding a Linear fractional Transformation that maps $A$ to $B$

I wonder if anyone has an idea how to construct a Linear fractional transformation that maps $A=\{z\in \mathbb{C}: |z-i|\geq 1\}$ to the unit disc $B=\{z\in \mathbb{C}: |z|\leq 1\}$. It is not ...
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2answers
44 views

Showing $(1 + z\omega)(1 + \overline{z\omega}) \leq (1 + |z|^2)(1 + |\omega|^2)$.

Consider the following inequality : $(1 + z\omega)(1 + \overline{z\omega}) \leq (1 + |z|^2)(1 + |\omega|^2)$. I was reading an article which uses this inequality, and the reading said this was an ...
2
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1answer
49 views

Is this complex set open and connected

Is the complex set $\operatorname{Re}(z) > 0$, $|z-2|>3$ an open and connected set?
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1answer
510 views

Chordal Metric - Showing it is in fact a metric

If I have $f(z_{1},z_{2}) = \displaystyle\frac{|z_{1} - z_{2}|}{\sqrt{1+ |z_{1}|^2} \cdot \sqrt{1 + |z_{2}|^2}}$, for $z_{1}, z_{2} \in \mathbb{C}$, how would I show that $f(z_{1},z_{2})$ is a ...
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0answers
42 views

Laurent series for f(z)

I need to find Laurent series for $f(z)$ at $z=-\frac{\pi}{2}$ where $$ f(z)=\frac{z^{\frac{1}{2}}}{1+\sin z} $$ I defined $z=-\frac{\pi}{2}+t$ and did the expansion of the denominator around $t=0$. ...
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1answer
50 views

A complex integral.

Evaluate $$\int_C \frac{e^z+\sin{z}}{z}dz$$ where $C$ is the circle $|z|=5$ traversed once in the counterclockwise direction. I can't find an antiderivative to this function, and I am not sure one ...
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233 views

Example of an analytic continuation for a function in integral form

Given $f(z) = \int_{-\infty}^\infty \frac{exp(-t^2)}{z-t}\,dt$, where $Im(z)>0$. Find an analytic continuation to the region $Im(z)<0$. Firstly the solution said that there is a branch cut on ...
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3answers
162 views

Describe the image of the set $\{z:|z|<1, Im(z)>0\}$ under the mapping $w =\frac{2z-i}{2+iz}$

Describe the image of the set $\{z:|z|<1, Im(z)>0\}$ under the mapping $w =\frac{2z-i}{2+iz}$ First I need to find the inverse which is $z=\frac{2w+i}{2-iw}$. Now let $w=u+iv$, we have ...
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2answers
110 views

Describe the image of the set $\{z=x+iy:x>0,y>0\}$ under the mapping $w=\frac{z-i}{z+i}$

Describe the image of the set $\{z=x+iy:x>0,y>0\}$ under the mapping $w=\frac{z-i}{z+i}$ So from this mapping , I can see that $a=1, b=-i, c=1, d=i$ thus $ad-bc=i+i=2i \not =0$ so this is a ...
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0answers
52 views

Cauchy Riemann Eqn

I need a bit of help as I'm stuck on this problem...this maybe simple..but.. I have to find all z ∈ C such that the function f(z) = z cos(z(bar)) satisfies the Cauchy-Riemann eqns. So far I have ...
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2answers
68 views

Prove identity related to nths root of unity

If $1=z_0,z_1,...,z_{n-1}$ are nth roots of unity, prove that $$(z-z_1)(z-z_2)...(z-z_{n-1})=1+z+z^2+...+z^{n-1}$$ I don't know what is meant by the condition given. If I substitute ...
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1answer
79 views

Does analytic continuation apply only to analytic functions?

I'm a high school senior attempting to do a project on the riemann zeta function. I've looked online, tried reading college textbooks but still don't have a completely clear idea of analytic ...
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0answers
49 views

analytic continuation on boundary of a power series

I've having some confusion. I think this statement is true, Suppose some power series is given say $P(z)$ in some neighborhood of $0$ whose radius of convergence is $R$ and then prove $P$ can't be ...
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2answers
49 views

An application of Liouville's theorem

I saw the following two theorems: 1) Let $f$ be an entire function such that $|f(z)|\leq k|z|^n$ for some $k>0$ and large $z$. Then $f$ is a polynomial of degree at most $n$. 2) Let $f$ be an ...
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1answer
45 views

Montel's Theorem WLOG statement: why the domain can be assumed to be the unit disk

I need help understanding the WLOG statement in a text I am working through. Here is the theorem as stated up until the statement that I am confused by: Theorem (Montel's Theorem): A family ...
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2answers
170 views

Residue theorem: When a singularity gives infinite to the residue

What if one of the singularity gives infinity to the residue. Consider this contour; $$X=\int_{\gamma} e^{i(\frac{z^{2}+1}{2z})}\frac{{(z^{2}-1)}^4}{2z^2(z-i)^{3}(z+i)^{3}}dz$$ I have ...
0
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1answer
46 views

Prove $|a-b| \ge \left||a|-|b|\right|$ using the triangle inequality

So I feel I'm right at the end of this proof... I just can't make the final step which is killing me. I'm given $b=z_2$ and $a=z_1+z_2$ and must prove $|a-b|\ge \left||a|-|b|\right|$ And I've ...
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3answers
101 views

Why is the series $\sum_n\frac{1}{n^x}$ not uniformly convergent on $x\in(1, \infty)$

I've been struggling with this problem for the past 5 days. I've tried to prove the series $\sum_{n=1}^\infty\frac{1}{n^x}$ is not uniformly convergent when $x$ belongs to $(1,\infty)$ but to no ...
0
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1answer
51 views

complex analysis exponential series evaluation

Evaluating the series $$f=\sum \frac{27^n}{(3n+1)!}$$ I could simplify this to $$\frac {1}{3}\sum \frac{3^{3n+1}}{(3n+1)!}$$ and use the chart $$\sum \frac{x^{an+b}}{(an+b)!}$$ to evaluate $f$. But ...
4
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1answer
285 views

Branch points of the Lambert W function

Let $W_{k}(z)$ be the kth branch of the Lambert W function. My question pertains to the branch point that the principal branch $W_{0}(z)$ shares with $W_{-1}(z)$ and $W_{1}(z)$ at $z = - ...
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2answers
118 views

$|\cos z|^2 = \cos^2 x + \sinh^2 y$

I am trying to prove the identity $|\cos z|^2 = \cos^2 x + \sinh^2 y$, where $z=x+iy$. I know $|\cos z|^2=(\cos z)(\cos \bar{z})$, since cosine is analytic for all $z \in \mathbb{C}$. Thus, $$|\cos ...
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1answer
14 views

Finding a certain residue

I wish to calculate the value of \begin{equation} \text{Res}\big((z+\pi/4)^2\tan(z);z=\pi/2\big)=\frac{1}{2\pi i}\int_C (z+\pi/4)^2\tan(z)\text{d}z, \end{equation} where $C$ is any counter-clockwise ...
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1answer
39 views

Finding a complex function from one open set to another

I am having problem with one specific type of sets, in the type of questions as the following: Find an analytic 1-1 and onto function from one open set to another, where the open set is missing a ...
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1answer
44 views

Extension of holomorphic function

Let $\Omega$ be a simply-connected domain in $\mathbb{C}$, and $A$ be a closed simply-connected set such that $A\subset \Omega$ and $\Omega\setminus A$ is connected. Let $f$ be a holomorphic function ...
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4answers
82 views

If $z=i^{2i}$ then $|z| =?$

I am given a complex number $z=i^{2i}$ so what would be $|z|$? Firstly I took $\log$ $$ \log(z)=2i\log i\\ $$ thus $$ \log(z)=2i\log(e^{-π/2}) = -i\pi; $$ then inverting the log $$ z=e^{ -i\pi} $$ ...
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1answer
91 views

Counterexample to Schwarz Lemma

Schwarz Lemma states the following: Let $D = \{z : |z| < 1\}$ be the open unit disk in the complex plane centered at the origin and let $f : D \to D$ be a holomorphic map such that $f(0) = 0$. ...
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2answers
50 views

Complex series where ratio test is not applicable, even though all coefficients are non-zero

I would like to see an example of a power series $\sum\limits_{n=0}^{\infty} a_nz^n$, ($a_i\in\mathbb{C}$), which satisfies: 1) the series has positive (and finite) radius of convergence. 2) ...
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1answer
39 views

Proof Janusz Algebraic number fields, convergence of Dirichlet Series.

The book Algebraic number fields, Janusz Please, Could you explain the proof of the part b) a little more? Thank you all.
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1answer
65 views

A question from Stein-Shakarchi- Complex Analysis

A simple exercise in the book (mentioned in title) is the following: Let $f\colon \Omega\rightarrow \mathbb{C}$ be holomorphic, where $\Omega\subseteq \mathbb{C}$ is open. If real part of $f$ is ...
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0answers
59 views

Show that an integral is analytic

Could somebody please help my with this problem, I am not sure how to do it, I never integrate function of two variables before et $U$ denote an open set in $\mathbb{C}$. Let $f :\thinspace U \to ...
0
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2answers
60 views

Prove that $\oint _{|z|=R} (f-g)' dz = 0$ (Residue Theorem)

I know that $f$ and $g$ have a pole or order $k$ in $z=0$. $f-g$ is holomorph in $\infty$. I need to prove that: $$\oint_{|z|=R} (f-g)' dz = 0$$ Any help? Note: $f$ and $g$ only have a singularity ...
2
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1answer
320 views

Residue theorem:When a singularity on the circle (not inside the circle)

This is the integration I am trying to solve $$\int_{0}^{\pi} \sin^{2}(\theta)\sec^{3}(\theta)d\theta$$ putting $$z=e^{i\theta}$$ $$\int_{\gamma} \frac{-2{(z^{2}-1)}^2}{i(z-i)^{3}(z+i)^{3}}d\theta$$ ...
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2answers
80 views

Image of circles under $f(z) = \frac{z}{(z-1)^2}$?

What is the image of circles $ |z| < 1$ under complex function $f(z) = \frac{z}{(z-1)^2}$? $$f(z) = \frac{z}{z^2 + 1 - 2z} = \frac{1}{z+\frac{1}{z}-2}$$ $z+1/z$ is an ellipse and $z+1/z - 2$ is ...
10
votes
1answer
189 views

Proving that a family of functions limits to the Dirac delta.

For each $\epsilon > 0$, define $f_\epsilon:\mathbb R\to \mathbb R$ as follows: \begin{align} f_\epsilon(k) = \frac{1}{\pi}\frac{\epsilon}{\epsilon^2+k^2}. \end{align} How does one rigorously ...
3
votes
2answers
96 views

Definite integral (in the complex plane?)

I want to prove that $$\int_{0}^{\infty} \frac{dx}{1+x^b} = \frac{\pi}{b \sin(\pi/b)} \ ,$$ where $b\in (1,\infty)$. I thought about doing it in the complex plane since the integrand is a ...