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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2
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1answer
20 views

Dirichlet character modulo p

How can I prove that if $\chi$ is a non-principal character modulo $p$ prime, then $\chi (-1) = \overline{\chi} (-1)= \pm 1$ and $\sum_{x=1}^p \chi (x) e^{2\pi i x}=0$? For the first question, I just ...
0
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2answers
27 views

Continuously extended holomorphic function on the unit disc.

Let $f$ be continuous on $\bar{\mathbb{D}}$ and holomoprhic on $\mathbb{D}$. How can we show that $$\int_{\partial \mathbb{D}}f(z)dz=0$$?
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2answers
16 views

Question about convergence in complex numbers field

It may be a simple question, but if we want to show that $(z_n)\subset\mathbb{C}$ is convergent to $z\in\mathbb{C}$ then we should just check that absolute value of $z_n$ is convergent to absolute ...
0
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1answer
35 views

Symmetry in complex plane

In a book I am reading, symmetry about a curve in complex plane is defined as follows: Let $F(x,y)=0$ be a simple curve. Then points $z, z_0$ are symmetric about this curve iff $ F \left( ...
0
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1answer
37 views

Convergence of $\sum_{n=1}^{\infty} \frac{1}{n^z}$

Let us consider $z\in \mathbb C$; what is the condition on modulus of z in order that $$\sum_{n=1}^{\infty} \frac{1}{n^z}$$ the series (zeta function?) converges? For example, if $|z|=1$, the series ...
1
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1answer
36 views

My attempt to prove an inequality get stuck——————where do I go wrong?

Hi, there. Bellow is my attempt. I don't know if I have gone in the wrong way and I am stuck. My attempt: Using Green's representation formula, $u(y)=\int_{\partial \Omega}u \frac{\partial ...
-1
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0answers
31 views

A problem on analytic function [duplicate]

Let $f(z)$ be analytic on $D=\{z\in\mathbb{C}:|z-1|<1\}$ such that $f(1)=1$. If $f(z)=f(z^2)$ for all $z\in D$, then which one of the following statements is not correct? (i) $f(z)=[f(z)]^2$ for ...
0
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0answers
27 views

Calculating the power series expansion about pi/2 of g(z)=tan[z/2]

Calculating the power series expansion about pi/2 of g(z)=tan[z/2]. Now calculate the expansion about 0. I'm having trouble doing this. I'm not even sure which is the best way to approach it, for ...
1
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1answer
27 views

Uniform convergence of the power series except at the point 1.

I couldn't solve the following problem from Lieb's Complex Analysis. Let $a_k$ be a decreasing sequence of real numbers that converge to $0$ and suppose that the radius of convergence of the series ...
0
votes
1answer
24 views

Orthogonal parameterization

Consider the function $$f(a,b,c,d):=\frac{\left(a^*\right)^2b^2-\left(b^*\right)^2a^2+\left(c^*\right)^2d^2-\left(d^*\right)^2c^2}{a^*a+c^*c}$$ With complex parameters $a,b,c$ and $d$ Now find any ...
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0answers
48 views

Can this be expressed by a contour integral?

Let $f(z)$ be a real entire function of the form $f(z) = a_1 z + a_2 z^2 + ...$ such that $0 < a_{n+1} < a_n$. Consider $g(x) = f^{-1}(f(x)-f(x-1))$ where $x$ is a positive real and $f^{-1}$ ...
0
votes
1answer
36 views

Calculate integral of $\ln(z)$ using the residue theorem

Please is it possible to calculate $\int_{C(0,1)}\ln(z)\,dz$ using the residue theorem? Thank you for your help.
2
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0answers
23 views

What does it mean to have an irrational/imaginary exponent and is there a way to calculate the latter?

In exponentiation, we are told that raising something to an integral power (n, say) means multiplying it with itself a total of n times, if n is non-negative. And we also learn fairly early on that ...
5
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0answers
69 views

Can a Power Series tell when to stop?

The naive description of the radius of convergence of a complex power series is as the largest radius so that the ball avoids poles and branch cuts. This makes sense in a world where analytic ...
0
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1answer
35 views

A holomorphic function with non-vanishing derivative

I really want to understadn the proof of the following theorem from Lieb's Complex Analysis: Let $f:U\rightarrow \mathbb{C} $ be a holomorphic function with non-vanishing derivative. Then: For ...
1
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3answers
30 views

Prove that $\phi(x,y)=e^{u(x,y)}cos(v(x,y))$ is harmonic

Suppose that $u,v$ are harmonic functions on doman $D$, and they are harmonic conjugate. Prove that function $\phi(x,y)=e^{u(x,y)}cos(v(x,y))$ is harmonic on $D$. What I've done was to take the ...
0
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0answers
36 views

Prove $U$ is subharmonic?

My attempt: Integration by parts says $\int u \triangle \varphi=\int\triangle u \varphi$. We know the left hand side is always $\ge 0$, and hence $\int\triangle u \varphi \ge 0$, since $\varphi \ge ...
0
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0answers
20 views

Determine f(z) by evaluating the sum

Determine an explicit expression for $f(z)$ by determining the sum of the series $f(z) = \sum_{n = 1}^\infty \frac{1}{n}$ $\cdot (\frac{z}{z-1})^n$ where $z\ne 1$ Yeah... I really don't know where ...
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0answers
20 views

conformal map of a portion of unit disk onto upper half plane

How do we construct a conformal map from $\{z=x+iy,x>1/2,|x+iy|<1\}$ onto the upper half plane? My idea is first create a sector sending one of the two intersection points to infinity.Any help ...
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2answers
42 views

$\tan(z)$ with residue theorem

Calculate $$\oint_{|z|=2}\tan(z)\,dz$$ because $\tan(z)=\dfrac{\sin(z)}{\cos(z)}$ the poles are when $\cos(z)=0$ at $z=\pm\pi/2\pm n\pi, \;n\in\mathbb{Z}$ Poles inside $|z|=2$ are $\pm\pi/2$ and ...
0
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1answer
23 views

$\log(z_1z_2)=\log(z_1)+\log(z_2)$ where $z_1,z_2\in \mathbb{C}$\{0}

I need to prove the set identity of the complex logarithm $\log(z_1z_2)=\log(z_1)+\log(z_2)$ where $z_1,z_2\in \mathbb{C}$. ...
0
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1answer
19 views

Function vanishes identically on the unit disc

Let $f$ be analytic in $\mathbb D$ and continuous in $\overline {\mathbb D}$ . Let $A$ $=$ {$ e^{i \theta}| |\theta - \pi |< \epsilon$} , for $\epsilon > 0$ small enough such that $f|_{A} = 0$ . ...
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0answers
25 views

Covergence to 0, while imaginary part is bounded

Given $f$ to be bounded and holomorphic on $\{z \in \mathbb C \mid -\pi < \operatorname{Im} (z) < \pi\}$. Let $\lim \limits_{x \to \infty} f(x) = 0$, where $x \in \mathbb R$ . Then prove that : ...
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1answer
15 views

Convergence in upper half plane.

Consider the upper half-plane $\mathbb H^{+}$ & let $f$ be a bounded holomorphic function on $\mathbb H^{+}$ . If $lim _{t \to \infty} f(it) = 0$ ; prove that: $lim _{t \to \infty} f(tz) = 0$ ...
0
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1answer
14 views

the crossover point of four complex points

If there is four complex points $z_1,z_2,z_3,z_4$ in complex plane $\mathbb{C}$, I want to get the crossover point of the line $z_1z_2$ and $z_3z_4$. If I use the $Re(z_i)$ and $Im(z_i)$, it is easy ...
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0answers
21 views

Let $f$ be entire in $\Bbb C$. If $Re(z)>0$ $\forall z \in \Bbb C$. Then prove that $f(z)$ is constant. [duplicate]

Let $f$ be entire in $\Bbb C$. If $Re(z)>0$ $\forall z \in \Bbb C$. Then prove that $f(z)$ is constant. Please read below before marking duplicate. If $Re(f(z))>M$ then taking $|\frac ...
1
vote
1answer
25 views

If w is a complex root of 1. Find the value of w^4+w^8

If $w$ is a complex root of 1. Find the value of $w^4+w^8$ Why does complex root of 1 always mean that $w^3=1$ ? Why not $w^2$ ? Back to the question, here's what I did: ...
0
votes
2answers
22 views

Integral with residues

Calculate integral $$\oint\limits_{\gamma}\frac{e^z}{z^4+5z^3}dz$$ Where $\gamma$ is parameterization of one rotation of circle $A(0,2)$ So if I write the integral like this ...
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votes
2answers
34 views

Complex integration along a curve

I have to calculate this integral: $$ \int_C e^z\,dz $$ where $C$ is the circle $|z - jπ/2| = π/2$ from the point $z = 0$ to the point $z = jπ$. I know how to calculate these with circles which ...
1
vote
1answer
23 views

Why the complex logarithm function$\ln(z)$ is not meromorphic on the whole complex plane

About meromorphic function, wiki says: In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D ...
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0answers
12 views

A conformal mapping from a sector to a strip

What is the simplest function that maps the sector $r < 1$, $0 < \theta < \pi$ conformally onto the strip $0 < u < \pi/2$, $v > 0$? Here, $r$, $\theta$, $u$, $v$ have their usual ...
0
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0answers
8 views

order and type of entire function $\exp(3z)\exp(-2\exp(z))$

what is the order and type of the following entire function? $$f(z)=\sum_{n=0}^{\infty}a_n z^n=\exp(3z)\exp(-2\exp(z))$$ The definition of order $\rho$ and type $\sigma$ from wikipedia.org are: ...
3
votes
1answer
40 views

inverse laplace transform by using complex integral

given function $$f(s)=\frac{1}{s}\frac{\sqrt{s}-1}{\sqrt{s}+1}$$ and $$\int_{0}^{\infty}{\frac{e^{-xt}}{\sqrt{x}(x+1)}dx=\pi e^t {erfc}(\sqrt{t})}$$ my steps: ...
6
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0answers
74 views

Area growth of harmonic functions

Can one construct a harmonic function $f$ defined in unit disk with condition $f(0)\geq1$ such that area of $\{z\in\mathbb{D}: f(z)>0\}$ is small enough?
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0answers
15 views

differential forms and index

The other question i can´t solve is this, If $\varphi$ a differential transformation such that $\varphi (x,y)=(f(x,y),g(x,y))$ and define $i(\varphi ,D)=\frac{1}{2\pi }\int _{\gamma }\theta _{0}$. ...
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0answers
33 views

1-forms and zero simple

Let $\varphi$ a differential transformation such that $\varphi (x,y)=(f(x,y),g(x,y))$ and $D\subset U$ such that $\varphi$ restricted to $\partial D=\gamma$ be distinct zero and we define $i(\varphi ...
0
votes
1answer
49 views

Why isn't Euler's formula multivalued?

So it seems that all complex exponential functions are multivalued except for ones with base $e$. Why? Shouldn't all exponentials be multivalued?
0
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1answer
35 views

Evaluating the integral of $1+z+1/\tan z$ over a circle

I am a beginner and I want to learn how to solve these kind of integrals: $$\int_{|z|= \pi/4}\left(1+z+\frac{1}{\tan z}\right)\,dz$$ So should I divide it in three integrals, calculate each integral ...
0
votes
1answer
50 views

If $w = e^{2i\pi/5} $, then $1 + w + w^{2} + w^{3} + 5w^{4} + 4w^{5} + 4w^{6} + 4w^{7} + 4w^{8} + 5w^{9}$=?

If $w = e^{i\frac{2\pi}5} $, then $1 + w + w^{2} + w^{3} + 5w^{4} + 4w^{5} + 4w^{6} + 4w^{7} + 4w^{8} + 5w^{9}$ =? I substituted $w$ into the expression and combined similar terms. I then tried to ...
2
votes
1answer
43 views

Suppose the limit of $f(z)$ as $z$ approaches $z_0$, exists and call it $w_0$. Suppose a sequence $(a_n)$ converges to $z_0$. Does $f(a_n)$ converge

Suppose the limit of $f(z)$ as $z$ approaches $z_0$, exists and call it $w_0$. Suppose a sequence $(a_n)$ converges to $z_0$. Does $f(a_n)$ converge to $w_0$ and $n \rightarrow \infty$? I would say ...
0
votes
2answers
43 views

Find the limit of a sequence $(\frac{z^n}{n!})_{n=1}^{\infty}$

I have to find the limit of a sequence $(\frac{z^n}{n!})_{n=1}^{\infty}$ where $z$ is a complex number. I think it is zero, because we know that $\sum_{n=0}^{\infty} \frac{z^n}{n!}$ is finite. Is this ...
0
votes
1answer
24 views

Express as a complex Fourier series

My function is $f(x)= \dfrac{1}{1-2e^{ix}} + \dfrac{1}{1-2e^{-ix}} $, which has been periodically extended by $2\pi$. I found $C_0$ to be $\pi$. I'm having trouble expressing $C_n$. All I have is ...
0
votes
1answer
15 views

A question on absolute values of line integrals

Let $f:U\rightarrow \mathbb{C}$ be continuous and $\gamma:[a,b]\rightarrow U$ be a smooth path where $U$ is open. Then we know that $$\int_{\gamma}^{}\ f(z)dz=\int_{\gamma}^{}\ ...
1
vote
1answer
29 views

How to calculate $\int_{-\infty}^\infty\frac{e^{ix}}{x}dx$

I need to calculate $$I_0=\int_{-\infty}^\infty\frac{e^{ix}}{x}dx=\lim_{R\rightarrow\infty}\int_{-R}^R\frac{e^{ix}}{x}dx=\lim_{R\rightarrow 0}(\lim_{\varepsilon\rightarrow ...
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0answers
22 views

boundary value of single complex variable holomorphic function

Is it right that there exists (one variable) holomorphic functions on the open unit disk, that are zero on the unit circle. If this is true, in what sense can we define this boundary value on the ...
0
votes
1answer
37 views

Show that a complex polynomial of degree $n$ doesn't have zeros in a unit ball [duplicate]

Let $n>1$ and $c_0>c_1>c_2>\dots >c_n>0$ and $f(z)=c_0+c_1 z+\dots + c_n z^n$. Show that this polynomial doesnt have zeros in a unit ball $B(0,1)$. Can you give me some feedback?
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0answers
42 views

A question on complex line integrals

The problem was as following Show that $|\int_{|z+1|=2} \frac{e^z}{z-2} dz| $ $\le$ $2\pi e$ I used that integral $\le |\frac{e^z}{z-2}|*4\pi$ where z is on the circle of radius 2 and -1 as center ...
1
vote
1answer
20 views

Taylor Expansion of complex function $\frac{1}{\sqrt{1-2tz+t^2}}=\sum_{n=0}^{+\infty}P_n(z)t^n$

Expanding $$\frac{1}{\sqrt{1-2tz+t^2}}=\sum_{n=0}^{+\infty}P_n(z)t^n$$ when $z \in \mathbb C$ I am trying to prove some prove that $P_n(z)$ satisfying: ...
0
votes
1answer
16 views

A question on branch of an inverse

Ignoring the 1st part of the 1st sentence of the question all I want to get is a branch $f$ of the inverse function of $g(z)=z^4. $ This is how I set about doing it, however, I need to verify this. ...
0
votes
0answers
17 views

A Direct Proof of Representation Theorem for Positive Harmonic Functions in the Half Plane?

Does anyone know a direct proof of this representation theorem for non-negative harmonic functions in the half-plane that doesn't appeal to a similar result in the unit disk? Also, does anyone who ...