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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6 views

Prove that $ \left|\sum_{k=1}^{n}z_kw_k\right|^2 z_k,w_k \in \mathbb{C}$

Prove that $$\left|\sum_{k=1}^{n}z_kw_k\right|^2=\sum_{k=1}^{n}|z_k|^2 \cdot \sum_{k=1}^{n}|w_k|^2-\sum_{1 \leq k , l \leq n}^{}|z_k\bar{w_l}-z_l\bar{w_k}|^2$$
0
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2answers
20 views

(complex analysis) Prove that: arg ((z3-z2)/(z3-z1)) = 1/2 arg z2/z1

if |z1|=|z2|=|z3| Urgent help needed. I have used: z1=x1+iy1 z2=x2+iy2 z3=x3+iy3 and obtained arg z3-z2/z3-z1 = arctan ((((y3-y2)(x3-x1))-((y3-y1)(x3-x2)))/((x3-x2)(x3-x1))) But, then I'm stuck.
5
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2answers
41 views

Showing that $\{z\in\mathbb{C}:|z-1|<|z+i|\}$ is an open set

Got stuck on some homework (from H. A. Priestley, Complex Analysis). My topology ain't quite up to speed yet. So, I want to show that $S=\{z\in\mathbb{C}:|z-1|<|z+i|\}$ is open. Geometrically it's ...
0
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1answer
42 views

Problems with understanding analyticity

I have a problem understanding the idea behind Analytic functions. (Please correct me on my terminologies while I state my problem). An analytic function, is a function that has a power series that ...
0
votes
1answer
24 views

Calculating the residue of a complex funciton with ln(z) at z=0

How can I calculate this residue: $$Res\left(\frac {z\ln(z)}{(z^2 +1)^3} , 0\right) $$ if it's possible at all. I know $0$ is a branch point for $\ln(z)$ and therefore isn't a pole, but when i plug ...
0
votes
1answer
33 views

Finding an Arg(w)

The question is: Describe (in words) and sketch the set of all $z \in \mathbb{C}$ such that $$\displaystyle 0<\arg\left(\frac{i-z}{i+z}\right)<\frac{\pi}{2}$$ I believe that I am supposed to ...
1
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1answer
19 views

Applying Cauchy theorem to the integral of $\overline{z}^2$ over two different curves.

I have solved the following exercise, in which I had to compute $$\int _\gamma \! \overline z ^2 \, \mathrm{d}z,$$ where $\gamma$ is The circumference $\left| z\right| =1$ The circumference ...
1
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1answer
25 views

Find the analytic function

$f(z)=1 $ satisfies the condition Using Identity Theorem $f(z)=1$ can be only function that satisfies this. so option (b) is NOT true. Am I on correct path?
2
votes
2answers
49 views

Sketch the complex function: $z\overline{z}+(1+2i)z+(1-2i)+1=0$

Tried sketching the complex function: $z\overline{z}+(1+2i)z+(1-2i)+1=0$ I first simplified it by converting $z=x+yi$ I got: $(x+yi)(x-yi)+(1+2i)(x+yi)+(1-2i)+1=0$ Which gave me this implicit ...
0
votes
1answer
21 views

Show that the imaginary part of $\frac{z^2}{z-z_p}$ is harmonic

Let $z\in\Omega \subset\mathbb{C}$ and $z_p\notin \Omega$. Show that $\text{Im}(\frac{z^2}{z-z_p})$ is harmonic in $\Omega$, where $\text{Im}(z)$ is the imaginary part of $z$. So far: For $z = \alpha ...
0
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1answer
21 views

why $f(z) = z^{(3/2)}$ does not have derivative at z = 0 in complex plane.

it seems that the $f'(z) = z^{(1/2)}$ means that this function has derivative for every complex value. But why $f(z) = z^{(3/2)}$ does not have derivative at z = 0
2
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0answers
15 views

Classify entire functions satisfying $|f(z)|\leq (1+|z|)^2$

I have to classify entire functions satisfying $|f(z)|\leq (1+|z|)^2$ for all $z\in \mathbb{C}$. Using Cauchy integral's formula, I've shown that $f^{(3)}=0$. Thus $f(z)=a+bz+cz^2$ for some $a,b,c ...
0
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0answers
20 views

Harmonic functions on $\{1<|z|<2\}$

I have to find all complex-valued harmonic functions on $\{1<|z|<2\}$ that extend continuously to $\{|z|=2\}$ and take value $0$ on that circle. My first idea was to map $\{1<|z|<2\}$ to ...
1
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3answers
34 views

Proving uniform convergence on disk within radius of convergence

Needham's Visual Complex Analysis 2.III.2 states that a power series $S_k=\sum{C_k z^k}$ with RoC $R$ converges uniformly on any disk $r<R$. He leaves the proof as an exercise to the reader. But ...
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0answers
36 views

Entire function satisfying $f(z + 1) = f(z)$ and $f(z + i) = f(z)$ is to be proven constant [on hold]

Show that an entire function satisfying $f(z + 1) = f(z)$ and $f(z + i) = f(z)$ is a constant.
2
votes
1answer
20 views

Find all functions with given property

Find all functions $f$ that are holomorphic on $B = \{z: -\pi/2 < \operatorname{Im}(z) < \pi/2 \}$ with $f(B) \subset B$ and such that $f(0) = 0$ and $f'(0) = i$. Thoughts so far: My first ...
2
votes
1answer
56 views

Solve complex integral with $\Gamma$-function

Let $s\in\mathbb C$ and $r\in\mathbb R$. In the integral $$\int_{-\infty}^\infty \frac{1}{z^{r+s}\overline{z}^s} dx$$ we have $z=x+iy$ where $y>0$ is fixed. I read that you can explicitly compute ...
1
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1answer
24 views

Let $\{P_n\}$ uniformly converge to $f$ on $T$. prove that exists analytic and continuous $F$ on $\bar{D}$ such that $F \equiv f$

Let $\{P_n\}$ be a series of polynomials which uniformly converges on the unit circle $ T = \{|z| = 1\}$ to $f$. prove there exists $F$ such that $F$ is analytic in the unit disk $D$ and continuous ...
1
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1answer
34 views

There exists a $M$ such that $\mid f^k (0)\mid \leq k^4 M^k$. Show that $f$ can be extended analytic on $\Bbb C$.

(a) Suppose that $f$ is analytic on the open unit disk $\{z: |z|<1 \}$ and there exists a $M$ such that $\mid f^k (0)\mid \leq k^4 M^k$ for all $k \geq 0$. Show that $f$ can be extended analytic on ...
0
votes
2answers
65 views

Why is Euler's formula a definition?

Even though there are proofs for Euler's formula for complex exponentials (see wikipedia for instance), it is mentioned as a "definition" in most textbooks. Why is that? My understanding is that a ...
0
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1answer
34 views

entire function problem in complex analysis [duplicate]

If f is an entire function satisfying f(z+1)=f(z) and f(z+i)=f(z) then f must be constant throughout the complex plane. I would appreciate if someone helps me know how to prove the above statement.
0
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3answers
30 views

Let $A$ be a complex number and $B$ be a real number. Prove that $\mid z^2\mid+Re(Az)+B=0$ can only have a solution iff $\mid A^2 \mid \ge 4B$.

Been stumped on this question for a while. I tried letting $z=\mid z \mid \cdot e^{i \alpha}$ and $A=\mid A \mid \cdot e^{i\beta}$ -- assuming that $\alpha$ and $\beta$ were the arguments of $z$ and ...
3
votes
2answers
42 views

Does the identity ${|\cosh z|}^2={\cos}^2x+{\sinh}^2y$ given in my text hold?

In my text book I saw that ${|\cosh z|}^2={\cos}^2x+{\sinh}^2y$ But when I tried deriving it myself I got this: $${|\cosh z|}^2={\cos}^2y+{\sinh}^2x$$ See my working below: $$\cosh ...
0
votes
1answer
29 views

Showing a given holomorphic function $f$ satisfies $|f''(0)| \leq 2$ and $|f(z)|\leq|z|^2$

Let $f$ be a holomorphic function from $D$ to $D$, where $D$ is the open unit disk. Suppose that $f(0) = 0$ and $f'(0) = 0$. Show that$|f''(0)| \leq 2$ and $|f(z)|<|z|^2$ for all $z \in D$. ...
1
vote
2answers
72 views

find the number of solutions to $p(z) = z^6 + 9z^4+z^3+2z+4$

Let $p(z) = z^6 + 9z^4+z^3+2z+4$ find then number of roots in each quadrant of the complex plane find in which quadrant exists a root which is inside the unit circle using the Argument priniciple ...
0
votes
3answers
45 views

Help regarding an exercise in complex analysis [on hold]

Let $f, g$ be entire functions and $ |f(z)| \le (1+|z|) |g(z)|$ for every $z \in \Bbb C$. Prove that there exist $λ, μ \in \Bbb C$ with $|λ|,|μ| \le 1$ so that $f(z)=(λz + μ)g(z)$. If $ g(a)=0 $ for ...
2
votes
1answer
21 views

A set $I$ of isolated complex numbers such that $[0,1]\subset\{Re(z):z\in I\}$

Is there a set $I$ of isolated complex numbers, such that $$[0,1]\subset\{Re(z):z\in I\},$$ where $Re(z)$ is the real part of the complex number $z$.
1
vote
1answer
24 views

Finding cubed roots of complex number

Is this correct? $a^3 =r^3e^{i3\theta}= 5\sqrt{5}e^{i\arctan(11/2)}$ $$\implies r=\sqrt{5}, 3\theta = \arctan(11/2)+2\pi n,n\in\Bbb Z$$ $$\theta = \frac{\arctan(11/2)+2\pi n}{3}$$ $$\theta = ...
0
votes
1answer
19 views

A billinear transformation

Let $$w(z)=(az+b)/(cz+d)$$.Then $w(z)$ maps a straight line of $z-$plane to the circle $|w|=1$ in $w-$plane if $1.|b|=|d|$ $2. |a|=|c|$ $3. |a|=|d|$ $4. |b|=|c|$ My work: I started by considering ...
1
vote
1answer
48 views

Branch cut of $\sqrt{z}$

In my complex analysis book, the author defines the $\sqrt{z}$ on the slit plane $\mathbb{C}\setminus (-\infty,0]$. I understand this is done because $z^2$ is not injective on the entire complex ...
2
votes
1answer
56 views

Compute $\int_0^\infty \frac{x \sin(ax)}{1+x^4} \, dx$

Compute $\displaystyle \int_0^\infty \frac{x \sin(ax)}{1+x^4} \, dx$. Thoughts so far: I see that the function is odd, so is one half the integral on the whole real line. So a half circle contour ...
0
votes
0answers
24 views

Show there's an analytic functions satisfying certain conditions

Suppose that $f$ is analytic on $\{z: |z|<r\}$ for some $r>1$. Suppose further that $|f(z)|<1$ for $|z|=1$ and $f\left(\frac{1}{2}\right) = \frac{1}{2} $. 1) Find a set $U$ so that $f(0) \in ...
2
votes
2answers
49 views

Does this set tend towards a disc?

Let $p$ be a complex polynomial \begin{gather*} p:\mathbb{C}\longrightarrow\mathbb{C},\\ \deg p = n,\quad n\in\mathbb{N}. \end{gather*} Define the set $\mathcal{R}=\{z\in\mathbb{C}:|p(z)|\leq R\}$, ...
1
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0answers
29 views

Visual understanding of radius of convergence for complex power series

I am examining Needham's proof (Visual Complex Analysis 2.III.2) that $\sum c_k z^k$ converges at $a$ implies absolute convergence (and hence convergence) for $|z| < |a|.$ Despite the book's ...
3
votes
2answers
51 views

Is the graph of $xy=1$ in $\mathbb C^{2}$ connected?

The graph of $xy=1$ in $\mathbb C^{2}$ is set of points $(x+iy,u+iv)$ that satisfies $$xu-yv=1$$ and $$uy+xv=0$$ How to find if this set is connected or not . I also have another ...
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0answers
19 views

Analysing a set in the complex plane

I would like you to follow my logic, confirming it if correct, suggesting change when flawed or suboptimal. Let $p$ be a complex polynomial \begin{gather*} p:\mathbb{C}\longrightarrow\mathbb{C},\\ ...
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1answer
22 views

Is the magnitude of the gradient non zero?

Let $f=u+iv$ be a holomorphic function on a domain $\Omega$. Suppose $x_{0}+iy_{0}=z_{0}\in\Omega$ such that $f^{\prime}(z_{0})\neq0$ and $\left\vert f(z_{0}) \right\vert > 0$. Let ...
0
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0answers
12 views

Questions about the inverse of Joukowski function

The inverse of Joukowski function is $ w=z+\sqrt{z^2-1} $. Please show me how to calculate the argument of w and why the branch points of this function are the same as that of $ f(z)=\sqrt{z^2-1} $.
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2answers
22 views

Proof using Identity theorem?

I need to prove that no two distinct holomorphic functions agree on all of $\frac{1}{n}$ where $n$ is an integer. So the identity theorem says that two functions $f,g$ are identical iff the set of ...
5
votes
3answers
177 views

All continuous functions are analytic

This might be very silly to ask, but somehow this sequence of results are leading me to this wrong result. I am dealing with complex analysis and the mistake I am making might be because I am using ...
0
votes
1answer
20 views

Trouble with integration in order to find analytic function

Let $u(x,y) = x/(x^2 - y^2)$ Find $v(x,y)$ such that $f(z) = u + iv$ I'm applying Cauchy-Riemann $u_x = -\frac{(x^2 - y^2)}{(x^2 + y^2)} = v_y$ But I don't see how to integrate that with respect ...
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2answers
33 views

An 11-gon with complex numbers

Let $A_1 A_2 \dotsb A_{11}$ be a regular $11$-gon inscribed in a circle of radius $2$. Let $P$ be a point, such that the distance from $P$ to the center of the circle is $3$. Find $[PA_1^2 + PA_2^2 ...
1
vote
1answer
23 views

Prove there are two points making two regions not conformally equivalent

Show that there exist distinct points $z_0,z_1$ in the open unit disk $D$ so that $D - \{1/2, 1/3\}$ and $ D - \{z_0, z_1\}$ are not conformally equivalent. Thoughts so far: I'm not sure where to go ...
3
votes
3answers
59 views

Geometric interpretation of inverse complex function?

Function $f\colon\mathbb{R}\to\mathbb{R}$ and its inverse $f^{-1}$ are symmetric over line $y=x$. It's easy to imagine inverse of real function, we just have to "flip" the plot over $y=x$. But what ...
2
votes
0answers
27 views
+50

Proof of conformal mappings onto polygons (Stein)

Recently I am reading Stein and Shakarchi's Complex Analysis and I find a great difficulty in understanding the proof of theorem 4.6 in Chapter 8 (p.242-244), which talks about conformal mappings onto ...
1
vote
1answer
34 views

If $f$ is non-constant and entire, prove that there exists a $z_0 \in C$ such that $f(z_0)$ is a positive real number

If $f$ is non-constant and entire, prove that there exists a $z_0 \in C$ such that $f(z_0)$ is a positive real number, without appealing to Picard's theorem. The obvious approach to this would be to ...
3
votes
1answer
124 views

Has anyone ever explored $(\sin{x})^x$ , $(\cos{x})^x$, etc?

I've come across a problem that involves something very close to: $$\int(\cos{x})^xdx$$ and I have no clue as to how to proceed with any kind of analysis for this type of equation. It occurred to me ...
0
votes
1answer
29 views

Show a certain analytic function must exist

Suppose that $f$ is holomorphic on $D - \{0\}$, where $D$ is the open unit disk. Suppose that $f$ has a pole of order one at $0$, with a residue equal to $n$ for some positive integer $n$. Show there ...
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3answers
52 views

Prove that if $f$ and $g$ are analytic at $w$, then so is $fg$

Prove that if $f$ and $g$ are analytic at $w$, then so is $fg.$ My main attempt was using the Cauchy-Riemann equations on the product in this manner but this did not work out. My thinking: ...
0
votes
1answer
16 views

Determine all open sets on which $f$ is analytic.

Given $$f(z) = \frac{1}{x} + i(-\frac{1}{y})$$ Determine all open sets in which $f$ is analytic. My attempt: $$f(z) = u(x,y) + iv(x,y)$$ where $u(x,y) = \frac{1}{x}$ and $v(x,y) = ...