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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35 views

Why is this a line equation?

Define $$L=\{z\in\mathbb{C} : cz + \overline{cz} + w = 0\}$$ Where $c$ is a nonzero constant. How does $L$ represent a line?
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1answer
41 views

Complex integration, help

I need help integrating $\int_{-\infty}^{\infty}\frac{z \sin (z)}{\left(z^2+1\right) \left(z^2+2\right)} dz$. I calculated the integral over the closed upper half circle in the complex plane which is ...
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1answer
29 views

Contour integral of $\frac{\bar{z}}{z-Z}$ on a square centered at the origin

I am having trouble calculating the following integral: $\oint_C \frac{\bar{z}}{z-Z} dz$ Here, Z is a complex constant and C is the contour of a square of side $2a$ centered at the origin. I ...
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1answer
47 views

Show that $f$ is identically zero.

Let $f$ be a entire function. Assume that there exist a real number $a$ such that $f^{(r)}(a)=0$ for all integer $r≥0$. My question is show that the function $f$ is identically zero.
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22 views

Maximum modulus principle, is it true?

Suppose f is analytic in an open set containing the open disk D(2+3i, 7) and its boundary circle C(2+3i, 7) such that |f(z) + 7i + 24|<25 for all z in C(2+3i,7). Then f has no zeroes inside D(2+3i, ...
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34 views

proving cauchy integral formula using integration by parts

I have found several proofs of Cauchy's generalized Integral formula, but I am looking to prove the first derivative case by using the fact that the first derivitave is analytic to state it as a ...
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27 views

Laurent series $\frac{\exp(z) - 1}{z^2}$

Anyone can help with this question? Find the Laurent Series for the given function about the indicated point. Also, give the residue of the function at the point $z=0$ $$ \frac{\exp(z) - 1}{z^2}$$
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1answer
47 views

a sequence of holomorphic functions with uniformly convergent derivatives

Let $(f_{n})_{n}$ be a sequence of holomorphic functions on a domain D which satisfies the following conditions: there exists some $z_{0}$ in D such that $f_{n}(z_{0})$ converges and the sequence of ...
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1answer
78 views

a sequence of polynomials converges to $0$

I am trying to show that there is a sequence $(P_{n})_{n}$ of polynomials such that $P'_{n}(0)=1$ for all $n$, $P'_{n}(z)\rightarrow0$ if $z \in \mathbb{C}^{\times}$ and $P_{n}(z)\rightarrow0$ ...
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1answer
48 views

Showing $\operatorname{Log}(z-i)$ is not analytic

Show that the function $\operatorname{Log}(z-i)$ is analytic everywhere except on the half line $y=1$ $(x\leq 0)$. I know that ...
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2answers
49 views

What is the definition of 'line' in $\hat{\mathbb{C}}$?

What is the definition of straight line in $\hat{\mathbb{C}}$? Is it defined as $\{x\in\mathbb{C}: \frac{Re(x-a)}{Re(b)} = \frac{Im(x-a)}{Im(b)}\}\cup \{\infty\}$? ($a,b$ are complex numbers and ...
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14 views

Continuity of Complex function and restrictions

I am trying the following question but am stuck at finding the restriction: Prove that $f(z)=1/z^2$ is continuous at $z_0= 1+2i$ Solution: I am trying the use epsilon-delta proof and got it down to: ...
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2answers
61 views

How to find branch points

I'm solving a set of exercises to understand how to find branch points and branch lines, but I'm having trouble with the more difficult('ish) ones. What I usually do in more simple exercises, is that ...
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0answers
18 views

Showing uniqueness of character identity

How would one show that any complex-valued C1 function satisfying the character identity must be of the form exp(cx) for c complex. Given a function f, it is said to satisfy the character identity if ...
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53 views

Whats the differences between the real-entire functions on $\mathbb R^{2}$ and complex entire functions on $\mathbb C$?

We note, as set of points, $\mathbb R^{2}= \mathbb C.$ A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point ...
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15 views

conflictions of analytic functions to the boundary and Schwarz reflection principle

Let $\Omega$ be an open subset of $\mathbb{C}$ and $f:\Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Then for any $z\in \Omega$ and any $r>0$ such that $D(z,r)\subseteq \Omega$, $f$ ...
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18 views

Cauchy integrals over a line

Can we generalize the Cauchy integral formula from a circle to a line? Since for real integrals, the following types of improper integrals do not converge, is it correct or not that for $z\notin ...
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2answers
46 views

Integral along $\Gamma_c := \{c + i t \mid c>0 , -\infty < t < \infty\}$

I have a Complex Analysis homework problem which I've been working on for some time, and have become stuck. I am asked to compute $$ I \equiv {1 \over 2\pi{\rm i}}\int _{\Gamma_c}{a^{s} \over ...
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20 views

Analytic Continuation of a Function Containing a Square Root to a Second Riemann Sheet

Consider the function $f(z) = g_1(z) + \sqrt{z} \, g_2(z)$, where $g_1(z)$ and $g_2(z)$ are entire functions, and we take the principal branch of the square root. $f$ is analytic on $\mathbb{C} / \{z ...
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1answer
23 views

Is it possible for a function to be differentiable at only one point?

I was taking a complex analysis class today, and we looked at the function f(z)=|z|^2 (with the domain over the complex numbers). It is continuous, but it satisfies the Cauchy-Riemann equations at ...
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1answer
37 views

Complex Analysis Computation

I'm not really sure how to tackle this problem, so any help/hints would be appreciated. Let $w=\cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2\pi}{n}\right)$ where n is a positive integer. ...
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1answer
30 views

Conformal mapping on two paths

GIven $ f (z) = z^2$ . Let $p = (0, −1)$ and take the curves $γ_1, γ_2$ passing through $p$ as $γ_1 = $arc of the unit circle through $(0, −1) $ counterclockwise and $γ_2 $ = a straight line ...
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42 views

Integral of Difference of Logs

I get the expansion of $h$ to be $$ h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r $$ $$ \Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}} $$ ...
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27 views

Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic?

Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic? If yes, in which domain?
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21 views

Value of a transcendental function

I am trying to transform really just evaluate or simplify: Log($\sinh(1+i))$ into Log($\sinh^2(1) + \sin^2(1)) + i\theta$ where $\theta=\arctan(\frac{\tan(1)}{\tanh(1)})$ I have tried the ...
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26 views

Show if for every convergent sequence $\{x_n\}$ with $\lim x_n = x_0$ and $\lim f(x_n)=c$ then $f(x) \rightarrow c$ for $x \rightarrow x_0$.

Let $f: \mathbb R \rightarrow \mathbb C, c \in \mathbb C$. Show if for every convergent sequence $\{x_n\}$ in $\mathbb R / \{x_0\}$ with $\lim_{n \rightarrow \infty} x_n = x_0$ and $\lim_{n ...
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1answer
61 views

Is there a name for the one-point compactification of $\mathbb{C}$?

Let $\hat{\mathbb{C}}$ be the one-point compactification of $\mathbb{C}$. This space $\hat{\mathbb{C}}$ is called the Riemann sphere. If I want to designate the topology $\tau$ on ...
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2answers
30 views

Show $e^z e^w = e^{z+w} \ \forall \ z,w \in \mathbb C$ by differentiation of $f(t):=e^{w+tz}e^{-tz}, \ t \in \mathbb R$.

Show $e^z e^w = e^{z+w} \ \forall \ z,w \in \mathbb C$ by differentiation of $f(t):=e^{w+tz}e^{-tz}, \ t \in \mathbb R$. I have already showed $e^z e^{-z} = 1$ for $z \in \mathbb C$. This result ...
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1answer
16 views

Finding residue with respect to closed path

Let $\gamma$ be the closed path consisting of straight line segments from $2+2i$ to $-2-2i$, from there to $-2+2i$, from there to $2-2i$ and finally back to $2+2i$. Evaluate $\int_{\gamma}f(z)dz$ for ...
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1answer
43 views

Residue theorem in evaluating complex integrals?

It's been a while since I used residue theorem to evaluate anything. I remember that whenever we have a real valued function, we can use residue theorem to evaluate its integral with associated ...
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1answer
38 views

$|z-a|+|z+a|=2|c|$ iff $|a| \leq |c|$

Full question: let a and c be complex numbers, prove that there exists a complex number z such that $|z-a|+|z+a|=2|c|$ iff $|a| \leq |c|$ so far I have this: ($\Longrightarrow$) |z-a| $\leq$ ...
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0answers
20 views

Very quick question on sesquilinear forms seen as bilinear maps.

This is a very quick question on sesquilinear forms when seen as bilinear maps. Let $V$ be a complex vector space, a sesquilinear map (or conjugate-linear in the first variable and linear in the ...
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1answer
28 views

analytic isomoprhism

hi everyone i am going to ask your help one of my homework problem. Let f be analytic isomorphism of D(0,1) that is f∈Iso(D(0,1),U). show that if D(f(0),R)⊂U then R≦|f'(0)|. so what i figured out so ...
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2answers
17 views

Proving $-u$ is a harmonic conjugate for $v$

Suppose $u$ and $v$ are real valued functions on $\mathbb{C}$. Show that if $v$ is a harmonic conjugate for $u$, then -$u$ is a harmonic conjugate for $v$. I know I have to use cauchy reumann here. ...
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1answer
24 views

Examples of vector field that is continuously differentiable but not conservative?

I am just curious what would be the case in which a vector field ($\vec f :\Bbb R^2 \rightarrow \Bbb R^2$) is well-defined and continuously differentiable on a region R enclosed by a simple closed ...
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2answers
33 views

Geometrical Meaning of derivative of complex function

What's the geometrical meaning of f'(z) in complex analysis, as we know in real analysis f'(x) has meaning ie. Slope of curve or gives max/ min. But what does derivative f'(z) has geometrical meaning ...
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1answer
31 views

Convergence of a sequence of holomorphic functions

I'm looking for a proof of the following fact: The sequence of entire functions $\{(1+\frac{z}{n})^{n} \}_{n\in \mathbb{N}}$ converges uniformly to $e^{z}$ on every compact subset of $\mathbb{C}$. I ...
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2answers
110 views

integral of sin(x) to the power 2014

For a course in Complex Analysis we're tasked to find the integral of \begin{align*} \int_0^{2 \pi} (\sin\theta)^{2014} d \theta \end{align*} but I'm a bit stumped so far on how to do this. What I've ...
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1answer
24 views

hyperbolic inequality

Calculating some contour integral, I have to prove that $\int^{R+i}_{R}\frac{cosh(az)}{cosh(\pi z)}dz$ goes to zero if R goes to infinity. And we know that $\left|a\right|<\pi$. I want to use the ...
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2answers
56 views

What is the Taylor series of $\frac{1}{\sin(z)}$ about $z_0 = 1$?

This was a exam question so I know it cannot take too long to write out the proof. Only I cannot see an answer. I would imagine you write $\sin(z) = \sin(1+(z-1)) = \sin(1)\cos(z-1) + ...
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7 views

Conceptual explanation for what happens to euclidean center and radius of pseudohyperbolic disk as pseudo hyperbolic center tends to unit circle

If $P(w, r)=\{ z \in \mathbb{D} | \rho(z, w)<r\}$ where $\mathbb{D}$ is the unit disk, and $\rho$ is the pseudohyperbolic metric, can someone please explain on a conceptual level how the Euclidean ...
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41 views

Weierstrass's elliptic $\wp$-function

For $\omega_1,\omega_2 \in \mathbb{C}$ we define $\Gamma:=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ and $\Gamma':=\Gamma\setminus\{0\}$. $\wp$ is weierstrass's elliptic $\wp$-function ...
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1answer
42 views

Almost everywhere convergence of some series

Let $\{r_n\}$ be an arbitrary numerical sequence. Prove that $\sum_{n=1}^\infty\frac{1}{2^n\sqrt{|x-r_n|}}$. Prove that it converges almost everywhere on $\Bbb R$.
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85 views

Calculating residue of $z\sin{ \frac {z+1}{z-1}}$

Let $f=z\sin{ \frac {z+1}{z-1} }$. Calculate the residue of $f$ in $z=1$. I think $f$ has an essential singularity at $z=1$ so the only way I can proceed is with Laurent series. I've defined ...
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3answers
86 views

Prove that $f$ is constant if $f'=0$

Suppose that $f$ is holomorphic on a domain $D$ and $f'=0$ on $D$. Prove that $f$ is constant on $D$.
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1answer
29 views

complex residue involving exponent of quotient of polynomials

I was trying to work out an integral and came to trying to find the complex residue of $$R = \exp\left(\frac{Ax^2 + Bix}{Dx + 1}\right)$$ at $x = -D^{-1}$. I used partial fractions to get: $$ = ...
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2answers
41 views

Limit concept under Complex analysis

Prove that $$ \lim_{z\to i} \dfrac{3z^4-2z^3+8z^2-2z+5}{z-i} = 4+4i $$
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1answer
106 views

One-one analytic functions on unit disc

Is the following statement true? Suppose, $ f:D\to \mathbb C $ is an analytic function where $ D $ is the unit disc of radius $ 1 $ around $0 $. Suppose, $ f $ is analytic on the boundary of $ D $ as ...
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23 views

Sketching sets in $\mathbb{C}$

This might seem too elementary but I'm teaching myself complex analysis and have very minor doubts about the following sets. I'm asked to sketch: A. $\{z \in \mathbb{C}: |e^{z^2}| \le e \}$ B. $\{z ...
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22 views

Is there a general notation for the Mobius group?

Let $a,b,c,d$ be complex numbers such that $ad-bc\neq 0$. Define $f(z)=\frac{az+b}{cz+d}$. Such function $f:\overline{\mathbb{C}}\rightarrow \overline{\mathbb{C}}$ is called a Mobius transformation. ...