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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Complex Analysis: Finding the level curves of a function?

Consider the function $f(z)=z^2$. Prove that level curves of $Re(f(z))$ and $Im(f(z))$ at $z=1+2i$ are orthogonal to each other. I am not sure how to apply level curves or contour lines for complex ...
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253 views

Field Extension problem beyond $\mathbb C$

There are lots of fields between $\mathbb C$ and Meromorphic Functions on $\mathbb C$. For example set of "All Even Meromorphic Functions on $\mathbb C$'' is a subfield between $\mathbb C$ and ...
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148 views

Estimates involving a holomorphic function on the unit disc

Assume that $f$ is an analytic function on the unit disc $\mathbb{D}$ and continuous up to the closure. Therefore $f(z)=\sum\limits_{n=0}^\infty c_nz^n$ for all $z \in \mathbb{D}$. If $f$ have $m$ ...
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396 views

Residue at $z=\infty$

I'm a bit confused at when to use the calculation of a residue at $z=\infty$ to calculate an integral of a function. Here is the example my book uses: In the positively oriented circle $|z-2|=1$, ...
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288 views

Characterization of Cauchy-Riemann operator

Let $U \subset \mathbf C$ be an open subset of the complex plane and suppose we have a differential operator of order 1, $L: \mathcal C^{\infty}(U) \to C^{\infty}(U)$ such that $Lu = 0$ if and only ...
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312 views

How to show set of all bounded, analytic function forms a Banach space?

I am trying to prove that set of bounded, analytic functions $A(\mho)$, $u:\mho\to\mathbb{C}$ forms a Banach space. It seems quite clear using Morera's theorem that if we have a cauchy sequence of ...
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300 views

Holomorphic Automorphism Group

By a domain I mean an open connected subset of ${\mathbb C}$. If $D$ is a domain, let $\operatorname{Aut}(D)$ denote the collection of holomorphic bijections $f:D\to D$. It is well-known that if $f$ ...
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111 views

Translation of entire functions along the real axis

Given an entire function $f(z)$, and $0\neq a\in \mathbb R$. We define the translation operator: $$T_{a}f(z)=f(z-a).$$ What properties the new function $f(z-a)$ could have? It is entire function! ...
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150 views

Limit conditions of a subharmonic function imply that it is constant

Let $u$ be a subharmonic function on $\mathbb{C}$. Suppose that $$\limsup_{z\to \infty} \frac{u(z)}{\log|z|}=0$$ I'm trying to prove that this implies $u(z)$ is constant. I have a feeling that it may ...
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4k views

Multiplying complex numbers in polar form?

Could someone explain why you multiply the lengths and add the angles when multiplying polar coordinates? I tried multiplying the polar forms ($r_1\left(\cos\theta_1 + i\sin\theta_1\right)\cdot ...
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259 views

Jensen's Inequality for complex functions

Jensen's inequality states that if $\mu$ is a probability measure on $X$, $\phi$ is convex, and $f$ is a real-valued function, then $$ \int \phi(f) \, d\mu \geq \phi\left(\int f \, d\mu\right).$$ Is ...
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452 views

Rectangular form of a complex number?

Why does rectangular form serve as an accurate description of a complex number? Why not $a * bi$(multiplication) or another operation? Why does addition describe the relationship between the real part ...
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230 views

Branch Points of Riemann Surfaces

Can a Riemann surface of a complex-valued function have three branch points? I've been learning about Riemann surfaces from Brown's complex analysis book and the exposition isn't too general, so if ...
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124 views

Bounds on unit disc imply boundedness at origin

Suppose $f$ is holomorphic on $D_{1}(0)$ the open unit disc. Let $\Gamma_{1} = \{z : |z| = 1, x>0, y>0\}$ where $z = x+iy$ and define $\Gamma_{2}, \Gamma_{3}, \Gamma_{4}$ similarly. On ...
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200 views

Construction of conformal mapping

Let $\epsilon>0$. I was asked to find a conformal mapping from $(\mathbb{R}\times(0,2))-((-\infty,i-\epsilon ] \cup[i+\epsilon,i+\infty))$ (An infinite horizontal strip but chopped a fine strip ...
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85 views

How to integrate $\int_{\gamma_1} \frac{dz}{z(z-i)}$ with $\gamma_1 = Re^{it}$, $R>1$?

I am stuck calculating the integral $$\int_{\gamma_1} \frac{dz}{z(z-i)}$$ over $\gamma_1 = Re^{it}, R>1$. If I had to integrate over $\gamma_2 = re^{it}, r < 1$, I could just expand the ...
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155 views

Finding the number of analytic functions which vanish only on a given set.

Let $S = \{0\}\cup \{\frac{1}{4n+7} : n =1,2\ldots\}$. How to find the number of analytic functions which vanish only on $S$? Options are a: $\infty$ b: $0$ c: $1$ d: $2$
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102 views

Difference of two subharmonic functions

Is it true that for a smooth real-valued function $h(z)$ on some neighborhood of the closure of a bounded domain, that $h$ can be expressed as the difference of two smooth subharmonic functions? If ...
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162 views

Question regarding an analytic function and a meromorphic one

Is it possible to have an analytic function on the unit disk $\mathbb{D}$ that has infinitely many isolated zeros? What is a good example? I guess then that would make this analytic function ...
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320 views

contour integral with rational and cosh

Here is a fun looking integral. $$\int_{0}^{\infty}\frac{1}{(4x^{2}+{\pi}^{2})\cosh(x)}dx=\frac{\ln(2)}{2\pi}$$. I rewrote it as $\frac{2e^{z}}{(4z^{2}+{pi}^{2})(e^{2z}+1)}$ It would appear there ...
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182 views

Prove that there is an unique $z$ s.t. $f(z) = z$ where $z$ is a complex number

Let $f$ be analytic on the closed unit disk centered at the origin and $|f(z)| < 1$ for $|z| = 1$. Show that $f$ has exactly one fixed point inside the open unit disk. That is, there exists a ...
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353 views

Application of Jensen's formula

While studying for an upcoming complex analysis qualifying exam, I found the following problem in Conway's Functions of One Complex Variable (XI.1 exercise #2). Let $f$ be an entire function, ...
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112 views

Problem on analytic function.

Let $f(z)$ be analytic function on $D = \{z\in C : |z-1|<1\}$ such that $f(1) = 1$. If $f(z) = f(z^2)$ for all $z\in D$. Then which of the following statement is not correct. 1-$f(z) = [f(z)]^2$ ...
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112 views

Residues - Which coefficients to choose?

When finding a residue, how am I to know which coefficient to choose? For instance, if I have, let's say three poles, which coefficients of the Laurent series do I choose to calculate the residue? ...
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120 views

Trouble with Cauchy Riemann…not sure which law to use?

I'm unsure of which Cauchy-Riemann law to use when I'm given either a real or imaginary function. For instance. I might be given a real function and asked to work out the imaginary part. For ...
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77 views

A simple Riemann mapping question

Let $\Delta$ denote the open unit disc. Let $G$ be a simply connected region and $G\neq\mathbb{C}$. Suppose $f:G\rightarrow\Delta$ is a one-to-one holomorphic map with $f(a)=0$ and $f'(a)>0$ ...
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225 views

Problem on Complex Analysis.

If $f(z)= u - iv$ is an analytic function of $z = x + iy$ and $u-v=\Large\frac{e^y-\cos x+\sin x}{\cosh y - \cos y}$, find $f(z)$ subject to condition, $f\left(\frac{\pi}2\right)=\large \frac{3 - ...
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88 views

Prove $2|f'(0)| \leq \sup_{z_1,z_2\in D} |f(z_1)-f(z_2)|$

Let $f:D\to \mathbb{C}$ be a holomorphic function where $D$ is the open unit disk. Then prove $$ 2|f'(0)| \leq \sup_{z_1,z_2\in D} |f(z_1)-f(z_2)| $$ I can show that $$2f'(0) = \frac{1}{2\pi i} ...
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331 views

Intuition for the Poisson kernel

The derivation of the Poisson kernel for a disc seems to involve a trick, and I don't really understand how one would come up with it. Let $f$ be a holomorphic function on a disc $D_{R_0}$ centered ...
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108 views

Analytic bijection preserves order of poles

If $f:U\rightarrow\mathbb{C}$ has a pole at $z$ of order $n$ and $\phi:V\rightarrow U$ is an analytic bijection with $\phi(w)=z$, show that $w$ is a pole of $f\circ\phi:V\rightarrow\mathbb{C}$ of ...
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148 views

For which simple closed curves $\gamma$, $\int\limits_{\gamma} z^{2}+z+1\, dz=0$?

For which simple closed curves $\gamma$ is $\displaystyle\int_{\gamma} z^{2}+z+1\, dz=0$ Could someone help me through this problem?
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351 views

Describe the set whose points satisfy the following relation.is region?

Describe the set whose points satisfy the following relation.is region? |z − 2| > |z − 3|. My atempt The open half-plane: Re z > 5/2; a region, My guess is that if this region takes all the ...
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178 views

Representation of Holomorphic Functions By Exponential

Let $f$ be holomorphic and nonzero on $D_{1}(0)$ the open unit disc. Can we write (for the given domain) $f(z) = e^{h(z)}$ where $h$ is holomorphic? This seems clear using a naive log argument but I'm ...
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65 views

How to evaluate this in the complex plane?

How to evaluate this in the complex plane? $\int_{\gamma}^{} z^{e^{z^{2}}}\, dz$ when $\gamma$ is the unit circle.
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191 views

how to evaluate this integral by considering $\oint_{C_{(R)}} \frac{1}{z^{2}+1}$

Consider the integral $I=\int_{-\infty}^{\infty} \frac{1}{x^{2}+1}\, dx$. Show how to evaluate this integral by considering $\oint_{C_{(R)}} \frac{1}{z^{2}+1}, dz$ where $C_{R}$ is the closed ...
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2k views

Entire function bounded by a polynomial is a polynomial

Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k>0$ is constant. Show that $f$ is ...
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344 views

Laurent series for an even function

Show that if the Laurent series $\sum_{n=-\infty}^{\infty}a_n(z-z_0)^n$ represents an even function, then $a_{2n+1}=0$ for $n=0,\pm 1,\pm 2,\ldots$, and if it represents an odd function, then ...
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48 views

Minimize a function in polar coordinates along a curve

Consider the line $\theta - \ln r = c$ where $-\pi < \theta \leq \pi$ and $c$ is a fixed real constant. How would I find the point $(r, \theta)$ which minimizes $1/r$? Alternatively, what is the ...
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300 views

Uniform convergence of sequence to the exponent function

Let $f_n(z) = (1-z^2/n)^n$, and let $f(z)=\operatorname{exp}(-z^2)$. I need to show that $f_n$ converges uniformly to $f$ in any closed disc. I saw this: Uniform Convergence of an Exponential ...
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87 views

Show that $\lim_{\varepsilon\to 0}I_\varepsilon = 0$

Consider $$I_\varepsilon :=\oint_{C_\varepsilon} z^αf(z)\,dz,$$ where $\alpha>−1$ is real, where $C_\varepsilon$ is a circle of radius $\varepsilon$ centered at the origin and $f(z)$ is analytic ...
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72 views

Show that $\lim_{R \to{+}\infty}{I_{R}}= 0$.

Consider $$\displaystyle I_{R}=\int_{C_{R}}^{} \frac{e^{iz}}{z^{2}}\, dz,$$ where $C_{R}$ is the semicircle with radius R in the upper half plane with endpoints $(-R,0)$ and $(R,0)$ $(C_{R}$ is ...
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197 views

Julia Set of polynomials

If $f$ is a polynomial and $z\in\mathbb{C}$, show that either $f^n(z)\rightarrow\infty$ or $\{f^n(z) : n\geq 1\}$ is a bounded set. Here, $f^2(z)=f(f(z))$ and $f^n(z)=f(f^{n-1}(z))$ for $n\geq 2$ ...
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239 views

Confusion about $\log(\frac{1+z}{1-z})$ being Analytic on $\mathbb{C}-[-1,1]$

Observe that: $\log(\frac{1+z}{1-z}) = -2\int\frac{dz}{1-z^2}$. (Not precisely true but read on) Supposedly this function is analytic on the domain $\mathbb{C}-[-1,1]$, despite the fact that it's ...
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245 views

Power Series Definition

What does it mean for a series to be centered around a number? I'm taking complex analysis and am suddenly very confused. I didn't have this explanation, or proof of taylor and power series in ...
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271 views

How can I show that arc length $L(\gamma)$ of a curve is unchanged after reparametrization?

Show that the arc length $L(\gamma)$ of a curve $\gamma$ is unchanged if $\gamma$ is reparametrized Can you help me please?
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140 views

Property of Analytic Function

If $f:\mathbb{C}\rightarrow\mathbb{C}$ is analytic and $Im(f(z))\neq 0$ whenever $|z|\neq 1$, show that $f$ is a constant. It sounds familiar but not so trivial at all...
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206 views

Cauchy's Integral Formula and Green's Theorem

I have been re-reading through my complex analysis text and wanted to try something different. Cauchy's Integral Theorem is typically proved using an application of Green's Theorem and then by virtue ...
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132 views

Singularity of Generating Function

Given $E' = (E^2 + E - x)/2xE$ $xF = E^3 E' + 2xE^3 E'' + E^2 - x^2$ where $E = \sum_{n > 0}{e_n x^n}$ with $e_n = (n-1) \sum^{n-1}_{i = 1}{e_i e_{n-i}}$ for $n > 1$ and $e_1 = 1$ I am ...
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109 views

How do I prove that $\sin(π/2+iy)=1/2(e^{y}+e^{−y})=\cosh y$?

How do I prove that $\sin(π/2+iy)=1/2(e^{y}+e^{−y})=\cosh y$? Can you help please?
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144 views

Maximum distance between images of two points under an analytic function

Let $z$ and $w$ be two points in the complex unit disk, and let $f$ be a holomorphic function from the unit disk to itself (i.e. $|f| < 1$). Intuitively, it seems that the maximum value of $|f(z) - ...