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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58 views

Use calculus of residues to evlauate

Use calculus of residues to evaluate the integral $$\int_0^{2\pi}\cos^{2n}\theta d\theta$$ My Ateempt : $$\int_0^{2\pi}\frac{(1+\cos2\theta)^n}{2^n}d\theta$$ $$=\frac{1}{2^n} \int_C ...
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62 views

Why in this case $f$ should be entire?

Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be a function. Assume $f$ satisfies follows: $f$ is analytic at a point $z_0$. $\limsup\limits_{n\to\infty} \left|\frac{f^{(n)}(z_0)}{n!}\right|^{1/n}=0$. ...
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48 views

What is the radius of convergence of this series?

Suppose we have this series: $$f(z) = \frac{1}{2z^3} + \frac{1}{12z} - \frac{z}{240}.$$ What is the radius of convergence?
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32 views

what would be an example of a function such that $\int_{|z|=r} f(z)=0$ for all $r>0$, but not analytic everywhere?

what would be an example of a function such that $\int_{|z|=r} f(z)=0$ for all $r>0$, but not analytic everywhere. I cannot think of one..
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434 views

Every non-trivial holomorphic involution on the open unit disc has a unique fixed point

I am trying to prove the following: If $f : \mathbb{D} \to \mathbb{D}$ is a non-trivial biholomorphism and $f\circ f = \operatorname{id}$, then $f$ has a unique fixed point. Uniqueness follows ...
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42 views

Laurent series of $\frac{e^{iz}}{z^2+p^2}$, $ p>0$.

I need help finding the main part of the laurent series of $f(z)=\frac{e^{iz}}{z^2+p^2}$ in $ip,-ip$ since these are the two poles of $f$. Due to the orders of the poles are 1 I just have to find ...
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80 views

Analytic bijective function is either $az$ or $\frac{a}{z}$

I am trying to solve the following problem: Let $\mathbb{C}^* = \{z: 0 < |z| < \infty\}$ and $f: \mathbb{C}^* \to \mathbb{C}^*$, analytic and bijective function. Show that $f(z) = az$ or ...
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2answers
38 views

How do i prove this equation? $\int_C y\,dz = i\int_C x \,dz$

Silverman - Complex Analysis (p.220) Let $C$ be any simple closed contour bounding a region having area $A$. Then $\int_C y \, dz = i\int_C x \, dz$ How do i prove this??
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1answer
113 views

Complex Integral Question

I'm trying to evaluate the following integral, in preparation for my exam tomorrow; $$\int_{0}^{\infty} \frac{\cos(2x) - 1}{x^2} dx$$ However, I'm having a lot of issues with it. I was initially ...
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74 views

When is a real-analytic function harmonic?

I recently learnt that every harmonic function occurs as the real part of a complex analytic function. We also know that every harmonic function is real analytic. So, when is a real-analytic function ...
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1answer
85 views

$\oint_{C}(A-\lambda I)^{-1}\,d\lambda=0$ implies interior of $C$ is in the resolvent.

Suppose that $A$ is a bounded linear operator on a complex Banach space $X$ with resolvent set $\rho(A)$. If $C$ is a simple closed smooth curve in $\rho(A)$ such that $$ ...
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287 views

Simple Branched covering over sphere.

A simple branched covering is a branched covering with branching points of degree at most 2, in some context, it is also required to have at most one branching point in each fiber. My question is ...
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1answer
141 views

Contour Integral of $\int_0^{\infty} \frac{1}{x^4+1} dx $ - Missing a factor of 2

I'm supposed to evaluate: $$ \int_0^{\infty} \frac{1}{x^4+1} dx $$ Consider $$ \oint \frac{1}{z^4+1} dz = \oint \frac{1}{(z - \frac{1-i}{\sqrt 2})(z + \frac{1-i}{\sqrt 2})(z - \frac{1+i}{\sqrt ...
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1answer
30 views

Holomorphic function and series converges in the unit disk $ (|z_{k}| < 1) $

$ f $ is holomorphic in the unit disc , bounded and not identically zero and $z_{1},z_{2},\ldots,z_{n},\ldots $ are its zeros$ (|z_{k}| < 1) $ , $a$ is a real number My question is :for which ...
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1answer
51 views

complex equations question

find all solutions of the equation: $w^4 = -8(1-i\sqrt{3})$ I dont wanna be that guy, but can someone tell me what the second solution to this equation is? cuz the solution manual says it's $-1 + ...
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1answer
72 views

Prove that the sequence ${f_n(z)}_{n=1}^\infty$ converges for all $z ∈ \mathbb{D}.$

Let $\{f_n : \mathbb{D} → \mathbb{D}\}_{n=1}^\infty$ be a sequence of analytic functions such that the sequences $\{f_n(1/k)\}_{n=1}^\infty$ converge for any $k ∈ N, k > 1.$ Prove that the sequence ...
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1answer
89 views

Elliptic functions surjective

Is it true that every nonconstant elliptic function $f:\mathbf{C}/\Lambda\rightarrow\mathbf{P}^1$ is surjective? (I take elliptic functions to be defined on the torus $\mathbf{C}/\Lambda$) For how is ...
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3answers
146 views

Find the principle value and all other values of $i^{2/\pi}$

I'm a little confused about how to go about this? Any help would be appreciated, thanks.
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35 views

Different ways of visualizing a certain classes of single variable complex functions

Single variable complex functions of a real variable are ubiquitous in engineering contexts such as control engineering and signal processing, and visualizing them is of utmost importance in designing ...
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204 views

Contour integration of trigonometric functions

I'm trying to show that: $$ \int_0 ^{\infty} \frac{\sin(px)\sin(qx)}{x^2} dx = \frac{\pi}{2}\min{(p,q)}$$ With $p,q \ge 0$ So far I have considered the function $f(z) = \frac{e^{itz}}{z^2}$ and ...
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178 views

Complex function to plane with non-negative real axis removed must be constant

I'm trying to prove that if I have a holomorphic function $f: \mathbb{C} \to \mathbb{C}\backslash A$ where A is the non-negative real axis then $f$ must be constant. My thoughts so far are to ...
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22 views

analyticity of an integral

Studying $$f(z)=\int_0^1 g(z,x)\ dx $$ where $g(z,x)$ is analytic in the open unit disk $D$ for all $x\in [0,1]$ and continuous for $|z| <1$ and $0\leq x \leq1$.Now, $$\lim_{n\to\infty} [1/n ...
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1answer
160 views

Corollary to Liouville's theorem

I have a fundamental doubt about a concrete step in this Corollary (I've copied the statement and its proof from Stein's textbook): Corollary to Liouville's Theorem Every non-constant polynomial ...
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92 views

Complex Integration: $\int_0^{\infty} \frac{\sin x}{x(k^2x^2 +1)} dx $

I'm supposed to evaluate: $$ \int_0^{\infty} \frac{\sin x}{x(k^2x^2 +1)} dx $$ Attempt Consider $ \int_0^{\infty} \frac{e^{iz}}{x(k^2x^2 +1)} dz $ Simple poles at $z = \pm \frac{i}{k} $, simple ...
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1answer
52 views

Holomorphic function with Taylor coefficients that tend to 0

Suppose $f$ is holomorphic on $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$ and continuous on $\overline{\mathbb{D}}$. If we can write $F(z) = \sum_{n = 0}^{\infty}a_{n}z^{n}$ for $z \in ...
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40 views

What does $|d x|$ denote in $\int_{\gamma} |f| |dx|$?

What does $|d x|$ denote in $\int_{\gamma} |f| |dx|$? I'm not sure how to interpret this notation. Is it $\int_0^1 |f(\gamma(t))| |\gamma'(t)| dt$? In the context where I see it that would give the ...
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1answer
219 views

Ahlfors' extension of Riemann mapping function, proof clarification

In the Ahlfors' Complex Analysis chapter about the Riemann Mapping Theorem section 6.1.3, page 233, he states and proves this theorem: Theorem 3. Suppose that the boundary of a simply connected ...
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2answers
139 views

How to show that $\int_{-\infty}^\infty\frac{t}{(a^2+t^2)(b^2+t^2)(e^{2\pi t}-1)}dt=\frac{1}{2ab(a+b)}+\frac{1}{b^2-a^2}\sum_{a<k\leq b}\frac{1}{k}$

I'm stuck on this problem. Here $a,b\in\mathbb{N}$ with $b>a$. I have already shown that $$-\lim_{\varepsilon\searrow 0}\int_{|t|>\varepsilon}\frac{\coth(\pi ...
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1answer
58 views

Operations with $\text{SL}_2(\mathbb{Z})$

We define $\Omega :=\left\lbrace z \in \mathbb{H}\colon -\frac{1}{2} \leq \operatorname{Re}z \leq \frac{1}{2} \wedge |z| \geq 1\right\rbrace$. I want to show that the following holds: $$ \forall \, ...
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2answers
198 views

Entire function real and imaginary part product

Let $f$ be an entire function such that $\Re(f(z))*\Im(f(z))\ge0$ for all $z$ then $f$ is constant. Prove or give contradicting example. I know about Louisville's theorem and Cauchy–Riemann equations ...
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32 views

The dual of all bounded and holomorphic functions

For an open subset $U \subseteq \Bbb C$, the set of all bounded and analytic functions $f:U \rightarrow \Bbb C$, is a Banach space with $\|.\|$$_{\infty}$. What is the dual of this space? Thanks ...
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1answer
62 views

If $\gamma$ is a path from $0$ to $1$, what do we know about $\displaystyle\int_\gamma\frac{1}{z\pm i}dz$?

Let $\gamma$ denote a path from $0$ to $1$ which doesn't cross $\pm i$. What can we say about $$\int_\gamma\frac{1}{z\pm i}dz$$
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1answer
42 views

Show that $\sum_{k=1}^N\frac{1}{(k+a)(k+b)}=\frac{1}{b-a}\sum_{a<k\leq b}\frac{1}{k}-\frac{1}{b-a}\sum_{a<k\leq b}\frac{1}{k+N}$

I am quite stuck on this problem and I don't know how to proceed. The question states: Let $a,b,N\in\mathbb{N}$, $b>a$, $N\geq b-a$. Show that ...
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1answer
138 views

Isolated Singular point at Infinity of tan z

Out of curiosity, does tan z have an isolated singular point at infinity and why? Thanks for any insight into this.
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1answer
123 views

Find all complex solutions to the equation

i) Find all complex solutions to the equation z^4 +1 -i*3^(1/2) = 0 I basically have no clue, any tips/advice/solutions would be great. I could also need some help with another question, this one ...
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3answers
174 views

Where does one use holomorphicity in the proof of Goursat's theorem?

Goursat's theorem: Let $f : U \to \mathbb{C}$ be a function that is holomorphic on the open set $U$. If $T$ is a triangle in $U$ and $\gamma$ is some smooth parametrization of that triangle, then ...
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2answers
248 views

Contour integral of $\int_0^{2\pi} \frac{1}{A - cos \theta} d\theta$

I'm supposed to evaluate $\int_0^{2\pi} \frac{1}{A - cos \theta} d\theta$ Using a contour of a unit circle, $z=e^{i\theta}$. This is the same as: $$2i \oint \frac{1}{z^2 - Az + 1 } dz $$ The ...
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1answer
135 views

Using the identity theorem to prove existence of non-identically zero function

If we consider the region $U = \{z\in\mathbb{C} : Im(z) \ne 0 \}$ and the sequence $z_n = (1+n^{-1})i$ can we find a holomorphic function that is not identically zero, but is zero at $z_n$? Now I've ...
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126 views

Prove that the functions $g_k(z) = f_k \circ h_k(z)$ form a normal family.

I am having a bit of trouble with the following complex analysis question which originates from a qual. Some help would be awesome. Let $f_k :\mathbb{D} \rightarrow \mathbb{C}$ be a normal family of ...
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1answer
138 views

Green's function for $\frac{\partial}{\partial\bar{z}}$

I've read some complex analysis texts and often there is some appeals to Green's theorems when proving facts about contour integrals of holomorphic functions yet there seems to be a lack of appeals to ...
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1answer
221 views

Calculate $\int_0^\infty\frac{\sin x}xdx$ by integration of a suitable function along given paths [duplicate]

How can I calculate $$\int_0^\infty\frac{\sin x}xdx$$ by integration of a suitable function along the following paths: where $R$ and $\varepsilon$ are the radius of the shown outer and inner ...
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1answer
29 views

Show that $f(a)=\frac{1} {2\pi}{\int_C {\frac{(R^2-a \overline a)f(z)}{(z-a)(R^2-z \overline a)}dz}}$

The function $f(z)$ is regular when $|z|<R'$ Show that if $|a|<R<R'$ then $$f(a)=\frac{1} {2\pi}{\int_C {\frac{(R^2-a \overline a)f(z)}{(z-a)(R^2-z \overline a)}dz}}$$ Where $C$ is the ...
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105 views

Poles of a function involing Gamma- and Beta function

I am interested in the poles of following function of $s$ where $0\leq x\leq1$ and $0\leq \delta < \infty$: $$M(s) = \frac{B(x;\delta+s-1,\delta)}{ \frac{\Gamma(2\delta)}{2 \Gamma(\delta)^2} + ...
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36 views

Linear homogeneous ODE system of first order

Good afternoon. I recently encountered the following problem to which I couldn't find a solution anywhere so far: Given $A:D\to\mathbb C^{2\times 2}$, $D\subset\mathbb C$ open, with holomorphic ...
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1answer
341 views

Möbius transformation image

Let $f(z)=\frac{az+b}{z+d}$, when $d\in\mathbb{R}$, $d\not=0$ $a,b\in\mathbb{C}$ and $f$ is not constant. I want to find the image of the real and imaginary axes under $f$. I've found that the image ...
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194 views

Finding the limit of sum $\sum \frac{1}{n^4}$

I'm trying to use the residue theorem to find the limit of $$\sum_{n=1}^{\infty} \frac{1}{n^4}.$$ So I am considering the function $$f(z) = \frac{\pi \cos(\pi z)}{\sin (\pi z)z^4}$$ on a square ...
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1answer
176 views

The limit function of decreasing sequence of subharmonic is also subharmonic

Let $u(z)$ be a continuous function on a domain $D \subset \mathbb{C}$ to $[−\infty, \infty)$. Suppose $u_n(z)$ is a decreasing sequence of subharmonic functions on $D$ such that $u_n(z) \to u(z)$ for ...
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2answers
87 views

The value of the itegral $\int_{\gamma} \dfrac{dz}{z-a}$ is a multiple of $2\pi i$

I am reading Ahlfors' proof of the lemma: Lemma If the piecewise differentiable closed curve $\gamma$ does not pass through the point $a$, then the value of the integral $$\int_{\gamma} ...
3
votes
1answer
54 views

Question of analysis

If we have a function $f$ which is analytic inside (but not on) the unit disc, can $f$ have infinitely many zeroes inside the unit disc? I feel like it would break some holy law if it did, but can't ...
3
votes
1answer
252 views

Contour integration of $\frac{(\ln z)^2}{z^2+1} $

I'm supposed to take the principal branch of $\ln z$ and evaluate this integral: $$ \oint \frac{(\ln z)^2}{z^2+1} $$ Attempt I suppose the integral they are talking about is something like ...