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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1answer
40 views

partial derivative with respect to $\overline{z}$

In my text on complex analysis, they give the definition of $\frac{\partial f}{\partial \overline{z}}$ for suitable $f : \mathbb{C} \rightarrow \mathbb{C}$. However, I do not understand how to make ...
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21 views

$\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$

$$\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$$ traversing the unit circle counterclockwise. So the singularities are $z=0$ and $z=-2$ but the second is outside the unit circle so it isn't ...
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3answers
45 views

$\int \frac{2+\sin(z)}{z} dz$

Please bear in mind that I am trying to teach myself complex integration having never taken a course in complex analysis, so assume I know very little. $\int \frac{2+\sin(z)}{z} dz$ traversing the ...
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1answer
24 views

Filling gaps in a proof

Consider the smooth map $\gamma:[a,b]\to\mathbb{C}$. Consider a partition of $[a,b]$ $$ P=\{a=t_0<t_1<\ldots,<t_m\} $$ Let's define $$ v=\sum_{k=1}^m|\gamma(t_k)-\gamma(t_{k-1})| $$ and $$ ...
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2answers
23 views

Holomorphic function which is not the derivative of a holomorphic function

I don't know how much knowledge of complex analysis is needed to find an open set $U\subset \mathbb{C}$ and a holomorphic function $f\colon U\to \mathbb{C}$, which is not the derivative of a ...
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0answers
22 views

Compute the order, type and genus of the entire function $\prod_{n=1}^\infty \left( 1-\frac{\sigma(n)}{n^3}z \right) $

Since $$\sum_{n=1}^\infty\frac{1}{(n^3/\sigma(n))}=\frac{\pi^2}{6}\zeta(3)$$ converges, where $\sigma(n)$ is the sum of divisor function (with maximal size a constant times $n\log\log n$), and ...
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0answers
38 views

Proving the uniform convergence of series from the infinite product

I have been trying to prove that this infinite product $(1)$ is uniformly convergent on every compact subsets of $\mathbb{C}\setminus\mathbb{Z}^-$. $$\prod_{m=1}^\infty \frac{\left ( \frac{1}{m}+1 ...
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1answer
42 views

how could we compute this infinite real integral using complex methods?

$\int^{\infty}_{-\infty} \frac{cos(x)}{x^4+1}dx$ I know a similar result, but I'm not sure if I can take it for granted, that $\int^{\infty}_{-\infty} \frac{cos(x)}{x^2+1}dx = \frac{\pi}{e}$ The ...
2
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0answers
29 views

function with branch cuts : the radius of convergence of its Taylor series

Let $f(z) $ being the analytic continuation of some holomorphic function, having many branch points and isolated singularities at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$ is the radius of convergence ...
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1answer
25 views

Suppose $f(z)$ is entire and $Re(f(z))$ is bounded. Show that $f$ is constant

There is a hint that states it might be helpful to consider $exp(f(z))$. I don't see why having a real part would imply that the function is constant.
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3answers
47 views

Finding the antiderivative of $\exp(\cos(z))$ on $\mathbb{C}$

Does this function have an antiderivative on the complex plane? How can this be proven? And if it does, how can we compute it?
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1answer
39 views

$\Gamma(1/2)$ and the Euler reflection formula

With Euler's reflection formula we can show that $\Gamma(1/2)^2=\pi$. Why can't $\Gamma(1/2)=-\sqrt \pi$ ?
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2answers
28 views

show that $f(z) = \overline{z}^2$ has no antiderivative in any nonempty region

It's been a few quarters since I've taken complex analysis and I'm reviewing for a comprehensive exam. I ran into this problem on a sample exam and it stumped me. I'm guessing I would have to do ...
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1answer
31 views

Characteristic function of a random variable $X$ is absolutely continuous

Let $(\Omega, \mathcal{F}, P)$ be a measurable space and $X:\Omega \to > \mathbb{R}$ a random variable. Assuming $E[|X|]<\infty$, prove that $\psi_X(t)=E[e^{itX}]$ is absolutely ...
2
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1answer
49 views

Determineing the largest number such that the Laurent series of converges for a trig function.

Question How to determine the largest number $R$ such that the Laurent series of $$f(z)= \dfrac{2sin(z)}{z^2-4} + \dfrac{cos(z)}{z-3i}$$ about $z=-2$ converges for $0<|z+2|<R$? Attempt : Its ...
1
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1answer
41 views

How to find the Laurent series expansion of an exp function.

Question: How to find the Laurent series expansion in powers of z of a) $f(z)= \dfrac{e^{z^2}}{z^3}$ $\text{where} \left| z \right| > 0$ Attempt: I know that the main idea is to ...
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0answers
71 views
+50

Inequality with analytic functions on the unit ball

Let $g(z) = \sum_{n\geqslant 0} a_nz^n$ be an analytic function where $a_n$ only take values in $\{0,1\}$ (not sure if it is a necessary condition, it is just the case I'm considering). Let ...
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0answers
24 views

Find the complex integral over a path

I have to integrate the complex function $$z+1/z$$ which is parameterized by $\gamma(t), 0 \le t\le 1$ and satisfies $Im\gamma(t) > 0$, $\gamma(0) = -4+i$ and $\gamma(1) = 6+2i$. Can I assume the ...
2
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1answer
53 views

Integrating a Complex Exponential Function

Suppose $w=\exp(2i\pi/3)$. How would I go about integrating $$\int\frac{3dx}{e^x+e^{wx}+e^{w^2x}}$$ Is there a transformation i can use? This is an entire function; there is no $x$ that will ...
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1answer
37 views

Prove that $f^{2016}(z)=z(z-1)\ldots(z-2015)$ for an analytic $f$.

Let $\Omega= \mathbb{C}\setminus \{x+iy:~x\in \mathbb{N},~y\geq 0\}$. Prove there there is an analytic function $f$ on $\Omega$, such that $$f^{2016}(z)=z(z-1)\ldots(z-2015).$$ I don't know ...
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0answers
24 views

Question about using the Cauchy-Riemann equation and showing complex differentiability?

Define $f(x+iy)=u(x,y)+iv(x,y)=xy^3$ then we have $u(x,y)=xy^3$ and $v(x,y)=0$. It follows that $u_x=y^3,u_y=3xy^2$ and $v_x=v_y=0$ so the Cauchy-Riemann equations hold for $(x,0)$ where $x \in ...
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34 views

How to prove the Complex problem? [closed]

For any complex $z_0$,show that $$\lim_{z \to z_0} (z^2+1)=z_0^2+1$$
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27 views

It is possible to prove that the inverse of every periodic function is a multivalued function?

As I state in the title, I wonder if is possibile to prove that the inverse of every periodic function is a multivalued function. First of all I can't found a counterexample for the statement, and ...
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1answer
37 views

How to prove $\lim_{z\to 0} \frac{z^2}{\overline{z}}=0$ using the definition of limit? [closed]

How to prove $$\lim_{z\to 0} \frac{z^2}{\overline{z}}=0$$? We can use definition of limit.
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23 views

Finding a function given its residues

I was given this question : The only singularities in C (the complex set) of the analytic function $f$, are simple poles at z=1 and z=2, with residues at these poles equal to -3 and 7 respectively. If ...
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27 views

Complex variable limit at infinity

Is $\lim\limits_{z\to\infty} \frac{4z^2}{(z-1)^2}$, $z\in\mathbb{C}$, evaluated the same way as a real variable function limit? Or does one need to show separate cases for $x\to\infty$ and ...
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3answers
35 views

Why does Dirichlet Series of Mangoldt Function has simple pole of order 1 at s = 1

Could someone explain why $\sum_{n=1}^\infty \frac{\Lambda(n)}{n^s} = -\frac{\zeta'(s)}{\zeta(s)}$ has a first order pole at $s=1$ with residue 1? That's what I found from Apostol's Introduction to ...
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1answer
28 views

$f_n$ converges uniformly on $\partial \Omega$ then $f_n$ converges uniformly on $\bar{\Omega}$

The problem states that $f_n$ is a sequence of functions which are continuous on the closure of $\Omega$ and holomorphic on $\Omega$ where $\Omega$ is a bounded region and were asked to show that if ...
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2answers
33 views

Laurent series and convergence [duplicate]

"Assume that a complex function $f(z)$ is regular in a neighborhood of $z = 0$ and satises $$f(z)e^{f(z)}= z$$ Write the polynomial expansion of $f(z)$ at $z = 0$ and find its radius of convergence." ...
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3answers
104 views

Is this an equivalent statement to the Fundamental Theorem of Algebra?

Is the following equivalent to the usual statement of the fundamental theorem of algebra: Let $$f(z)=c_nz^n+\cdots+c_1z+c_0$$ be a polynomial with complex coefficients. For all but finitely many ...
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0answers
25 views

Bounding of series

Please help - I tried writing the LHS term as a sum and was hoping to find a 'known' convergent sequence that would bound it, but haven't had any luck. Let R > 1. Show that there is some M > 0 such ...
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2answers
65 views

Unique solution of the equation $ f (x) f (y)=f (p) f (q) $ given $x+y=p+q$ and $ x^2+y^2=p^2+q^2 $

Unique solution of the equation $ f (x) f (y)=f (p) f (q) $ given $x+y=p+q$ and $ |x|^2+|y|^2=|p|^2+|q|^2 $ $\quad$ $f(x) \ge 0$ $ x, y,p, q $ are vectors (Euclidean space) that is each vector has ...
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0answers
18 views

Determining the E.G.F from an Umbral Type Recurrence Formula

Suppose I have the recurrence formula $$\left(A+\frac{2}{3}\right)^n+w_3^2\left(A+\frac{1+w_3}{3}\right)^n+w_3\left(A+\frac{1+w_3^2}{3}\right)^n=0; A_0=1, A_1=-\frac{1}{3}$$ where ...
1
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2answers
57 views

Can a holomorphic function vanish on an infinite 2D rectangular grid?

Consider a set of values $x_{n,k}=n+ik$, where $n,k\in\mathbb Z$. Does there exist a non-constant holomorphic function $f$, such that $f(x_{n,k})=0$ for all $n,k$? I've tried to construct such a ...
3
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1answer
38 views

Characteristic functions of random variables are non-negative definite

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X:\Omega\to\mathbb{R}$ a random variable. How to prove that the characteristic function $\varphi_X(t) = E[e^{itX}] ...
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0answers
18 views

Bound for series in Gamma proof

Please help. I am rather stuck with this problem (and our lecturer says she doesn't know how to do this either). It is part of a proof that the Gamma function is holomorphic, but was omitted from the ...
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0answers
26 views

Jacobi modulus and Weierstrass $\wp$

Let $0 < k < 1$ and $$K := \int_0^{\pi/2} \frac{1}{\sqrt{1 - k^2 \sin^2(\theta)}} \, \mathrm{d}\theta, \; \; K' := \int_0^{\pi/2} \frac{1}{\sqrt{1 - (1-k^2) \sin^2(\theta)}} \, ...
2
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0answers
40 views

Question about proving Cauchy's integral theorem

In my course we were given several proofs of Cauchy's theorem, each at various points in the course, each version stronger than the previous. I'd like to learn a proof of the theorem, so naturally ...
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1answer
63 views
+50

closed path, winding number, Jordan contour

If $ D$ is a domain in $\Bbb C$, $z_0\in \Bbb C\setminus D$, and $\gamma$ is a closed, piecewise smooth path in $ D$ for which the winding number $n(\gamma, z_0)\ne0$, show that there is a Jordan ...
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43 views

An analytic function is the identity function

Let $\Omega$ be a bounded domain and $\phi:\Omega \rightarrow \Omega$ a conformal mapping. Let $P\in \Omega$ be such that $\phi(P)=\phi'(P)=1$. Show that $\phi$ must be identity.
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45 views

$\int_{-\infty}^\infty \frac{dz}{z - z_0}$ by contour integration

Consider the integral $\int_{-\infty}^\infty \frac{dz}{z - z_0}$. It has a simple pole at $z = z_0$. Assume $\Im (z_0) < 0$ so the pole is in lower half-plane. Divide $$ \oint_{C_0} = \int_{-R}^R ...
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0answers
26 views

Biholomorphic map to a belt. Why?

The exercise is this: We have a set A = {z | |z-i|>1 and Im(z)>0} We want to find a biholomorphic map f that would map that area A into a belt. So the solution is apparently f(z) = 1/z which maps it ...
2
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1answer
27 views

Does there exist a conformal $\phi: D\rightarrow\Omega\cup\{\infty\}$?

Let $\gamma$ be a Jordan curve and $\Omega$ the unbounded connected component of $\mathbb{C}\setminus\gamma$. $\Omega$ is not simply connected in $\mathbb{C}$, but $\Omega\cup\{\infty\}$ is simply ...
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0answers
21 views

Complex variable, multiplication of numbers

Question: Let a and b be complex numbers with $a \neq 0.$ Describe the set of points $az + b $ as $z$ varies over the first quadrant, $\{z = x+iy: x>0 \,and \,y>0\}$ Solution: Let $a = ...
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1answer
39 views

“Direct” derivation of exponential form of the Riemann zeta function.

There is the identity $$ \zeta(s) = \exp\left(\sum_{n=2}\frac{\Lambda(n)}{\log(n)} n^{-s} \right) $$ for $\Re(s)>1$. Apparently there are quite a few possibilities to derive this. I am out to "try ...
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0answers
46 views

Claim about holomorphic extension

Prove or disprove the following claim. "For all continuous $f : S(0, 1) \to R$, there is a holomorphic $g : B(0, 1) \to C$ which extends to a continuous ${h : \overline {B(0, 1)}} \to C$ such ...
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1answer
47 views

Conformal map is an isometry

I have the upper half-plane $\mathbb H$ with the metric given by $$\mathrm ds^2=\frac{1}{y^2} (\mathrm dx^2+\mathrm dy^2)$$ and the unit disk $\mathbb D$ with the metric given by $$\mathrm ...
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1answer
24 views

Resolvent $R(\lambda,A)x \to 0$ as $|\lambda| \to \infty$

If I have a closed operator $A:D(A) \to X$, not necessarily bounded on a Banach space $X$, and the resolvent is unbounded, can I show for a fixed $x \in X$ that $$R(\lambda,A)x \to 0$$ as ...
1
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3answers
60 views

Does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$?

Let $S^1:=\{z \in \mathbb C:|z|=1\}$ ; does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$ ?
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1answer
36 views

On a simple application of Paley-Wiener theorem and related doubts

Let $$F(x)=\frac{ \left\{ x \right\} }{e^{\sqrt{x}}},$$ be supported on $ \left( 0,\infty \right) $, where $ \left\{ x \right\} $ is the fractional part function. Then $F\in L^2(0,\infty)$ and the ...