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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Analytic function on a domain

If $f(z) + \sin(z)$ is an analytic function on a domain D and $f(z) + \cos(z)$ is analytic on D, then $f(z)$ is constant on D. Is this true?
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25 views

Differentiable cauchy riemann equation

If f is differentiable and |f(z) = 7| in D(0,5) then f(z) is a constant function on this disk D(0,5). Is this true?
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52 views

What is an complex interval $[a,b]$?

I'm asked to give a parametric equation for $[a,b]$ where $a,b \in \Bbb C$. But I don't know what is meant with a complex interval ? Is this is a line? Is this a rectangular? I have no idea, and it ...
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41 views

What is the series expansion of $f(z)\cdot\exp\left({s\,\log(z)}\right)$?

For analytic $f$, how can I represent the expression $f(z)\cdot\exp\left({s\,\log(z)}\right)$, i.e. $f(z)\cdot z^s$ in the form $$\sum_{n}^\infty\left(\sum_{k}^\infty a_k s^k\right)z^n,$$ at least ...
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45 views

Complex Analysis analytic function

If $f$ is an analytic function on a domain $D$ and $\mathrm{Im} f$ takes on only the value $71$ then for some constant $C \in \Bbb{R}$, is it true that $f = C + 71 i$ on $D$?
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27 views

Is this an appropriate strategy for evaluating a complex limit?

$$\lim_{z\rightarrow 0} \frac{\overline{z}^2}{z}=\lim_{z\rightarrow 0}\frac{r^2e^{-2i\theta}}{re^{i\theta}}=\lim_{z\rightarrow 0}re^{-3i\theta}=0$$ I wanted to avoid an $\varepsilon$-$\delta$ proof.
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79 views

compactness in the space of analytic functions

I am always getting confused by the idea of compactness so I would like some help to see whether a set is compact. (I need this to prove existence of a solution of a map) So let $D\in\mathbb{C}$ be ...
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1answer
94 views

The transformations on the nome and Landen's transformation

Could someone please explain how to transform the nome $q = e^{-\pi K'/K}$ from $q^2$ to $q$ and then to $-q$? In other words, how does changing $q^2$ to $q$ and then $q$ to $-q$ affect $k$ and $K$. ...
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49 views

confluent hypergeometric equation

I just proved that by writing $w(z)$ in the form of an integral equation the hypergeometric equation $zw''+(c-w)w'-aw=0$ has solutions in the form $\int_\gamma t^{a-1}(1-t)^{c-a-1}e^{tz}dt$ providing ...
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47 views

Cauchy principal value of $\int_{-\infty}^{\infty}\frac{e^{-x^2}}{x} dx$

I would like to show that $P.V \int_{-\infty}^{\infty}\frac{e^{-x^2}}{x} dx=0$ I do not know why this cant be shown by means of a large semicircle, I proved that $P.V ...
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74 views

Determine whether the following complex limits exist and find their value if they do: $\lim_{z\to 0}\frac {e^z -1}{z}$ [closed]

What is the method of approaching such a problem? $\lim_{z\to 0}\frac {e^z -1}{z}$
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46 views

What is a good book that focuses on the applications of complex analysis and spectral theory?

My research involves a great deal of complex analysis and spectral theory, and I always feel a bit flustered when non mathematicians ask me what I study. It's hard to explain the math in layman's ...
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32 views

Finding Partition, Riemanns Integral

Define $f:[0,2]\rightarrow\mathbb{R}$ by setting $f(x)=1$ if $x\not=1$ and $f(1)=3$. Find a partition $D$ of $[0,2]$ for which $S_D-s_D<2^{-1000}$.
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111 views

Trying to evaluate a complex integral?

I have a question where I must evaluate the following integral over a circle where $\lvert z \rvert = 2$. $$I = \oint \frac{z^3e^{\frac{1}{z}}}{1+z}dz$$ I have tried the $z^3 = 8 \cdot ...
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34 views

What are the poles of this function?

This past exam question has me a little lost. I know I should pick a value for $z$ for which the denominator is zero, but cannot think of more than one pole (there should be more I think because of ...
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35 views

Real-valued Irreducible Representations of Lie Groups

I'm interested in the real-valued irreducible representations of a number of Lie groups. For concreteness I'll use the group $M(2)$ of Euclidean motions, which can be parameterized as follows: $$ g(t, ...
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53 views

How to determine if $f(z)=\frac{i}{z^8}$ is analytic?

How can I prove that a function like $$f(z)=\dfrac{i}{z^8}$$ is analytic or not? I have to use Cauchy–Riemann equations but I can't find the $u$ and $v$ functions.
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28 views

About complex exponential summation

Let $f:\,\mathbb{R}^{+}\rightarrow\mathbb{R},\, f\in C^{\infty}\left(\mathbb{R}^{+}\right)$ and such that $f\left(n\right)>0\,\forall n\in\mathbb{N}$. Let $c>0$ a real number, $N>0$ a large ...
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2answers
29 views

Do I need $(z-3)$ in computing my residue?

I'm asked to evaluate $$\int_{\gamma} \frac{1}{(z^4+1)(z-3)}$$ if $\gamma$ is the circle of radius 2 centered at the origin and travelled once in the counterclockwise direction. Forgive me if my ...
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1answer
45 views

Why is an admissible function from a non-compact surface non-surjective?

I'm at the end of the proof of uniformization for simply connected manifolds in Farkas-Kra's Riemann Surfaces text. I feel like I'm missing something really obvious here. (The proof in question is on ...
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56 views

existence of primitive - diff. of log function

I have this homework problem which i need your help about how to start? g is holomorphic in a simply connected domain U. show that there is a f which is holomorphic in U without zero such that g=f'/f ...
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35 views

A complex integration problem

The problem is: Let $\gamma$ be the circle of radius $R$ centered at $0$. Let $m$, $n$ be positive integers. Prove that, as $R$ goes to infinity, $\int _\gamma\frac{z^m}{z^n+1}dz=O(R^{m+1-n})$. And ...
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188 views

What insight is supposed to be gained from this complex analysis exercise?

Let $C_0$ denote the circle centered around some point $z_0\in\mathbb{C}$ with radius $R$. We can parametrize this circle like this: $$\begin{array}{cc} z(\theta)=z_0+Re^{i\theta}, & \theta \in ...
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31 views

The Group of Complex Continuous Functions?

Let $C(\mathbb{C},\mathbb{C})=\{f:\mathbb{C} \rightarrow \mathbb{C}\,|\,f $continuous $\}$ be the set of all continuous functions from the complex plane to itself and consider the composition ...
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47 views

Primitive of $f(x)$ on the punctured plane where $f(x)$ is analytic

Suppose $f$ is analytic on $D = C\setminus\{0\}$ i.e. the punctured at zero complex plane. Show that there exists a constant $A$ such that $f(z) - \frac{A}{z}$ has a primitive on $D$. Find the value ...
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34 views

About complex sum and modulus

Let $\left(a_{n}\right)_{n},\,\left(b_{n}\right)_{n}$ two succession of non negative real numbers, $\left(c_{n}\right)_{n}$ a succession of complex numbers and $N$ a large natural number. Suppose that ...
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184 views

Closed form for $_2F_1\left(\frac12,\frac23;\,\frac32;\,\frac{8\,\sqrt{11}\,i-5}{27}\right)$

I'm trying to find a closed form (in terms of simpler functions) for the following hypergeometric function with a complex argument: ...
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1answer
64 views

Laurent Series of Weierstrass P-function

This is problem 2.1 in O. Forster, Lectures on Riemann Surfaces, Springer-Verlag, 1981. Let $\Gamma\in\mathbb{C}$ be a lattice. Let ...
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17 views

what is relation between the image of the closure of the unit disk and the closure of the image of the unit disk under an analytic function?

what is relation between the image of the closure of the unit disk and the closure of the image of the unit disk under an analytic function? That is what is the relation between $f(\overline U)$ and ...
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70 views

Bessel functions: proof that $J_0(z)=\frac{1}{\pi}\int_0^\pi e^{i z \cos(\theta)}d \theta$.

I encountered the above when dealing with the Bessel functions of the first kind, $J_n(z)$, specifically $n=0$. Using the differential-equation definition of the Bessel function, I obtained the above ...
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1answer
51 views

Question on $2^N$th Roots of Unity within a function.

Prove that, if $w$ is a $(2^N)$th root of unity, where $N \in \mathbb N$, then: $$\lim_{r\to 1^-}|f'(rw)| = \infty$$ Where: $$f(z) = \sum\limits_{j = 1}^\infty 2^{-j}z^{2^j}$$ I haven't done left ...
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2answers
78 views

Asymptotics of coefficients

This is a question that asks the reader for a $strategy$ to solve a particular problem. I cannot solve this problem myself so I am looking around for general methods one might use to confront it with. ...
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22 views

Complex trigonomteric functions

If $z=x+iy$, I need to prove $$\sin z=\sin x\cosh y+i\cos x\sinh y$$ $$\cos z=\cos x\cosh y-i\sin x\sinh y$$ and from here I need to prove that these are unbounded functions. How should I do that. ...
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1answer
42 views

Calculating a complex integral by rewriting as a contour integral on |z|=1.

I need to show that $\int_0^{2\pi}\frac{d\theta}{2+i\:sin\theta}=\frac{2\pi}{\sqrt{5}}$ I used $sin\theta=\frac{1}{2i}(e^{i\theta}-e^{-i\theta})$ and substituted $z=e^{i\theta}$. I ended up with ...
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1answer
82 views

holomorphic functions, such that $|f|$ depends only on $Re \ z$ and $arg \ f$ depends only on $Im \ z$.

I need to describe all holomorphic functions, such that $|f|$ depends only on $\text{Re}\, z$ and $\text{arg}\, f$ depends only on $\text{Im}\, z$. My thoughts: Let $f=u+vi, z=x+iy$, then $0=d ...
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26 views

Does a lattice in $PSL(2,\mathbb{R})$ stabilizing $\infty$ have a domain with vertex at $\infty$?

Suppose $\Gamma$ is a lattice in $PSL(2, \mathbb{R})$ acting on the upper half plane. Suppose that the stabilizer in $\Gamma$ of the point at infinity is nontrivial. Does it then follow that the ...
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47 views

Show that $\ c_X(p,q) \le d_X(p,q)$, for $ p, q \in X$

Update I'm trying to show the Corollary, but I have stuck...That is: For any complex space $X$, we have: $$\begin{align} (1).\ c_X(p,q) &\le d_X(p,q),\ \text{for}\ p, q \in X \\ (2).\ ...
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1answer
17 views

Convergence of a sequence with nonincreasing real terms on the unit disk

Let $a_0 \geq a_1 \geq ... \geq a_n \geq ...$ Then $\sum_{n=0}^{\infty} a_n z^n$ converges for all $|z|=1, z \neq 1$. My take was this: let $z \in \delta D(0,1)- \{1\}$. Then ...
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1answer
54 views

Nth root of complex number (z)

I have to prove that: Prove that $\displaystyle z^{\frac{1}{n}}=e ^{\frac{1}{n}(\text{Log }z+2k\pi i)}$ gives the $n$th root of $z$, taking $k=0,1,2, \ldots$ Well, with the suggestion of Gerry ...
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94 views

Is this analytic continuation possible?

I'n new to complex analysis and am a little flustered by the following function. I would like help understanding whether or not it is possible to analytically continue it outside of the unit circle. ...
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58 views

Interpretation of the Argument Principle

Recall that the argument principle states that given a meromorphic function $f$ and a compact region $K \subseteq \mathbb{C}$ whose boundary determines a simple contour and on which $f$ has no ...
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33 views

Soft Question about Mobius Transformations

Very soft question and I may be completely wrong about this, but does it make any sense to think about the Mobious transformation matrix as a change of basis for C?
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89 views

On the subject of holomorphic functions on an open disc, D.

The question I am pondering over is an interesting one: If $f(z) = u + iv$ is holomorphic on an open disc $D$, and the range of $f$ lies in either a straight line or a circle, prove that $f$ is ...
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50 views

MVT doesn't extend to complex derivatives

Let $f(z)=z^3$. For $z_1=1$ and $z_2=i$ prove that there doesn't exist any complex number $c$ on the line segment joining $z_1,z_2$, such that $$\frac{f(z_1)-f(z_2)}{z_1-z_2}=f'(c).$$ A general point ...
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1answer
160 views

Cauchy Integral Formula on Boundary

Suppose that I'm trying to evaluate the following integral: $$\frac{1}{2\pi i}\ \int_{C} \frac{cos(\pi z)}{z^2-1}dz $$ And further suppose that C is a rectangle going over $ 2+i,2-i,-2+i,and -2-i$. ...
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1answer
93 views

Showing that a mobius transformation exists

I'm trying to show that a specific Mobius transf. exists, where I have some points that map to some other points. (I don't wanna be too specific here about what goes where, as I don't wanna run the ...
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2answers
79 views

Prove that the entire function $f$ is linear.

Suppose $f=u+iv$ be an entire function such that $u(x,y)=\phi(x)$ and $v(x,y)=\psi(y)$ for all $x,y\in\mathbb{R}$. Prove that $f(x)=az+b$ for some $a\in\mathbb{C},b\in\mathbb{C}$. My approach was: ...
2
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1answer
47 views

Holomorphic function interpolating $e^{-n}$

Consider the question: does there exist a holomorphic function $f$ on the unit disk in the complex plane such that $f\left({1 \over n}\right) = e^{-n}$ ? I came up with an answer but I'd like to know ...
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1answer
61 views

Complex Conjugation question

I had a complex analysis exam yesterday, and one of the questions is bothering me. Suppose $f(z)$ is an entire function. Show that $g(z) = (f(z^*))^*$ is also entire. Here $^*$ indicates complex ...
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3answers
118 views

If $f$ is holomorphic and $\lvert f\rvert$ is constant, then $f$ is constant.

Let $\Omega$ be a connected open set and $f:\Omega\rightarrow\mathbb{C}$ is holomorphic. If $\lvert f\rvert$ is a constant function then we need to show, $f$ is also a constant function. I tried to ...