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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Nontrivial homomorphisms from G to T

Let $G$ be a compact metric abelian group. $T$ be the circle group. Let $\mathcal{A}$ be the set of all finite linear combinations of continuous homomorphisms from $G \to T$. I want to show ...
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49 views

Is this function, a sum of one term and a convergent series, analytic?

$$(\frac{1}{z} + \sum z^n)$$ for 0<|z|<1. This is for complex variables. So, the series, convergent for the above domain of definition, always represents an analytic function. What about the ...
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50 views

A Question on Maximum Modulus Principle

Let $h : \mathbb C \to \mathbb C$ be an analytic function such that $h(0) = 0; h( \frac{1}{2 }) = 5$, and $|h(z)| < 10$ for $|z| < 1.$ Then, conclude that (a) the set $ \{z : |h(z)| ...
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71 views

Using Rouche to find zeros of a polynomial

Given $P(z)=z^6+3z^4+z^2+z+9$, prove that all its zeros are contained in the annulus $1<|z|<2$, and find how many of them are in the first quadrant. I have been able to prove that they are ...
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81 views

L'Hospital's rule for analytic functions of complex variable

Is there L'Hospital's rule for analytic functions of complex variable? If yes where can I find it, in what books, and how to prove it?
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104 views

Does convergence of power series on radius of convergence imply absolute convergence?

Let $R$ be radius of convergence of power seires $\displaystyle\sum_{k}a_kz^k$. If the power series converges for all $|z|=R$, can we say that it converges absolutely on the radius of convergence? I ...
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102 views

Should a certain entire function be a polynomial?

Assume $f$ is an entire function such that $$\lim_{z\to\infty}\frac{|f'(z)|}{1+|f(z)|^2}=0,$$ then should $f$ be a polynomial? Picard's Theorem proves this instantly; which states: Let $f$ be a ...
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67 views

Does a Plancherel-style theorem for the Hardy space $\mathcal{H}^2(\mathbb{T})$ exist?

I am working on a problem regarding Toeplitz operators, and it involves trying to prove $\mathcal{H}^2$ boundedness of the operator (defined in terms of its Fourier coefficients). Now normally when I ...
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46 views

If $f$ is harmonic on $G$ and $f\big|_U=0,$ then $f\equiv0$

Let $G\subset\mathbb C$ be a connected open set, and let $f:G\to\mathbb R$ be harmonic. If there is an nonempty open set $U\subset G$ s.t. $f\big|_U=0$ then $f\equiv0.$ In the proof $N:=\{z\in ...
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29 views

Using term-by-term Integration to solve LaPlace Transforms

I am attempting to use term by term integration to find the LaPlace transform of $$u(t) = \frac{sin(t)}{t}H(t)$$ The LaPlace transform is going to be $\int_0^\infty \frac{sin(t)e^{-st}}{t}$. Every ...
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89 views

Is there a spherical coordinates system for vectors of complex numbers?

Suppose I have a scalar field $f(\vec{x})$, where $\vec{x}\in\mathbb{R}_3$, and I wish to average $f$ over a sphere $|\vec{x}|=R$: $\displaystyle\langle f\rangle_{R} = \frac{\int_{S} f(\vec{x})\, ...
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69 views

Computing an integral using residues

I am trying to find an integral: $$\int_{-\infty}^{+\infty}\frac{e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2}\,\mathrm dx$$ I went about applying contour integral over a semicircle with diameter along $ x = ...
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49 views

Interpreting and understanding the identity $e^{iz} = \cos(z) \pm \sqrt{\cos^2(z) - 1}$

A question in my complex analysis book (Gamelin's "Complex Analysis", question I.8.7) asks me to prove that $e^{iz} = \cos(z) \pm \sqrt{\cos^2(z) - 1}$. Using the identity $\cos(z) = \frac{e^{iz} + ...
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71 views

Construct holomorphic function from harmonic function

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω=\Bbb C \setminus \{0, 1\}$. Show that there exist unique real numbers $a_0, a_1$ such that $$u(z)=h(z)−a_0 \log |z|−a_1 \log ...
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46 views

Finding the coefficients of the Weirestrass $\wp$ function.

I am trying to find the coefficients of the $\wp$-function. Right now I have the Laurent series about the pole $ z = 0$: $$\wp(z) = \frac{c_{-n}}{z^n} + \cdots + \frac{c_{-1}}{z} + c_0 + c_1 z + ...
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136 views

On the importance of the Riesz–Markov–Kakutani representation theorem.

I am following big Rudin and I have arrived at the representation theorem. Before doing the full long proof I would like to know what results are based on this theorem that for completeness I state ...
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351 views

Prove Laurent Series Expansion is Unique

Suppose that $f$ is holomorphic on $A=\{r<|z|<R\}$, where $0\le r<R\le \infty$. Suppose that there are two series of complex numbers $(a_n)_{n\in{\mathbb Z}}$ and $(b_n)_{n\in\mathbb Z}$ such ...
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130 views

what is the the value of $\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}$

If $\frac{a}{a+i}+\frac{b}{b+1}+\frac{c}{c+1}=1$ then what is the the value of $\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}$ here I got $a=0$ and $bc=1$, when $ bc\neq 0$ but then I cant ...
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73 views

Finding residues of rational functions with extremely large powers

$h(z)=\frac{5z^{2015} + 7z^{2010} - 38z^5 + z^4 - 2z^3 + 5}{7z^{2016} + 2z^{2013} - 6z^8 + 3z^7 + z^5 - 4z^2 - z + 111}$ Find the sum of the residues of h at its poles in $C$ How do I find the ...
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31 views

How to recover complex function on $\mathbb C$ from integral equation?

Let $f:\mathbb C \to \mathbb C$ be a continuous function with the form $f(z)= z\tilde{f}(|z|)$ for all $z\in \mathbb C,$ where $\tilde{f}$ is a real function defined on $(0, \infty).$ We define ...
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38 views

A complex analysis problem regarding the stereographic projection

Let $S^1 = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 < 1 \} $. Let $X = (x_1,x_2,x_3), Y = (y_1,y_2,y_3)$ be in $S^1$. Suppose that the angle between the great circular segment from $X$ to $Y$ ...
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120 views

Why is $1/z$ analytic at infinity?

I was given this proof: Let $w(z)=1/z$, so $w$ maps origin to inifinity and infinity to origin. Consider $f(z) = z$. It has no singularities in finite $z$-plane. So $f(w) = 1/w$ has a pole at the ...
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92 views

Real analytic function with radius of convergence 1 at non-negative integers

So, as the title states, the problem I was confronted with was to find a real-valued everywhere analytic function $$f:\mathbb{R}\to \mathbb{R}$$ s.t. at every non-negative integer, k ...
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68 views

Complex Analysis (Complex Mapping) stuck on professor's method of simplification in math notes

I'm having an exam this university semester and need some help with my math notes. I've come accross some problems with the section "Complex Mapping." Link to Image of my Notes: Click Me (see first ...
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111 views

Finding Laurent Series of a function

I've been assigned to write a computer program which then calculates the Laurent series of a function. Of course I'm familiar with the concept, but I've always calculated the Laurent series in an ad ...
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251 views

Is this a legit way to visualize complex functions?

I am doing laplace transform in a class and I hate how there seems to be no graphical support when things are transformed to laplace domain i.e. nobody cares what they look like in laplace domain But ...
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56 views

Cauchy Goursat, why triangles?

Cauchy Goursat: Let $f$ be analytic in a simply connected domain $D$.If $C$ is a simple closed contour that lies in $D$ , then $$\int_C f(z) dz = 0.$$ I've been reading a lot of proofs on this ...
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51 views

Help evaluating this seemingly simple integral using residue theorem

$f(z)$ = $\int$$\frac{1}{2e^z+1}$, this is a path integral along $|Z| = 4$ and i know from my textbook that: My attempt: singularity is at: $2e^z = -1$ therefore at $z = ln|0.5|+2ik\pi$ ...
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39 views

If $f$ is entire and $f \circ (\_)^{-1}$ has a pole at $z = 0$, then $f$ is a polynomial.

I was wondering if somebody could help me finish off the proof of this statement; I'm not sure if my approach can be salvaged, but here's what I've got so far: Since $f \circ (\_)^{-1}$ has a pole at ...
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89 views

Invert a somewhat tricky characteristic function to find density function

I am interested in find the probability density function corresponding to the characteristic function $\phi(t) = \left(\frac{1 - i b t}{1 - i t}\right)^c$ where $c > 1$ and and $0< b < 1$. ...
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50 views

Reversing an “inverse Fourier transform”

Let $g$ be the Fourier transform of an unknown function $y\in L_1(-\infty,\infty)$:$$g(\lambda)=\int_{\mathbb{R}}y(x)e^{-i\lambda x}d\mu_x$$Let $f$ be defined as ...
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64 views

Problems on Hardy's Introduction to Divergent Series

In the introduction to Hardy's last work, "Divergent Series", he writes that $$\frac{1}{1-e^{ix}}$$ generates this sort of power series: $$1+ e^{ix} + e^{2ix} + \ldots $$ According to (1.2.2) of the ...
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60 views

Construction of an explicit series of meromorphic functions.

I have to construct an explicit series of meromorphic functions that converges locally uniformly on the unit disk, and such that it has poles of first order at the points $a_k:=\frac{k-1}{k}$ with ...
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84 views

How to integrate $\int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}$?

In Lancaster & Blundell's QFT book they show that \begin{equation}A:= \int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}\end{equation} returns a nonzero value for $x$, $t$ and $m$ ...
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136 views

Approximation of holomorphic functions and topological properties

So, in the last couple of lectures of my complex analysis class we've proved some approximation theorems of holomorphic functions. Eventually, we showed the following propositions: Theorem 1. Let ...
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69 views

Describing generators for the fundamental group of an elliptic curve given by an equation

Say you're given an equation in the form $y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6$. If the $a_i$'s are complex numbers, the subset $E^*\subset\mathbb{C}^2$ satisfying this equation is a ...
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115 views

Using complex analysis, calculate $I_m = \int_{-\infty}^\infty \frac{dx}{1+x+x^2+\cdots+x^{2m}}$ for $I_2$ and $I_3$

Question: Using complex variables, calculate $I_m = \int_{-\infty}^\infty \frac{dx}{1+x+x^2+\cdots+x^{2m}}$ for $I_2$ and $I_3$. Attempt: With some help, I have determined that the integral is ...
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a question related to using Hurwitz theorem to bound the locations of zeros of Riemann zeta function

Let $F(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$ be the Gamma-complete version of the Riemann $\zeta$ function. Let $f(z)=F(1/2+i z)$. So it is known that all zeros of $f(z)$ are in the strip $S_{1/2}$ ...
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Is the exponential function the one this problem is hinting at?

Suppose that $f$ is holomorphic on all of $\mathbb{C}$ and that $$\lim_{n\rightarrow \infty} \left(\frac{\partial}{\partial z}\right)^nf(z)$$ exists, uniformly on compact sets, and that this limit ...
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162 views

How can one derive Stokes lines of the Stokes phenomenon of asymptotics from a differential equation?

Is there a standard technique to calculate Stokes lines and anti-Stokes lines of the Stokes phenomenon of asymptotics for a function defined as the general solution to a differential equation without ...
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67 views

Klein's invariant and negative discriminant

Let $J = J(\tau)$ be Klein's invariant and let $0 < k < 1$ be the elliptic modulus. It is known that $$J = \frac{4}{27} \frac{(1 - \lambda + \lambda^2)^3}{\lambda^2 (1 - \lambda)^2},$$ where ...
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141 views

Confused about Pochhammer contour?

I know some theorems about complex analysis such as the argument principle. But I do not get the Pochhammer contour. I read about it on the wiki page of the beta function , but I do not understand a ...
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57 views

Problem on Complex numbers involving a point on a Circle

Question: The Complex number $z$ is represented by the point $T$ in the Argand Diagram.Given that $$z =\frac{1}{3+it}$$ where $t$ is a variable, show that i) as $t$ varies, $T$ lies on a circle, and ...
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324 views

contour integral over line segment and parabola

Let $f:z \mapsto \bar{z}^2$ Calculate integral along line segment parabola $y=x^2$ From origo $z=0$ to $z=1+i$ The first one I parametrized $z(t)=t+it, t\in[0,1]$ and $z'(t)=1+i$. Then used ...
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320 views

Rank, degree and slope of a general coherent sheaf

Let $(X,\mathcal O_X)$ be a ringed space and $\mathcal F$ be a coherent sheaf of $\mathcal O_X$-modules on $(X,\mathcal O_X)$. Are there the definitions of rank, degree and slope of $\mathcal F$ in ...
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47 views

Gaussian curvature of a complex projective curve

Let $X \subset \mathbb CP^2$ be a complex curve inheriting metric from $\mathbb CP^2$. Suppose that locally $X$ is given by a holomorphic map $z \to [h_1(z) \colon h_2(z) \colon h_3(z)]$. What is the ...
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47 views

Analytic cohomology on Zariski site vs analytic cohomology on analytic site

If I have an affine algebraic complex manifold (in fact it is Stein), what is known relating the cohomology of analytic sheaves using only Zariski opens vs the cohomology of analytic sheaves using the ...
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137 views

Infinitely many roots $z e^z = a, a\neq 0$

Spent some time trying to tackle this problem. It is supposed to use Rouche's Theorem, but not sure how. Show that $ze^z = a$ for $a \neq 0$ has infinitely many roots. Rouches: (1) $f$ and $g$ ...
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105 views

What is the limit of this sequence of complex numbers?

Let $z_1$ and $z_2$ be two complex numbers in the upper half-plane. Does the sequence $c_n = \exp^n\left(\sqrt{\log^n(z_1) \cdot \log^n(z_2)}\right)$ converge to a fixed point as $n\to\infty$? If so, ...
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86 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...