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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What is reflection across parabola?

Reflection across a line is well familiar, reflection across a circle is the inversion, the point at a distance $d$ from the center is reflected into a point on the same ray through the center, but at ...
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Relation between runge domain and polynomial convexity

Are these concepts the same? Just to state the definitions Definition 1 A domain $\Omega \in \mathbb{C}^n$ is a Runge domain if every function $f \in H(\Omega)$ can be approximated, uniformly on ...
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71 views

When is $F(a)=\int_0^af(x)\mathrm{d}x$ holomorphic?

Let $f : \mathbb{C} \rightarrow \mathbb{C}$ and let $\gamma_a$ be a continuous family of paths in the complex plane going from $0$ to $a$. Which restrictions have to be imposed on $f$ to make ...
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1answer
17 views

Sequence of holomorphic functions and approximation by polynomials.

Let $\Omega=\{ z\in \mathbb{C}:$ $Im$ $z>0,$ $|z|>1\}\cup\{z \in \mathbb{C}:$ $Im$ $z<0$ $|z|>1\}$ I know that since $\hat{\mathbb{C}}\setminus \Omega$ is connected there's a sequence of ...
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26 views

Apply the Cauchy-Goursat theorem to show that $\int_C \operatorname{Log}(z+2)\, dz=0$ on a unit circle.

Cauchy-Goursat theorem. If a function $f$ is analytic at all points interior to and on a simple closed contour $C$, then $$\int_C f(z) \,dz=0.$$ This is a problem from Churchill's Complex Variables. ...
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28 views

Find and classify the singularities of $f(z)$

Let $$f(z) = \frac{e^z\sin(3z)}{(z^2-2)z^2}$$ Find and classify the singularities of $f(z)$. So far I have that there are singularities at $0$, $\sqrt{2}$ and $-\sqrt{2}$. Are these correct? ...
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26 views

Differentiability on $Re(z)^2$

Suppose that $l(z)=Re(z)^2$. Is $l$ differentiable at $z_0 \in \mathbb{C}$? What is $f'(z_0)$? Where is $l$ differentiable? Write $w=u+iv$ and $z_0=x_0 +iy_0$. So $$\lim \limits_{w \rightarrow 0} ...
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1answer
21 views

Find and classify singularities

Let $f(z)=\frac{e^{-z}\sin(2(z-1)^2)}{(z^2-4)(z-1)^2}$. Find and classify the singularities of $f(z)$. So far I have that there are singularities at $1$, $2$ and $-2$ $2$ and $-2$ are simple ...
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369 views

Lie group heuristics for a raising operator for $(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$

Consider the fractional integro-derivative $\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=FP\frac{1}{2\pi ...
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1answer
16 views

Boundary preserving map

Let $K\subseteq\mathbb{R}^2$ be a compact set. Is it true that for a continuous map $p:K\to\mathbb{R}^2$ we have: $p(\partial K)=\partial p(K)$? Are there any generalizations? P.S. Note that ...
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16 views

Runge's Theorem for meromrophic functions

Is there a name for this extension of Runge's theorem? Theorem: Let $K\subset\mathbb{C}$ be compact, and let $A\subset K^c$ be a set which intersects each component of $K^c$. Let $f$ be meromorphic ...
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28 views

Multivalued Functions for Dummies

I have been studying complex analysis for a while, but I still cannot "get" how multivalued functions work. Despite having it explained to me many times, my brain cannot process it. For example, I ...
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1answer
29 views

Existence of analytic function on a unit dsic (Converse of Schwaraz Pick Lemma )

(Schwarz - Pick Lemma) Suppose that $f$ is analytic on the Unit Disk $\triangle$ and satisfies the following two conditions : (1) $|f(z) \leq 1$ for all z $\in \triangle$ (2) $f (a) = b $ for some ...
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1answer
25 views

Function poles and divergence of series

Yesterday I tried to calculate the residues of a function the way below, but soon I realized it won't work. Now I have a question about the poles of a function, and a series representing it. $$z\in ...
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22 views

Complex limit proof

Definition: Say $f: S \longleftrightarrow \mathbb{C} \longleftrightarrow \mathbb{R}^2$, $z_0 \in S$, $l \in \mathbb{C}$. We say $$\lim \limits_{z \rightarrow z_0} f(z) = l_0$$ if $\forall \varepsilon ...
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1answer
73 views

Proving an entire function is constant

I'm trying to prove that the entire functions such that \begin{equation*} n^2f(1/n)^3+f(1/n)=0 \end{equation*} for all $n\in\mathbb{N}$, are constant. I suppose I should prove that $f$ is bounded ...
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1answer
16 views

Derivative of a complex conjugate

I anticipate that this is a stupid question, but suppose $c \in C$. What is $\frac{\partial c^{*}}{\partial c}$? I've been trying and failing for about an hour to figure it out from the definition of ...
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49 views

residue of a contour integral with a branch point on the boundary

I am considering a problem where I would like to find the contour integral given by \begin{align} \oint_C f(z) dz \end{align} where $f = u+iv$. $C$ is the wedge shaped contour where $0 \leq r \leq ...
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37 views

problems with singularity $0$ of $\int_{W} \frac{e^{\frac{1}{z}}}{(z-3)^3} dz$.

I have the complex integral \begin{equation*} \int_{W} \frac{e^{\frac{1}{z}}}{(z-3)^3} dz \end{equation*} on a circle with radius $6$. Obviously we have two singularities, one in $0$ and one in $3$. ...
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3D surface intersections

I tried to look at 3D Hypersurface intersections of 4D this way based on four Mathematica (circular) trigonometric parametrization combination selections. No hyperbolic functions are directly ...
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1answer
325 views

Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
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1answer
330 views

Equality in the Cauchy-Bunyakowsky-Schwarz inequality for a semi-inner product

I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product. Definition. Suppose that $\mathscr X$ is a vector space over ...
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3answers
37 views

Radius of convergence of power series of complex $\log$

Let $f(z) = \log(z)$ for $z\in \Bbb{C}\setminus (-\infty,0]$. Since $f$ is holomorphic on its domain, we know it has a power series development about each point $z_0\in \Bbb{C}\setminus (-\infty,0]$. ...
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1answer
13 views

Contour integral of non analytic exponential function

The value of the integration of the function $f(z)$ over the circle of radius 3 centered at $z=1$, where $f(z) = e^{\frac{-1}{(z-1)^2}}$ this function has a pole at $z=1$ of $2$nd order. I don't ...
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1answer
15 views

$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$ onto upper half space $\{\,z\in \mathbb C : \operatorname{Im}(z)>0 \,\}$

I am search an one to one mapping that maps the domain $$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$$ onto upper half space $$\{\,z\in \mathbb C : \operatorname{Im}(z)>0\, ...
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2answers
74 views

Is entire function a polynomial? [duplicate]

Let $f:\mathbb{C}\to \mathbb{C}$ be an entire function, and suppose that for every $z\in \mathbb{C}$ there exists $n_z\in \mathbb{N}$ such that $f^{(n_z)}(z)=0$. Is $f$ necessarily a polynomial?
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2answers
64 views

Entire function with vanishing derivatives?

Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be an entire function. And assume that at each point, one of it's derivatives vanishes. What can you say about $f$? A hint suggests that $f$ must be a ...
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1answer
71 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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1answer
26 views

General Solution of ODE (complex eigenversion)

I am trying to figure out the general solution to the following matrix: $ \frac{d\mathbf{Y}}{dt} = \begin{pmatrix} -3 & -5 \\ 3 & 1 \end{pmatrix}\mathbf{Y}$ I got a solution, but it is so ...
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1answer
11 views

Classification of conformally equivalent annuli via periods

How does one define the periods that appear in this question and show they are conformally equivalent? Are the details worked out in a textbook somewhere? Presumably we do something like take the ...
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1answer
17 views

Normal Family complex

I search about a family of holomorphic complex function that not normal but their derivative is normal. the definition of normality similar to Montel's Thrm
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1answer
31 views

Given $z$ show that $\left | z\right | = 2\sin\theta$ and $\arg z = \theta$

I've been attempting this complex-related question but couldn't quite crack the challenge. (b) Given that $$z={1-\cos 4\theta+i\sin 4\theta\over\sin 2\theta+2i\cos^2\theta},$$ show that ...
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1answer
16 views

Complex Analysis Triangular Inequality

I recently started learning Complex Analysis and as of now don't have much command over it, I am stuck up with this assignment Question of mine which is as follows: If $|Z_i| < 1$ and $V_i ≥ 0$ ...
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1answer
13 views

Orthogonal complex matrices: polar decomposition

Is there a decomposition of $SL_n(\mathbb C)$ as a product of $O_n(\mathbb C)\times Sym_n(\mathbb C)$ ? I mean is there a result as the polar decomposition but with orthogonal (not unitary)? thanks ...
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1answer
295 views

Solution to equations involving Plasma dispersion function

I am trying to solve an equation involving a complex argument for the plasma dispersion function as: $z = x + \iota y$, $ x = \omega / \sqrt2 k v_{Ti} $ $ y = \nu_i /\sqrt{2} k v_{Ti} $ $S[z] = ...
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0answers
25 views

Complex Analysis Modulus Property

My teacher has asked an extra question apart from the one mentioned below in the picture: What will happen if |z| = |w| = 1 ? With respect to following answer: I don't see any point in that ...
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3answers
83 views

Given $Re\{f'(z)\}$, to find $Im\{f(z)\}$

An analytic function $f(z)$ is such that $\Re\{f'(z)\} =2y$ and $f(1+i)=2$. Then the imaginary part of $f(z)$ is $-2xy$ $x^2-y^2$ $2xy$ $y^2-x^2$ Here, by using Milne Thomson's ...
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1answer
46 views

Integration of hyperbolic functions. [on hold]

Kindly solve this integral. I shall be very grateful. $$ \int_0^{\infty}\frac{\mathrm{e}^{-x}}{\mathrm{e}^{ax}-1}dx $$ Thanks.
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1answer
46 views

Computing $\int_{\partial S} \frac{1}{1+z^n} dz$

Let $S=\{re^{it} : 0<r<R, 0< \varphi < 2\pi/n\}$ for some $R>1$ and $n\geq 2$. How can we compute $$\int_{\partial S} \frac{1}{1+z^n} dz?$$ I can't compute it directly, so I assume I ...
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1answer
142 views

Are multi-valued functions a rigorous concept or simply a conversational shorthand?

In Brown and Churchill's book, the concept of multivalued functions is not discussed in a very rigorous way (if at all). But I can see that branch cuts have importance in complex analysis, so I want ...
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24 views

Beyond Schwarz Lemma

Let $f(z)=a_1z+a_2z^2+a_3z^3... $ be a Schwarz function then by lemma $|a_1|\leq 1 $.But what is known about the higher coefficients? For example ; what can be said about $ max [a_1+a_2]$ ? Is there ...
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1answer
17 views

Establishing a Variant of the Mean Value Property of Harmonic Functions

Let $u:U\to \mathbb{C}$ be harmonic and $\overline{D}(P,r)\subset U$. Verify the following variant of the mean value property of harmonic functions: $$u(P)=\frac{1}{2\pi r}\int_{\partial ...
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31 views

Computing an exponential generating function from the first few terms

The current question is related to this one, and this other one. I have a number sequence, and I want to find generating ...
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1answer
33 views

Non-constant entire function-bounded or not? [duplicate]

Show that if $f$ is a non-constant entire function,it cannot satisfy the condition: $$f(z)=f(z+1)=f(z+i)$$ My line of argument so far is based on Liouville's theorem that states that every bounded ...
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1answer
45 views

A consequence of Schwarz lemma

Suppose that for some $\epsilon>0$ the function $f$ is holomorphic on $B(0,1+\epsilon)$ such that $f(a) = 0$ and $|f(z)|\leq1$ if $|z| \leq 1$. Prove for $|z| \leq 1$: $$|f(z)|\leq ...
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1answer
23 views

Maximum /Minimum Modulus theorem for Harmonic Function ( Corollary 6.16 )

Suppose thatt $u(x,y)$ is a real valued non constant harmonic function on a bounded domain D. Then $u(x,y)$ can not attain its maximum or minimum value in $D$. I am studing complex ...
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2answers
1k views

Evaluating $\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1}$ using the inverse Mellin transform

Inspired by this post, I'm trying to evaluate $\displaystyle \sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1} = \frac{1}{24} - \frac{1}{8 \pi}$ using the inverse Mellin transform. But my answer is twice ...
3
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1answer
183 views

How to find the inverse Mellin transform?

On the wikipedia page http://en.wikipedia.org/wiki/Mellin_transform The second formula is an integral transformation for the inverse Mellin transform. Being new to integral transforms, I wonder how ...
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2answers
5k views

Show that $\sin(z)$ is analytic

I have show that the function $\sin(z)$ satisfies the Cauchy-Riemann equations but don't know where to go from here. If it saves some working for you they are $$du/dx=-\sin(x)\cosh(y)$$ ...
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21 views

Hölder continuity of measure associated to Nevanlinna function

Let $F$ be a Nevanlinna function and $\mu$ the (via Stieltjes inversion formula) associated measure, which is a finite Borel measure on $\mathbb R$ and let $C(\lambda)$ be the function ($\alpha \in ...