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

How to prove this?.

let $f:D\to D$ be a holomorphic function on the unity disk $D$. If $f(0)=0$, prove that $|f'(0)|\le1$.
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32 views

Special polynomials having atleast one root on the unit circle

I have the following problem: For each $w\in\mathbb{T},$ ($\mathbb{T}$ denotes the unit circle), consider the polynomial $P_{w,n}(z)=z^{n+1}+z^n-2w$ of degree $n+1,$ where $n\in\mathbb{N}.$ Does there ...
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0answers
33 views

How does $e^{az}$ change when $z$ is shifted $2i\pi$?

I want to evaluate an real-valued integral using residue theory, and the approach is to shift the fraction $\displaystyle\frac{e^{ax}}{1+e^x}$ from the real line up $2i\pi$ in the complex plane and ...
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1answer
123 views

Entire Function

I am facing the following problem. Let $f$ and $g$ be analytic functions in $|z|<1$, with $$f(z)=\sum_{n=0}^{\infty}a_nz^n,\quad g(z)=\sum_{n=0}^{\infty}b_nz^n$$ such that $a_n\geq 0$, $b_n\geq 0$ ...
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1answer
110 views

The sheaf $\mathfrak{S}$ of germs of analytic functions over $D$ is a topological group (Ahlfors)

In Ahlfors' complex analysis text, page 286 he gives the following definition: Definition 1. A sheaf over $D$ is a topological space $\mathfrak S$ and a mapping $\pi:\mathfrak S \to D$ with the ...
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1answer
70 views

Analytic function or not?

Is $f(t) = 1 + e^{2\pi i \phi t}$ a complex analytic function? $t\in\mathbb{R},\phi\in\mathbb{Z},i=\sqrt{-1}$. I know this could be an easy question, but I just want to make sure that the 1 does not ...
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1answer
53 views

Proving that a metric on space of analytic functions is equivalent to compact convergence

Let $U\subseteq \mathbb C$ be open and $\mathscr A(U)$ consist of all analytic functions on $U$. I can easily prove that there exists a sequence $K_n$ of compact sets in $U$ so that ...
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1answer
29 views

Prove that $f(x)=(e^{ix}-e^{iz_0})f_1(x)$ where $f_1(x)$ is also a trigonometric polynomial

Let $f(x)=\sum_n c_ne^{inx}$ be a trigonometric polynomial. It then makes sense to define $f$ on $\mathbb{C}$ by allowing $x$ in this formula to be any complex number. Suppose $f(z_0)=0$ for some ...
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1answer
64 views

$\varphi$ the Laurent series near pole $z_0$, how prove that $f-\varphi$ is analytic near $z_0$?

I'm reading the proof that a meromorphic function on the Riemann sphere must be a rational function, and I think I need to understand Laurent series better. The idea of the proof I'm reading is that ...
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1answer
61 views

Cauchy's Integral Formula and evaluation of an integral

Let D be a simply connected region in $\mathbb{C}$ and let C be a simple closed curve contained in D. Let $f(z)$ be analytic in D. Suppose that $z_0$ is a point enclosed by C. Then ...
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65 views

Weierstrass and Borel summation

In the Wikipedia article on Borel summation, there is the following quote attributed to Gösta Mittag-Leffler: Borel, then an unknown young man, discovered that his summation method gave the ...
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0answers
102 views

Computing the logarithmic derivative of the numerator and denominator of a rational function.

Consider the rational function $R(z)=N(z)/D(z)$ where $N(z)$ and $D(z)$ are polynomials of $z$ with real coefficients. Furthermore, $N(0) \neq 0$, $D(0) \neq 0$, and $N(z)$ and $D(z)$ are relatively ...
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97 views

Establish $\int_0^{\infty} \frac{x^a}{x^2 + b^2}dx = \frac{\pi b^{a-1}}{2 \cos(\pi a /2)}$ when $-1 < a < 1$

My attempt at a solution: (this is homework, btw) Let $f(z) = \frac{z^a}{z^2 + b^2}dz$ then the singularities of $f$ occur at $\pm ib$. $$ Res(f; ib) = \frac{z^a}{z + ib} \biggr |_{ib} = ...
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41 views

Determine residues

Suppose $f(z)=\frac{sin(\pi z)}{z^4-z^2}$. Now I have to determine the residues of all isolated singularities of $f(z)$. The isolated singularities are $z=1$, $z=-1$ and $z=0$ I think. Then I thought ...
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138 views

Deep reason why infinite sheet means logarithm while finite sheet means polynomial?

When reading the "Lectures on Riemann Surfaces" by Otto Forster on page 37, he claimed that Suppose $X$ is a Riemann surface and $f:X\to D^{*}$( $D^*$ is the punctured unit disk ...
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1answer
247 views

Orders of poles/zeros of an even elliptic function

I am reading a proof of the fact that every even elliptic function $f$ with periods $1$ and $\tau$ is a rational function of the Weierstrass $\wp$ function. The proof seems to use this fact often, ...
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2answers
66 views

Defining a branch of a logarithm

Write down a branch of $log(z+i)$ which is holomorhpic in the plane with ${z: Re z = 0, Im z < 1}$ removed. Do I just need to say that $log((1-i)+i) = 0$? Is this enough for a definition? Thanks ...
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1answer
48 views

Is a function-valued matrix analytic if its entries are analystic functions?

Let $\boldsymbol{A}(t):\mathbb{R}\rightarrow\mathbb{C}^{n\times n}$ be a function valued Hermitian matrix. If the entries $a_{ij}(t):\mathbb{R}\rightarrow{C}$ of $\boldsymbol{A}(t)$ are analytic ...
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54 views

Approximation argument for Cauchy's theorem?

In a simply connected region $U$, do functions of the form $\frac{1}{z-a}$, for $a\notin U$, generate all analytic functions in $U$ in some way? For example as limits of linear combinations of these ...
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1answer
274 views

Does De Moivre's Theorem hold for all real n?

I have seen the proof by induction for all integers, and I have also seen in a textbook that we can use Euler's formula to prove it true for all rational n, but nowhere in the book does it say its ...
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1answer
39 views

Holomorphic function and real part of a function

We have $f : U \rightarrow C$. $f$ is holomorphic on $U$ ; the question is : if $f$ is holomorphic on $U$, is $Re(f)$ holomorphic on $U$?
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30 views

An identity with line integrals in complex analysis.

Show that for each $x \in \mathbb R$ and $n \in \mathbb N$, the second equality holds: $$f_n(x) \equiv \frac{1}{\pi}\int_{-\infty}^\infty \frac{e^{-itx}}{2}\left(\frac{\sin t}{t} \right )^n \,dt = ...
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68 views

Convergence of sequence of partial sums using convergence of term

This is Exercise 2 on page 37 of Complex Analysis by Ahlfors. Sorry if this is an easy question. I'm slowly teaching myself using this book, but I cannot seem to figure this one out with so little ...
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1answer
178 views

Show there does not exist a certain holomorphic function with the unit disc?

Show that there does not exist a holomorphic function $f$ on $D(0, 1)$ such that $$ f\left(\frac{1}{n} \right) = \begin{cases} 1+\frac{2}{n} & \text{if $n$ even}\\ 1 & \text{if $n$ odd} ...
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0answers
42 views

perturbative series expansion of integral via complex integration

Define for real $x>0$ and $\epsilon>0,$ the function $$ f(x,\epsilon):= \int_{\epsilon}^\infty \frac{\mathrm{d}s}{s} \frac{1}{\sinh^2 s/2} e^{-sx}. $$ Question: is it possible to compute ...
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183 views

Show whether $\log r$ has a conjugate harmonic function on $\mathbb{C} \setminus \{0\}$

Can someone help me understand this passage in a student-written wiki article? The question is whether $u(x+iy) = \log \sqrt{x^2+y^2}$ has a conjugate harmonic function on $\mathbb{C}\setminus \{0\}$. ...
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81 views

Show that $f(z)=z+a_2 z^2$ is univalent in $\mathbb{D}=\{z∈\mathbb{C}:|z|<1\}$ if and only if $|a_2 | \leq 1/2.$

Show that f(z)=z+a_2 z^2 is univalent in D={z∈C:|z|<1} if and only if |a_2 |≤1/2. My solution: (If part): Suppose f(z)=z+a_2 z^2 is univalent in D. By definition, we know that f(z_1 )=f(z_2 ) ...
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161 views

integrating $\sin(px)\sin(qx)/x^2$

Show that the integral from $0$ to infinity of $\sin(px)\sin(qx)/x^2$ equals $\pi\cdot\min(p,q)/2$, where $p,q>0$. I need to use Cauchy's Residue theorem i think, but I can't see what function to ...
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2answers
160 views

How to find this integral using Cauchy integral formula

How to obtain that $$\int\limits_{|z|=r} (\bar{z})^{-m} z^{-n-1}\, dz = \begin{cases} 2\pi ir^{-2m} &\text{if}\,\,n=m, \\ 0 &\text{if}\,\,n \neq m, \end{cases}$$ for $r>0$. I suppose I ...
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5answers
123 views

Residues at singularities

I have the following question: Show that the integral $$\int_{-\infty}^{+\infty}\frac{\cos\pi x}{2x-1}dx = -\frac\pi2$$ Clearly there is a singularity at $z=1/2$ but I think this is a removable ...
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1answer
82 views

Integration in complex analysis using Residue Theorem

Prove that for $a>0$, $$ \int_{-\infty}^{\infty} \frac{1}{x^4+a^4}dx=\frac{\pi}{a^3\sqrt{2}} $$ I think I'm supposed to use Cauchy's Residue Theorem somehow, but I don't know what closed path to ...
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1answer
314 views

Complex integral over a semi circle

Let $f(z) := \frac{\mathbb{e}^{iz}}{z}$ $z \in \mathbb{C}$ where $0 \notin \mathbb{C}$ I need to show that when $C_k$, a semi circle of radius e is traversed in the clockwise direction is traversed ...
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4answers
172 views

I am trying to show $\int^\infty_0\frac{\sin(x)}{x}dx=\frac{\pi}{2}$

I am trying to show $\int^\infty_0\frac{\sin(x)}{x}dx=\frac{\pi}{2}$ It was an exercise from a book about complex analysis, so I've gone through the complex plane to do it! Consider a semi-circle ...
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2answers
50 views

Answer check $\int_{\lvert z\rvert=1}(z-1)^2\lvert dz\rvert$ = $2\pi$

$$ \int_{\lvert z\rvert=1}(z-1)^2\lvert dz\rvert $$ This one is either simpler or I've have my experience jogged. Let $z=e^{j\theta}$ for $\theta\in[0,2\pi]$ (this is the unit circle, $|z|=1$), then ...
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1answer
44 views

evaluating $\int_{_C}cothz \,dz $ using residue theorem

Evaluate $\int_{_C}cothz \,dz $ where $C$ is the Circle $|z| = 1$. Now $cothz=\frac{cosh}{sinhz} $ And $coshz=\frac{e^z+e^{-z}}{2}, sinhz=\frac{e^z-e^{-z}}{2}$. expanding $e^z=1+z+z^2/2+...$ and ...
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1answer
36 views

Complex integral verification

Z = 4 is not in the circle |z| = 2 so that integral to 0, correct?
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1answer
66 views

An application of Rouché for $z^4 - z + 5$

This is an assignment question I submitted but didn't recieve full marks for, so I'm trying to correct it. My reference is Stein and Shakarchi's Complex Analysis. The question is in three parts, ...
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2answers
77 views

Answer check: $\int_{\lvert z\rvert=1}\lvert z-1\rvert^2 dz$ - stuck, I (HOPE) I've missed a trick

$\int_{\lvert z\rvert=1}\lvert z-1\rvert^2 dz$ where z is a complex number (so the integral over a circle effectively) I'm still working it out (I'm doing lots of show thats in the margin, I've got ...
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52 views

Is there a way to formally describe the (complex) transformation $z\rightarrow\frac{1}{z}$

The transformation $z\rightarrow\frac{1}{z}$ does what I can only (over $\mathbb{C}\backslash\{0\}$ of course) describe as "turning space inside out" (then reflecting it in the real axis) This ...
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1answer
27 views

Can't interpret a matrix multiplication (that somehow applies to the Cauchy-Riemann equations - that is $u_x=v_y$ and $v_x=-u_y$)

NOTE: this is self learning Let $\mathbb{B}$ denote the unit disk in $\mathbb{C}$ centred at the origin. let $f:\mathbb{B}\rightarrow\mathbb{C}$ be holomorphic Suppose that $\lvert f(z)\rvert$ is ...
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1answer
117 views

Residue Theorem with winding number

Let $\gamma$ be a closed path in a domain $D$ such that $W(\gamma,\zeta)=0$ (winding number) for all $\zeta\notin D$. Suppose that $f$ is analytic on $D$ except at isolated singularities ...
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1answer
321 views

Analytic continuation of the Riemann zeta function using contour integration

To find the analytic continuation of the Riemann zeta function using contour integration one can integrate $\displaystyle f(z) = \frac{z^{s-1}}{e^{-z}-1}$ around a contour that consists of rays just ...
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1answer
73 views

Entire function as product of meromorphic and entire functions

Let $\{a_n\},\{b_n\}$ be complex numbers such that $|a_n|\rightarrow\infty$ as $n\rightarrow\infty$. Let $g$ be an entire function with simple zeros at the $a_n$'s, and let $h$ be a meromorphic ...
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1answer
197 views

Two circles intersect orthogonally

Suppose I have two circles in the complex plane $|z - a| = r$ and $|z - b| = s$ (with $a, b \in \mathbb{C}$), are there any ways I can test to see if these two circles intersect orthogonally without ...
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1answer
54 views

Derivative of exp with definition of differentiability

Prove with the definition of differentiability that $\exp(z)$ is differentiable in $\mathbb C$ and $(\exp(z))' = \exp(z)$ for all $z \in \mathbb C.$ I tried: \begin{align*} \frac{\exp(z+h) - ...
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2answers
98 views

Find Laurent series

Let $$f(z):=\frac{e^{\frac{1}{z}}}{z^2+1}$$ and let $$\sum_{k}a_kz^k$$ with k in Z the Laurent series of $f(z)$ for $0<|z|<1$. I have to find a formula for $a_k$. I've tried a lot, but I'm ...
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1answer
66 views

Complex differentiability equivalent to linear approximation

Let $G \subset \mathbb C$ be an open set and $f: G \to \mathbb C$ a complex function on $G$. Prove that the function $f$ is complex differentiable at a point $z \in G$ if and only if there exists a ...
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2answers
44 views

Cauchy formula for polynomials

I'm stuck with this problem, any help appreciated. It says: "If $P(z)$ is a polynomial, prove that $\int_{|z-a|=r}{P(z)d\bar{z}} = -2\pi i r^2 P'(a)$. So far I'm using that $P'(a) = \frac{1}{2\pi ...
3
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2answers
160 views

Roots of $e^z=1+z$ on complex plane

What are the roots in the complex plane of $e^z=1+z$? Clearly $z=0$ is one root. On the real line, we can show that $e^x>1+x$ for all $x\neq 0$. But what about the rest of the complex plane?
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1answer
179 views

How to prove that a complex power series is differentiable

I am always using the following result but I do not know why it is true. So: How to prove the following statement: Suppose the complex power series $\sum_{n = 0}^\infty a_n(z-z_0)^n$ has radius of ...