Tagged Questions

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
306 views

Laurent expansion of $\csc^2(\frac{\pi}{z})$ about $\frac{1}{3}$ for $|z-\frac{1}{3}| \lt \frac{1}{12}$

The question is: Find the Laurent expansion of $\csc^2(\frac{\pi}{z})$ about $\frac{1}{3}$ for $|z-\frac{1}{3}| \lt \frac{1}{12}$. In particular what is the coefficient of $(z-\frac{1}{3})^{-2}$. I ...
2
votes
1answer
59 views

Radius of convergence of $ f(z)=\frac{z^2} {e^z+1} $ without expansion

I want to find the radius of convergence of $ f(z)=\frac{z^2} {e^z+1} $ w.r.t $0$ without using Taylor's expansion. My work, $f(z)$ has poles at $z=(2k+1)i\pi,k \in \mathbb Z$, so the poles nearest ...
3
votes
1answer
155 views

Prove the inequality?

Let $f$ be an analytic function in the unit disc without zeros satisfying $|f|\leqq 1$. Prove that $$ \sup_{|z\leqq{1/5}|}|f(z)|^2\leqq \inf_{|z|\leqq{1/7}}|f(z)| $$ Help me please. These questions ...
0
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1answer
198 views

Radius of convergence of $\sum_{-\infty}^{\infty}3^{-|n|}z^{2n}, z \in \mathcal{C}$

I want to find out the radius of the following power series of a complex variable: $\sum_{-\infty}^{\infty} 3^{-|n|} z^{2n}, z \in \mathbb{C}$ The ration test $\lim_{n \to ...
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2answers
1k views

Taylor series expansion of $\log[z]$ about $z=1$ (different branches)

I realize this is not the fastest way of getting a Taylor's series expansion of $f(z)=\log(z)$ about $z=1$. But here goes. I am assuming I am working on the principal branch of the logarithm ...
2
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1answer
215 views

Error in Proof of Residues?

I wanted to prove that the function $$F(z) = \frac{z-\sum_{j =2}^{n-1} z^j}{1-\sum_{k=1}^{n} z^k} $$ will only contain simple poles. Is the following proof correct? Which implies that $z_o$ ...
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2answers
1k views

Determine and classify all singular points

Determine and find residues for all singular points $z\in \mathbb{C}$ for (i) $\frac{1}{z\sin(2z)}$ (ii) $\frac{1}{1-e^{-z}}$ Note: I have worked out (i), but (ii) seems still not easy.
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1answer
55 views

Interpolation with analytic functions

For $i \in \mathbb N$, Let $x_i, y_i \in \mathbb C$ with the $x_i$ distinct and having no limit point in $\mathbb C$. Is it true that there is an entire function $f$ such that $f(x_i) = y_i$ for all ...
1
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0answers
126 views

Series expansions of branches of multivalued functions

In David Wunsch's Complex Variables with Applications, Example 3 on page 266 asks the reader to find a Maclaurin expansion of $f(x)=(z+1)^{1/2}$ where the principal branch is used. The principal ...
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1answer
199 views

Is manifold mapping degree equal to algebraic degree for polynomials?

If $M$ and $N$ are oriented $n$-manifolds and $f: M \to N$ then the degree of $f$ is given by $$ \deg f = \sum_{p \in f^{-1}(q)} sign_p f $$ where $q$ is a regular value and the sign is $+1$ if $f$ is ...
1
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1answer
53 views

Intuition Behind Krantz Theorem

The theorem I'm referring to is as follows: Let $z_0$ be a root of a nonzero holomorphic function $f$ , and let $n$ be the least positive integer such that, the $n$-th derivative of $f$ evaluated ...
2
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4answers
92 views

Asymmetry in the complex plane.

When $\mathbb{C}$ is viewed as a topological field, it has precisely two automorphisms, namely the identity automorphism, and complex conjugation. Thus we can flip the complex plane upside down ...
3
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1answer
315 views

If a rational function is real on the unit circle, what does that say about its roots and poles? Clarification

I'm also self studying the Ahlfors Complex Analysis book. A question asks: Suppose $R(z)$ is some rational function which is real on the circle |z|=1 in the complex plane. The question asks, how ...
0
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1answer
90 views

Example of analytic function in unit disk in no class $H^p$ but with non-tangential limits at almost every point of unit circle

Given an example of an analytic function in the unit disk which is in no class $H^p$ but which has non-tangential limits at almost every point of the unit circle?
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1answer
117 views

Show $z\to\frac{2}{1-z}$ sends unit semicircle to $x=1$ and line $x=1$ to $x=0$ [duplicate]

How can we show that $z\to\frac{2}{1-z}$ sends the unit semicircle to $x=1$ and the line $x=1$ to $x=0$? $x=1$ implies $z=1+iy$. Then the transformation sends it to i*(2/y).
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2answers
70 views

Integral between $-\pi$ and $\pi$ [duplicate]

How can I show that $\int_{-\pi}^{\pi}\sin (mt) \sin (nt) {dt}=\begin{cases}\ 0 \mbox{ if } m \neq n\\ \pi \mbox{ if } m=n \end{cases}$. I want to prove the above property by expressing sinAsinB as a ...
1
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1answer
1k views

Using Cauchy's Inequality to prove a function's second derivative is zero

Let $f$ be an entire function (i.e. analytic everywhere, i.e. holomorphic) such that $\left\vert f(z) \right\vert \leq A \left\vert z \right\vert$, $\forall z \in \Bbb{C} $, where $A$ is a fixed ...
4
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0answers
306 views

extension of Cauchy's Integral formula

This question is from Brown and Churchill's Complex Variables and Applications, 8ed., Section 52, Question 6. Let $f(s)$ denote a continuous function taken along a simple contour, $C$ enclosing a ...
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2answers
338 views

Let $\displaystyle f$ be an analytic function defined on $\Bbb D=\{z \in \Bbb C:|z|<1\}$

I came across the following problem : Let $\displaystyle f$ be an analytic function defined on $\Bbb D=\{z \in \Bbb C:|z|<1\}$ such that range of $f$ is contained in the set $\Bbb C \backslash ...
3
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1answer
180 views

Discontinuities of $\sum \frac{x^{\rho}}{\rho}$

H. Edwards in his book on the zeta function says that $\sum\frac{x^{\rho}}{\rho}$ converges conditionally "even when $\rho ,1-\rho$ are paired." I tried calculating some terms (n = 500 or so) and ...
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2answers
61 views

Find limit as $y\to \infty$

Find the limit of $$\frac{-(iy - 1/2) -|iy-1/2|}{(iy - 1/2) -|iy-1/2|}$$ as $y\to \infty$. The answer is $i$ but I dont know how to show step by step. if i divide by y on top and bottom, I get ...
2
votes
2answers
242 views

Why does the non-euclidean distance between the lines $x=0$, $x=1$ approach $0$ as $y \to \infty$?

Why does the non-euclidean distance between the lines $x=0$, $x=1$ approach $0$ as $y \to \infty$? Please see http://books.google.ca/books?isbn=0387290524 on pg 191 for more information. My ...
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votes
2answers
238 views

Improper integral of a rational function whose denominator is of degree at least two greater than that of the numerator

There's a technique in complex analysis (involving residue calculus) to solve the improper integral (from $-\infty$ to $\infty$) of a rational function whose denominator is of degree at least $2$ ...
2
votes
2answers
401 views

Find a Möbius transformation sending 0 and infinity, to -1 and 1.

Find a Möbius transformation sending 0 and infinity, to -1 and 1, hence mapping $y$-axis onto unit semicircle My thoughts: Since we need to send three points to define the Möbius transformation, we ...
2
votes
1answer
179 views

What does Schwarz's lemma tell us in this case?

Find $\max |f'(i)|$ and mapping for which it reaches a maximum, if $f:\mathbb{H} \to \mathbb{D}$ is analytic function and $f(i)=0.$ Notation: $\mathbb{H}=\{z\in \mathbb{C}: ...
1
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2answers
1k views

Power series by partial fractions

I am trying to find a power series centered at the origin for the function $f(z) = \frac {1}{1-z-2z^2}$ by first using partial fractions to express $f(z)$ as a sum of two simple rational functions. If ...
1
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1answer
98 views

To show that a partial dertivative (of a piecewise function) is continuous at $0$

$$f(z)=\cases{\frac{x^4-6x^2y^2+y^4}{x^2+y^2} +i\frac{4xy(x^2-y^2)}{x^2+y^2},& $z\ne0$\cr 0, &$z=0$}$$ Let $u=\Re(f)$. I have shown from first principles that $\frac{\partial ...
0
votes
1answer
233 views

Proving that $f$ is analytic if $f$ and $z f(z)$ are harmonic

If $f$ is harmonic and $zf(z)$ is harmonic, then $f$ is analytic. Please help me prove this. Thanks.
1
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1answer
164 views

Express in Rectangular Form

a) $(-1+i)^{-i}$ so I know that the answer is $9.92-3.58i$. My track getting there is off. I know that $x=-1$ and $y=1$, so $r = \sqrt{2}$, also that $\displaystyle \theta=-\frac{pi}{4}$. I've ...
3
votes
1answer
99 views

Moduli Spaces of Higher Dimensional Complex Tori

I know that the space of all complex 1-tori (elliptic curves) is modeled by $SL(2, \mathbb{R})$ acting on the upper half plane. There are many explicit formulas for this action. Similarly, I have ...
2
votes
1answer
74 views

Evaluate $\int_C\frac{dw}{e^w-1}$ over some loop C contained in the annulus $0<|z|<2\pi$.

Evaluate $\int_C\frac{dw}{e^w-1}$ (counterclockwise) over some loop C contained in the annulus $0<|z|<2\pi$. Considering the coefficient of $1/z$ in the Laurent series for $\frac{1}{e^z-1}$ by ...
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2answers
3k views

How to integrate complex exponential??

Consider $$\int^{\frac{1}{2}}_{-\frac{1}{2} } e^{i2\pi f} \,df = \int^{\frac{1}{2} }_{-\frac{1}{2} } \cos(2 \pi f)\, df$$ Why do we only look at the real part? What about the imaginary part ...
0
votes
1answer
105 views

Contour Integrals

Evaluate: $\int_C \hat{z} dz$ where $C$ is the straight line from $i$ to $2-i$. $\int_C \frac{dz}{z}$ where $C$ is the straight line from $3$ to $4i$ $\int_C (z-z_0)^{n-1}dz $ for any integer $n$, ...
0
votes
1answer
82 views

Imaginary complex numbers

Let $z=x+iy$ and $v=2xy$, show that $v=Im[z^2]$ and find a harmonic conjugate of $v$ on domain $D$. Also find the largest domain $D$ on which $v$ is harmonic.
1
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2answers
263 views

How to construct this Laurent series?

How do I construct the following Laurent series (clipped off Wolfram Alpha)? I know that the numerator can be written as $-1+\frac{\pi}2 z-...$ Alternatively (without the Laurent series), how can I ...
0
votes
1answer
341 views

Show that this piecewise function is differentiable at $0$

I have shown (from first principles) that the Cauchy-Riemann equations for the following function are satisfied at $z=0$. But to properly prove differentiability at $z=0$, what should I do next? Do I ...
1
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1answer
77 views

Unique Conformal map satisfying normalization at $\infty$

Let $H$ be the upper half plane in $\mathbb C$. Let $A$ be a compact subset of $\bar H$ with $H\setminus A$ simply connected. I am trying to prove that there is a unique biholomorphism $f : H\setminus ...
0
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2answers
163 views

How to show that $z^4$ is not uniformly continuous?

$f$ maps $z$ in $\mathbb C$ to $z^4$ in $\mathbb C$. How do I show that this function is not uniformly continuous?
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4answers
541 views

Why is the Riemann sphere the compactified complex plane?

Physicist here, although math inclined but I certainly won't brag about it on a math forum, I am encountering a confusion with the compactification of the complex plane. I am learning conformal field ...
5
votes
3answers
122 views

Two trivial questions about zeta function

I have two questions concerning the Riemann zeta function which are rather trivial so if anyone can give me the answers that would be nice, here is what I`m interested in: 1) In the equality ...
5
votes
2answers
426 views

Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$

Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$Given that $|1-zw^*|^2 - |z-w|^2 = (1-|z|^2)(1-|w|^2)$I think the first part can be proven by ...
1
vote
1answer
29 views

Extension of real functions to Riemann surface

Let $f:\mathbb{R}^*_+\to \mathbb{R}$ be a function that is locally the restriction of an holomorphic function. Notating $R$ the Riemann surface of the complex logarithm, with coordinates ...
5
votes
3answers
398 views

Contour Integration of $\sin(x)/(x+x^3)$

How should I calculate this integral $$\int\limits_{-\infty}^\infty\frac{\sin x}{x(1+x^2)}\,dx\quad?$$ I have tried forming an indented semicircle in the upper half complex plane using the residue ...
0
votes
1answer
33 views

find $f(1+i)$ if $\Im[f'(z)]=6x(2x-y), f(0) = 3-2i, f(1) = 6-5i$

Given that $\Im[f'(z)]=6x(2x-y), f(0) = 3-2i, f(1) = 6-5i$, where $z = x+iy$ how to find $f(1+i)$. The answer in answer sheet is $6+3i$. Any hints will be appreciable.
1
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1answer
216 views

using power series expansion to find a holomorphic function which solves a differential equation

Using power series expansions, find a function $f$ which is holomorphic on the unit disk $D:=$ {$z\in\mathbb C:|z|<1$} and solves the differential equation $(1-z^2)f''(z)-4zf'(z)-2f(z)=0$ for ...
2
votes
2answers
317 views

Laurent series, complex analysis.

I am working on a complex analysis exercice. I need to find the Laurent series of the function: $$f(z) = \frac{e^z}{z + 1}$$ about $z = −1$. I know that the result is ...
1
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1answer
49 views

Open sets and holomorphic functions in Complex analysis

Let $U \subset \mathbb{C}$ be an open set such that {$ z\in\mathbb{C};|z|\leq2$}$ \subset U$ and $f:U\rightarrow\mathbb{C}$ a holomorphic function. Show that there exists infinitely many natural ...
2
votes
1answer
327 views

Evaluate a complex integral using power series expansions

Using power series expansions, evaluate the integral $$\int_{\gamma_r}\sin\left(\frac{1}{z}\right)dz.$$ where $\gamma_r:[0,2\pi]\rightarrow \mathbb C$ is given by $\gamma_r(t)=r(\cos t + i\sin ...
3
votes
1answer
96 views

Does $\frac {z^5}{\sin z^2-z^2}$ have a non-isolated singularity at $0$?

Does $\frac {z^5}{\sin z^2-z^2}$ have a non-isolated singularity at $0$? If so, is it not meaningful to discuss its residue at $0$?
4
votes
0answers
64 views

number of zeros of complex waves

Does anybody know about any type of methods how to calucalte/estimate the number of the zeros of complex waves (periodic functions as superposition of many harmonic waves) within a given period [0,x] ...