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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3
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2answers
387 views

$uv$ is harmonic if and only if $u+icv$ is analytic for some real c

Let $u$ and $v$ be non constant harmonic functions on a complex domain. Prove that $uv$ is harmonic if and only if $u+icv$ is analytic for some real $c$. I can prove the "if" part. I am having some ...
0
votes
1answer
267 views

Prove meromorphic function can be written as product of holomorphic and rational function

I'm not able to prove this. Any help would be welcomed ! Let U be a simply connected domain and let $f$ be a meromorphic function on U with only finitely many zeroes and poles. Prove that there is ...
0
votes
1answer
64 views

Evaluating a series with some given formula [duplicate]

I have a formula for the power series corresponding to the function $$\frac{z^{3k}}{(3k)!}$$ and I need to evaluate a new series with it but I can't see how to manipulate it even though I've had some ...
1
vote
1answer
119 views

Concatenation with continuous function is entire

Apologies. I have to ask two questions in one and I will give you the reason below. The questions are: If $f$ is entire and $g$ is continuous does it follow that $g\circ f$ is entire? If ...
4
votes
2answers
240 views

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem.

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem, just as the title says. I have used rectangles, circles to do, but without any progress. By changing variable ...
2
votes
2answers
93 views

Defining a branch of $(1-\zeta^2)^{-1/2}$

In this question I brought up a passage from Stein/Shakarchi's Complex Analysis page 232: ...We consider for $z\in \mathbb{H}$, $$f(z)=\int_0^z \frac{d\zeta}{(1-\zeta^2)^{1/2}},$$ where the ...
0
votes
2answers
102 views

Show that $\log|\sin(z)|$ is the real part of a holomorphic function

$D$ is a connected, simply connected domain with $\sin(z)$ never zero on D. Show that $\log|\sin(z)|$ is the real part of a holomorphic function. My question is: how to show $\sin(z)$ maps a simply ...
0
votes
1answer
62 views

Hermitian Matrix n x n

Sea M an Hermitian matrix that satisfies the condition : $$M^5 + M^3 + M = I $$ with I the identity matrix n x n. How can i prove that $M = I$. Please help...
1
vote
4answers
225 views

“philosophical” question about the transcendence of $\pi$

I don't have any knowledge on transcendence proofs. I just heard that Lindemann proved that for any $\alpha \in \mathbb R^*$ algebraic, $e^\alpha$ is transcendental. Then, since $i$ is algebraic, and ...
2
votes
3answers
90 views

Help with a contour integration

I've been trying to derive the following formula $$\int_\mathbb{R} \! \frac{y \, dt}{|1 + (x + iy)t|^2} = \pi$$ for all $x \in \mathbb{R}, y > 0$. I was thinking that the residue formula is the ...
2
votes
1answer
168 views

Möbius transformation on Upper half plane

If I have the function $\phi (x)= \frac{-z+i} {-iz+1} z \in \mathbb{C} $ from D to H where $ D = { z \in \mathbb{C} | |z| <1} $ and H is the upper half plane It's not that hard to see that ...
3
votes
2answers
151 views

calculation of Stefan's constant

In the calculation of Stefan's constant one has the integral $$J=\int_0^\infty \frac{x^{3}}{\exp\left(x\right)-1} \, dx$$ which according to Wikipedia is equal to $\frac{\pi^4}{15}$. In this page of ...
3
votes
1answer
112 views

Rank of homology basis in Ahlfors' Complex Analysis

In Ahlfors' Complex Analysis book Section 4.4.7, he decomposes the complement in the extended plane of a region $\Omega$ into connected components. He then constructs a collection of cycles $\gamma_i$ ...
1
vote
2answers
70 views

Finding power series

I need to find the power series for $e^z + e^{az} + e^{a²z}$ where $a$ is the complex number $e^{2πi/3}$. I know that $1 + a + a² = 0$. I have tried to differentiate the expression and give values ...
2
votes
2answers
61 views

Integral Of $\oint_{0}^{1+i} f(z)dz$ , $f(z)=y-x-3x^2 i$

I want to evaluate the following: $$\int_{0}^{1+i} f(z)\ dz$$ $f(z)=y-x-3x^2 i$ I need to present $z$ with some $t$, I dont know how to make the connection between them. Hints? Thanks.
1
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0answers
36 views

A power series in the space complex

I heve question. Is this series converges in the complex and please justify. $$\arcsin z=\sum_{n=0}^\infty \frac{(2n)!z^{2n+1}}{(2^n n!)^2(2n+1)}$$ where $|z|\leq 1$ This is a power series $\arcsin ...
4
votes
2answers
196 views

Describe the set of harmonic functions $h(x,y)$on $\mathbb{C}$ s.t. $(x^2-y^2)h(x,y)$ is harmonic.

The following is a qual-prep question: Describe the set of harmonic functions $h(x,y)$on $\mathbb{C}$ s.t. $(x^2-y^2)h(x,y)$ is harmonic. I've tried using the definition of harmonic function from ...
5
votes
2answers
452 views

Bound for Analytic Function on Unit Disk

The following is an old qualifying exam problem that I can't seem to piece together: Suppose we have an analytic function $f$ on the unit disk $\mathbb{D}$ s.t. $|f| \leq 1$. Show $$ ...
1
vote
1answer
94 views

Ratio of coefficients for Laurent series expansions [duplicate]

Let $f$ be analytic in the disk $D(0,2)$ except for a pole of order $1$ at $z=1$, and let $$f(z)=\sum_{k=0}^\infty a_k z^k$$ be the series expansion for $f$ in the disk $D(0,1)$. Prove that ...
2
votes
1answer
167 views

Is Fourier transform of a $L^{1}$ integrable function is $L^{1}$ integrable?

Let $f:\mathbb R \to \mathbb R$ such that $$f(x)= \frac{\sin \pi x}{x (x^{2}-1)}$$ for $x\in \mathbb R - \{ 0, -1, 1 \}$ and $f(x):= \pi $ for $x=0$ and $f(x)=-\frac{\pi}{2}$ for $x= -1, 1$. Let ...
3
votes
0answers
194 views

How to calculate this residue

How to calculate this residue $$Res\left(\frac{\ln z}{z(z+1)},0\right).$$ Is it $\infty$? And if this could not be calculated, then how to calculate $$\int_0^\infty \frac{x}{e^x+1}dx$$ by changing ...
2
votes
1answer
66 views

Help with a complex integral

Evaluate $$\int_{|z|=R} \frac{z^{10}-4z^8-6z^3-5}{(z-1)(z-2)(z-5)^9}$$ for all positive $R \neq 1, 2, 5$. My attempt is to break the solution into four pieces and to apply Cauchy's Integral ...
1
vote
2answers
61 views

What area in complex plane is $\{ z \in \mathbb{C}: |z-1|<|z+1| \}$? Is it open and connected?

What area in complex plane is $\{ z \in \mathbb{C}: |z-1|<|z+1| \}$( shade it if possible or describe? Is it open and connected? I ask to comment my solution: $|z-1|<|z+1| \Leftrightarrow$ ...
5
votes
2answers
576 views

Intuition behind the residue at infinity [duplicate]

The residue at infinity is given by: $$\underset{z_0=\infty}{\operatorname{Res}}f(z)=\frac{1}{2\pi i}\int_{C_0} f(z)dz$$ Where $f$ is an analytic function except at finite number of singular points ...
0
votes
1answer
84 views

Line integral of a rational function

Let $P(z)$ be a polynomial such that all the roots of $P(z)$ are inside a circle around the origin with radius $R$. Calculate $\int_{|z|=R}\frac{1}{P(z)}dz$. I know you can use partial function ...
0
votes
1answer
153 views

singularity of $f(z)={(z-1)\over (e^{2\pi i\over z}-1)}$

$f(z)={(z-1)\over (e^{2\pi i\over z}-1)}$ Then which of the following is/are true? $f$ has an isolated singularity at $0$ $f$ has an removable singularity at $1$ $f$ has infinitely many poles each ...
3
votes
0answers
51 views

In complex analysis, is there a special name for functions that can be written as an infinite product of linear factors?

Let $Z$ denote a subset of $\mathbb{C}$. Then some functions $f : Z \rightarrow \mathbb{C}$ have the property that there exist sequences $a,b : \mathbb{N} \rightarrow \mathbb{C}$ such that for all $z ...
0
votes
1answer
60 views

Complex Fourier Transform

Quick question, say I have a complex series of data $f(t)$, so that at each data point $t_i$ have a real and imaginary number, is it correct to calculate the power spectrum of that series (so I want ...
1
vote
2answers
538 views

$f$ be a nonconstant holomorphic in unit disk such that $f(0)=1$. Then it is necessary that

$f$ be a non-constant holomorphic in unit disk such that $f(0)=1$. Then it is necessary that there are infinitely many points inside unit disk such that $|f(z)|=1$ $f$ is bounded. there are at most ...
1
vote
1answer
81 views

$h(z)=|z-a|. |z-b|. |z-c|$, max value of $h$ is attained

Let , $a,b,c$ be non-collinear points in complex plane, $\Delta$ be the closed triangular region of the plane with vertices $a,b,c$. for $z\in\Delta$, let $$h(z)=|z-a|. |z-b|. |z-c|.$$ Then max value ...
0
votes
1answer
31 views

The residue of $\sin \left(\frac{z}{z-1}\right)$ around $1$.

I've tried to expand the function by first using the standard expansion of sinx and then plugging in $x = \frac{z}{z-1}$. The answer is supposed to be $\operatorname{Res}(1) = \cos 1$.
1
vote
1answer
46 views

Can an ordinary point be a fixed point?

Is it possible that a point on the unit circle which is an ordinary point (that is, a point which is not a limit point of any set of the form $\Gamma z$ for $|z| <1$) for a Fuchsian group $\Gamma$ ...
2
votes
0answers
52 views

How to compute next expectation?

$x$ and $y$ are normal random variables, $x\in N(\mu_x, \sigma_x^2), y\in N(\mu_y, \sigma_y^2)$ How to compute next expression $$ \mathbb{E} \arg (e^{ix} + e^{iy}) $$ In English, what is expected ...
2
votes
3answers
120 views

power series expansion of $z^a$ at $z = 1$

I'm working through some problems in a complex analysis book, and one asks to compute the power series expansion of $f(z) = z^a$ at $z = 1$, where $a$ is a positive real number. The series should ...
1
vote
1answer
274 views

(Updated) How to compute expectation and variance of an argument of a complex random variable?

Assume that $\xi$ is a complex random variable. Its argument $\arg \xi$ is a real random variable. I am interested in how to computed expectation and variation of $\arg \xi$. Edit: I add more ...
2
votes
3answers
332 views

Gamma function can be extended as a meromorphic function

Let $$\Gamma (z)= \int_{0}^{\infty} e^{-t}t^{z-1}dz$$ for $\Re z\gt 0$ Can be extended as a meromorphic function to the entire complex plane with simple poles at non positive integers. How can I ...
4
votes
2answers
107 views

$f$ be a non constant entire, which of the following is possible?

$f$ be a non constant entire, which of the following is possible? Re(f(z))=Im(f(z)) $|f|<1$ Im(f(z))< 0 $f\ne 0$ as $f$ non constant so all $1,2,3$ are false as they would imply $f$ as ...
0
votes
1answer
30 views

Find an integral expression for $\Gamma'(z)$ for $\Re z\gt 0$

Find an integral expression for $\Gamma'(z)$ for $\Re z\gt 0$ I know the result. But I dont know how to show this step by step.
1
vote
1answer
58 views

Local representation $f(z_{0})=w_{0}$ and the open mapping theorem

Let $f(z)$ be a non-constant analytic function on a domain $D$ with $f(z_{0})=w_{0}$. I want to show that $f(z)-w_{0} = (z-z_{0})^{n}k(z) = g((z − z_{0})^{n}) = (h(z − z_{0}))^{n}$ with $k,g,h$ ...
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votes
1answer
3k views

Prove the improper integral of the Gamma function $\Gamma(t)$ converges for $z \in \mathbb C$ with $Re(z) > 0$:

Prove the improper integral of the Gamma function $\Gamma(t)$ converges for $z \in \mathbb C$ with $Re(z) > 0$: The gamma function $\Gamma(t)$ is defined by the following improper integral ...
1
vote
1answer
104 views

Mirror point with respect to Riemann circle (Möbius transformation)

The problem is "Find a Möbius transformation $w(z)$ that maps the area $Re( z) > 0, |z-1| > 1$ to the strip $0<Re(w)<2$." I realize there are many ways to skin a cat, but what I wanted ...
9
votes
1answer
607 views

Error on Wikipedia: Nelson's proof of Liouville's theorem works only for bounded modulus?

On Wikipedia, it is stated: If $f$ is a harmonic function defined on all of $\mathbb{R}^n$ which is bounded above or bounded below, then $f$ is constant...Edward Nelson gave a particularly ...
7
votes
1answer
122 views

Find $\sum_{k=1}^{\infty}\frac{1}{z_k^2}$

Let $z_1, z_2,\dots, z_k,\dots$ be all the roots of $e^z=z$. Let $C_N$ be the square in the plane centered at the origin with siden parallel to the axis and each of length $2\pi N$. Assume that ...
3
votes
1answer
337 views

Definition of a branch point (Ahlfors)

This is a continuation to my previous question here. In Ahlfors' complex analysis text, page 299 he says (defines) that a global analytic function $\mathbf{f}$, which can be continued along all arcs ...
3
votes
1answer
1k views

Conformal map from unit disk to strip

I have the following question: Write down the solution $u(x, y)$ to the Dirichlet problem for the following region and boundary conditions: $U = \{x + iy : 0\le y\le1\}; u(x, 0) = 0, u(x, 1) = 1$. ...
1
vote
1answer
71 views

Zeta function in complex analysis.

Show that $$\frac{\zeta'(z)}{\zeta(z)}=-\sum_{n=2}^{\infty}\frac{f(z)}{n^z}$$ for $\Re z\gt 1$ Where $f(z)= \ln p$ if $n=p^m$ for some prime $p$ and some $m\in \Bbb N^+$ Or $f(z)=0$ otherwise. ...
1
vote
1answer
30 views

$\frac{1}{2\pi i}\int_C \frac{f'(z)}{z}z =3$ and $\frac{1}{2\pi i}\int_C \frac{f'(z)}{z}z^2 = 4$ in a rectangle with boundary $C$

Let $C$ be a positively oriented contour forming the boundary of the rectangle $1\leq \text{Re}(z)\leq 4$, $-2\leq \text{Im}(z)\leq 2$. An entire function $f(z)$ is known to have no zeros on ...
6
votes
1answer
436 views

How to imagine zeros of an analytic function of several variables

Let $f(z_1,\cdots, z_n)$ be a holomorphic function of several variables in an open subset of $\mathcal C^n$. Let $Z(f)=\{ (z_1,\cdots, z_n) \: | \: f=0\}$ be the zero set of $f$. If $n=1$, the zeros ...
0
votes
1answer
55 views

If $f(1) = i$ what is $f(i)$ for a function with $|f(z)| \leq K|z|$ ? GATE 2011 [duplicate]

Let $f(z)$ be an entire function s.t. $|f(z)| < K|z|$, $\forall$ $z \in \mathbb{C}$, for some $K > 0$. If $f(1) = i$ what is the value of $f(i)$ and why? The value of $f(i)$ will be one of these ...
0
votes
1answer
49 views

Analytic functions in two variables

Let $f$ be an analytic function in two complex variables. It is well known that we can expand $f$ in a convergent series of two variables. Can we separate the variables in such a manner that $f$ ...