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, ...

learn more… | top users | synonyms (2)

9
votes
2answers
37 views

If $f(a+re^{it})\in \Bbb{R}$ for all $t\in \Bbb{R}$ then $f$ is constant.

I would like to prove that if $f(a+re^{it})\in \Bbb{R}$ for all $t\in \Bbb{R}$ then $f$ is constant. Of course $f$ is holomorphic on a domain $U$ and $r>0$ such that $\overline{D(a,r)}$ is ...
0
votes
1answer
7 views

which of the following is/are true for the entire function $f$?

Let , $f$ be an entire function. Let, $g(z)=\overline{f(\bar z)}$. Let, $D=\{z:Im(z)=0\}\cup\{z:Im(z)=a\}$ for some $a>0$. Then which are correct ? (A) If $f(z)\in \mathbb R$ for all $z\in \mathbb ...
2
votes
1answer
23 views

If $\lim_{|z|\to \infty}\frac{f(z)}{g(z)}$ exists then either $f\equiv0$ on $\Bbb C$ or $f(z)\not =0$ for all $z\in \mathbb C$.

Let , $f,g:\mathbb C\to \mathbb C$ be analytic such that $g(z)\not =0,\forall z\in \mathbb C$. If $\lim_{|z|\to \infty}\frac{f(z)}{g(z)}$ exists then prove that either $f\equiv0$ on $\Bbb C$ or ...
4
votes
2answers
41 views

Proving that a doubly-periodic entire function $f$ is constant.

Let $f: \Bbb C \to \Bbb C$ be an entire (analytic on the whole plane) function such that exists $\omega_1,\omega_2 \in \mathbb{S}^1$, linearly independent over $\Bbb R$ such that: ...
0
votes
0answers
12 views

Fuchs type equation [on hold]

How to show for any second order equation $u''+p(z)u'+q(z)=0$, with finitely many singularities at $z_0,\ldots,z_n,\infty$ of Fuchs type is of the form $$p(z)=\sum_{j=0}^n\frac{p_j}{z-z_j}, \quad ...
-6
votes
3answers
46 views

Evaluate $\int_0^{2\pi} e^{ e^{i\pi} } d\theta $ by rewriting this as an integral about a suitable contour. [on hold]

Evaluate $\int_0^{2\pi} e^{ e^{i\pi} } d\theta$ by rewriting this as an integral about a suitable contour. The integral became intense. I would write my work here, but I am not the best at LaTeX. ...
1
vote
0answers
25 views

a function defined as an integral can be continued analytically

I am trying to solve the following question: Verify that the integral $\int_{0}^{\infty} \, \frac{t^{z}}{e^{\,t\,}+1}dt$ represents an analytic function in the half plane $Re(z)>-1$. Show also ...
2
votes
0answers
29 views

integral of harmonic function

I'm having trouble with this one: Let $u$ be a real-valued harmonic function on $D(0,1)$, and let $\gamma$ be a closed curve in that disk. Then $\int_\gamma u=0.$ I'm supposed to prove or disprove ...
0
votes
2answers
28 views

Evaluating a complex integral (Hints please)

I am supposed to be able to show that, given $f(z)=\frac{1}{\pi}\int_0^1r\int_{-\pi}^\pi\frac{d\theta}{re^{i\theta}+z}dr$ then $f(z)=\overline{z}$ for $|z|<1$ and $f(z)=1/z$ if $|z|\geq1$. (This ...
1
vote
1answer
25 views

Use Cauchy's Integral Formula to evaluate the following integrals.

Use Cauchy's Integral Formula to evaluate the following integral: $$\int\limits_\Gamma \frac{1}{{(z-1)^3}{(z-2)^2}}dz$$ where $$\Gamma$$is a circumference of radius $4$ centered at $-2+i$ and ...
-1
votes
0answers
9 views

Covariance and cross spectrum

A bivariate process $(x_t, y_t)$ is called stationary if each component is a univariate stationary process and $cov (x_s , y_{s+j}) =cov (x_t , y_{t+j}), \forall s,t,j$. The autocovariance function ...
2
votes
0answers
19 views

Finding the Laurent series given the poles and residues

I am working on the following problem, suppose that $f$ has a simple pole at $-1$ with $Res(f,-1) = 1$. A double pole at $2$ with $Res(f, 2) = 2$. Also $f(0) = 7/4$ and $f(1) = 5/2$. I am supposed ...
0
votes
1answer
70 views

The importance of being real

Let $\Sigma$ be a collection of holomorphic, one-to-one function from some simply connected region $\Omega$, which map $\Omega$ into the open unit disc $U$. Fix $z_0 \in \Omega$ and put $$\eta = ...
0
votes
0answers
33 views

Solve the complex euqtions

I have a question from complex calculus. How to solve this two equations: a) $$ sin(z)=2015 $$ I know that sin(z) equals to $$ \frac{e^{iz}-e^{-iz}}{2i} $$ And i don't know whats next. b) $$ ...
1
vote
1answer
24 views

Calculate complex integral $\int_\Gamma\frac{\ln(z+5)}{z^3+iz^2+6z}$

$\Gamma$ is a circle of radius 2 around the point $1+i$. I've parametrized the circle as $\gamma(t)=2e^{it}+1+i$ substituting $z$ in te integral for that expression gets really ugly really quickly. ...
2
votes
1answer
41 views

Is there any condition while applying law of exponents?

${[(-3)^2]}^\frac{1}{2}$ = ${(-3)^2}^\frac{1}{2}$ = $-3^1$ = $-3$ But counted other way it is $9^\frac{1}{2} = \surd{9} = 3$ where I went wrong?
0
votes
1answer
25 views

Corollary of Riemann Mapping Theorem

I was trying to prove the uniqueness of the map in the Riemann mapping Theorem. I'm not sure if the proof I wrote is right. Let $\Omega \subset \mathbb{C}$ be a simply connected open subset such that ...
3
votes
0answers
29 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})\, ...
1
vote
3answers
22 views

Some complex logarithms: please could somebody check my work?

I am doing some exercises from my book, this one asks me to find suitable $z \in \mathbb C$. Please could someone check my work? 1) $z$ such that $e^{z}=-2$: This means that $-2 = iArg(z) + ...
1
vote
1answer
44 views

Show that f and e^f can not have a common pole

Let $f$ be holomorpic on a punctured neighborhood of $z_o$. Show that $f$ and $e^f$ can not have a common pole. My attempt at solution is WLOG let $z_o =0$ be a pole of $f$. Then the Laurent series ...
1
vote
2answers
39 views

How do I find $\frac{d}{dz}\left(\frac{2z-i}{z+2i}\right)\text{?}$

How do I find: $$\frac{d}{dz}\left(\frac{2z-i}{z+2i}\right)\text{?}\quad\quad z\in\Bbb C$$ Do I turn it into an $x+iy$ form and use the Cauchy-Riemann equations? I couldn't get it into such a ...
0
votes
0answers
14 views

Where $|f| <\infty$ a.e. condition is used in Vitali Convergence Theorem

Vitali convergence theorem_Wiki Here above is a Wiki article about Vitali convergence theorem, which is referred to Rudin, Real and Complex Analysis. And I'm wondering where the fourth condition is ...
2
votes
1answer
32 views

Divergent succession, but with convergent sum average.

An example of a sequence $a_n$ such that: $$a_n\rightarrow\pm\infty$$ but $$b_n=\frac{\sum_{k=1}^{n}a_k}{n}$$ converge.
1
vote
1answer
27 views

Showing $f(z)=x^2+iy^3$ is not analytic anywhere

I want to show that the following function is not analytic anywhere. $$f(z)=x^2+iy^3$$ Now I don't really understand the Cauchy-Riemann equations, but it seems we take: $$u(x,y)=x^2,v(x,y)=y^3$$ as ...
3
votes
4answers
42 views

Finding $\lim \limits_{z\to i} \frac{1}{(z-i)^2}$ rigorously

I want to find the limit of the following: $$\lim \limits_{z\to i} \frac{1}{(z-i)^2}$$ And to me, I can see that the denominator is clearly $0$, and since we are in the extended complex plane, I feel ...
2
votes
5answers
51 views

Solving $\cos z = i$ for $z$

Solve $\cos z = i$ for $z$. What I have tried: $$\cos z = i$$ $$\frac{e^{-zi}+e^{zi}}{2}=i$$ $$e^{-zi}+e^{zi}=2i=2e^{\frac\pi 2 + 2\pi k},\quad k\in \Bbb Z$$ I would take logs, but then I would ...
1
vote
1answer
23 views

Where does the imaginary unit dissapear in the Fourier transform of $f(t)= \exp(iat)$?

So I make the Fourier transform of$ f(t)= e^{iat} $on $[- \pi, \pi]$ for some real $a$ and i get: $$a_n=\frac{2a \sin(a \pi)(-1)^n}{\pi(a^2-n^2)}$$ $$b_n=\frac{2i(n\sin(a \pi) (-1)^n)}{\pi(a^2 - ...
0
votes
0answers
25 views

Compute radius of convergence and the first three coefficients of a function

Let $\displaystyle f(z) = \frac{z+1}{(2z+1)(1+ \sin z)}$, with serie expansion $\sum_{n=0} ^\infty a_n z^n$ around zero. Now I want to compute the radius of convergence and the first three ...
0
votes
0answers
18 views

Harmonic Function Cauchy implication

Let $b$ be harmonic real valued on unit disk. Then I wish to prove that $\int_\alpha b =0$. I know that there exists $f$ holomorphic such that $\Re(f)=b$, and I know from Cauchys result that ...
1
vote
1answer
23 views

What is the condition for Morera's theorem to be true?

The answer could be chosen from a) simply connected domain b)connected domain c)no conditions(true for any complex domain) I chose c because the theorem(in our textbook, at least) does not imply ...
0
votes
1answer
25 views

Residues, singularities

For $t\in\mathbb R$ and $n=1,2,3,\dots$ let $$f_n(z)=\frac{z^n}{1-2z\cos t +z^2}$$ Find the singularities of $f_n$ inside $B_2=\{z\in\mathbb C:|z|<2\}$, determine their types, and compute ...
1
vote
1answer
25 views

Schwarz Lemma, an onto map with $f'(0)>0$ is the identity

Let $f$ be $1-1$ holomorphic on unit disk onto itself. It satisfies (a) $f(0)=0$, (b) $f'(0)>0$. We need to prove that $f(z)$ is equal to $z$. I am stuck here, because I can prove using Shcwarz ...
0
votes
0answers
28 views

Prove locally uniformly convergence of a sequence

Let for $n = 0,1,2,...$ , $f_n : [0,1] \rightarrow \mathbb{R}$ defined by $f_n (x) = x^n$. 1) Is the convergence of {$f_n$}$_{n=0} ^\infty$ to $f$ locally uniformly on the interval $[0.1]$? 2) And ...
1
vote
1answer
27 views

Modular forms: What is $\mathbb{H} / SL_2(\mathbb{Z})$?

I am beginning to understand the very basics of modular forms, in that I understand the concept of a weakly modular function, I have seen the examples of $G_k(z)$ and $E_k(z)$ as weakly modular ...
0
votes
1answer
15 views

Two holomorphic functions which have a simple roots at the origin

I am trying to solve the following question: Let $f$ and $g$ be functions holomorphic on the closed unit disk. Assume that f and g have simple zeros at the origin and that g has no any other root in ...
0
votes
1answer
25 views

Cauchy formula in Polydisks

I don't understand a remark after the proof. Here's the theorem: The proof is done by induction on $n$; starting from $n=1$ on the unitary disk in $\Bbb C$, which is the well known Cauchy formula. ...
-1
votes
0answers
16 views

contour integral

I am looking for help in find the contour integrals of I want to know what is a good theorem to use in this integral I do not know who to deal with power 1/3 and 2/3 when I need to find the ...
2
votes
1answer
23 views

Radius of convergence of a complex function with a Taylor Series expansion

The function $$f(z)=\frac{1}{1+i-\sqrt{2}z}$$ has a Taylor series expansion around $z_0=0$. What is its radius of convergence? So far I have computed the singularity point to be $z = ...
0
votes
0answers
15 views

Proof of Contour Integrals and Limits

I've been thinking about the following proof and I'm simply not sure where to start, so any help is appreciated. Thanks in advance. Proof: Let $E$ be a domain in $\mathbb{C}$ and let $f$ be an ...
2
votes
3answers
236 views

Geometry with complex numbers.

Let $a$, $b$, $c$, and $d$ be four complex numbers on the unit circle, such that the line joining $a$ and $b$ is perpendicular to the line joining $c$ and $d$. Find a simple expression for $d$ in ...
0
votes
2answers
16 views

Find the image of the set $\{z \in \mathbb C | -3\lt Re(z) \lt 5, -1 \lt Im(z) \lt 6 \}$, under the function $e^z$

Find the image of the set $\{z \in \mathbb C | -3\lt Re(z) \lt 5, -1 \lt Im(z) \lt 6 \}$, under the function $e^z$ So I know that I should like it as $e^z=e^{x+iy}=e^x(\cos(y)+i \sin(y))$. And I ...
0
votes
0answers
17 views

holomorphic function writen as a serie

To each complexe number $\alpha\in \mathbb{C}$ we associate a function defined by: $$ f_\alpha(z)=1+\sum_{n\geq 1}\frac{\alpha(\alpha-1)...(\alpha-n+1)}{n!}z^n $$ I want to show that this function is ...
1
vote
1answer
19 views

A little guidance on finding the limit

How do I find the limit of $f(z) = \frac{x^2y}{x^3+y^3} + ixy$ as $z \to0$ ? What I think is if $z\to0$, that implies $x ,y\to0$. But since the $f(z)$ has both variables $x$ and $y$ mixed together, ...
0
votes
2answers
23 views

All solutions of $z^2 + (i-2)z + 3-i =0$

I want to find all solutions of $$z^2 + (i-2)z + 3-i =0$$ Now this is what I do: $$x^2 - y^2 +2xyi +(i-2)(x+iy)+3-i =0$$ $$x^2-y^2 +2xyi + xi-y-2x-2iy+3-i=0$$ $$x^2-y^2-y-2x+3+i(2xy+x-2y-1)=0$$ Now ...
3
votes
2answers
36 views

Finding roots of $z^3 = 8$

I am having trouble finding the cubed roots of $8$ as a complex number. $$z^3 = 8+0i$$ $$z^3 = r^3 e^{3\theta i}=8e^{2i\pi k},\quad k\in \Bbb Z$$ $$\implies r=2,3\theta = 2\pi k\implies \theta = ...
2
votes
1answer
14 views

Upper bound for Estimation Lemma

I am struggling with the following question using the Estimation Lemma: Let $ \gamma$ describe the semi-circle $Re^{it}$, where $ 0 \le t \le \pi$, and $ R \gt 3$. Show that $$\int_\gamma ...
1
vote
2answers
38 views

$4^{th}$ root of $-6i$

I want to find the $4^{th}$ roots of $-6i$. What I do is: $$z^4 = -6i$$ $$z^4 = r^4 e^{4i\theta} = 6e^{-i\frac\pi2}$$ $$\implies r=\pm 6^{\frac14}, 4\theta = -\frac\pi2\implies \theta ...
4
votes
4answers
44 views

Simplifying$\left|\frac{z-3}{z+3} \right|=2$

I want to graph the following, but simplifying is the question here: $$\left|\frac{z-3}{z+3} \right|=2$$ Now I can do this : $$\frac{|z-3|}{|z+3|}=2 $$ $$|z-3|=2|z+3|$$ $$|x+iy-3|=2|x+iy+3|$$ What ...
0
votes
2answers
24 views

Finding the limit of complex function

I am trying to check the continuity of this complex function at the origin. $f(z)=\begin{cases} \operatorname{Im}( \frac{z}{1+|z|} ) \qquad &\mbox{when } z\neq0,\\ 0 \qquad ...
2
votes
3answers
37 views

How is this step done? $\left|\frac{i\overline{z}}{2} -\frac i2\right|=\frac{|z-1|}{2}$

Absolutely everything makes sense other than what is in red. How is this step completed? Let us show that if $f(z)=\dfrac{i\overline{z}}{2}$ in the open disk $|z|\lt 1$, then$$\lim ...