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)

0
votes
0answers
11 views

construct a function which is differentiable at infinitely many points in the complex plane but is nowhere analytic

I tryed many times. but It is very hard to me to find such a function. I need some help or hint. It is not the infinitely differentiable but the infinetly many points and nowhere analytic in the ...
1
vote
0answers
22 views

Poles of power series

This may be a trivial question, but I haven't been able to find an answer. Given a power series about $x_0$ $F(x)=\sum_{n=0}^\infty a_n (x-x_0)^n$, how do we find its (complex) poles? What about the ...
1
vote
2answers
30 views

Fourier transform of $e^{-4\pi ^2 x^2}$

How do you prove $$\int_{-\infty}^{\infty}e^{-(2\pi x + i\xi/2)^2}dx=\int_{-\infty}^{\infty}e^{-(2\pi x)^2}dx$$ for $\xi \in \mathbb{R}$. The Question arises from calculating the Fourier Transform ...
0
votes
1answer
48 views

Prove: $\int_a^b e^{z_0t}dt=\frac{1}{z_0}e^{z_0t}|_a^b$

From a complex variables online course, and I need to prove that $$\int_a^b e^{z_0 t}dt=\frac{1}{z_0} e^{z_0 t}|_a^b$$ For every $0\neq z_0\in \mathbb{C}$ and for every $a,b\in\mathbb{R}$. Do I need ...
-1
votes
0answers
32 views
0
votes
0answers
14 views

Contour integration along a contour containing two branch points

I need to compute following contour integrations: $$f(u)=\oint_\alpha dz \sqrt{z^3+z+u} \qquad ; \qquad g(u)=\oint_\beta dz \sqrt{z^3+z+u}$$ In which $\alpha$ and $\beta$ are two contours in ...
1
vote
0answers
22 views

Finite groups of Mobius Transformations

Let $M_2(\mathbb{C})$ be the group of all Mobius transformations $z\mapsto \frac{az+b}{cz+d}$ from $\mathbb{C}\cup\{ \infty\}$ to itself. Let $PSU(2,\mathbb{C})$ be the group of all Mobius ...
1
vote
0answers
19 views

Derive chain rule for complex functions, from the chain rule for real functions

I'm trying to obtain the chain rule for complex (not necessarily holomorphic) functions $\mathbb{C} \to \mathbb{C}$, using the known chain rule for functions $\mathbb{R}^2 \to \mathbb{R}^2$. The ...
2
votes
0answers
26 views

$\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{1}{s^2}e^{s(t - \frac{1}{2}x^2)}ds$ - different answers depending on value of $t$?

After taking an inverse Laplace transform I have the following - $$y = \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{1}{s^2}e^{s(t - \frac{1}{2}x^2)}ds$$ In my notes it says if $t ...
0
votes
0answers
8 views

Composition of a holomorphic function with a normal family of holomorphic functions.

Let $\Omega_1$ and $\Omega_2$ (open) $\subset \mathbb{C}$ and $\mathcal{F}$ a normal family of holomorphic functions in $\Omega_1$ such that $f(\Omega_1) \subset \Omega_2$ $\forall f \in \mathcal{F}$. ...
-1
votes
2answers
46 views

Showing $f$ is a constant function

If $f$ is an entire function across the complex plane, how can I show that $Im(f(z)) \gt Re(f(z))^2 - 2$ is constant?
1
vote
0answers
12 views

Find domain of $f$ where $f(z)=\ln(z^2-5z+6)$

$\ln$ is the principal branch of the complex natural logarithm. I think I've solved it, but I don't know if I covered everything. Here's what I did: $Im(z^2-5z+6)>-\pi$ and $Im(z^2-5z+6)\leq\pi$ ...
1
vote
1answer
31 views

Develop the Taylor series of $\ln(z^2-5z+6)$ in $z=0$

Also, determine the radius of convergence. $\ln$ is the principal branch of the complex logarithm. What I've tried is splitting the function into $\ln(z-3)+\ln(z-2)$ and then finding the formula for ...
11
votes
2answers
69 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 ...
1
vote
1answer
24 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
25 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
43 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
13 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 ...
-5
votes
3answers
52 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
28 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
30 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
29 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
26 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 ...
0
votes
0answers
11 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
22 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
77 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
34 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
44 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
35 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
53 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
40 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
28 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
26 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 ...
2
votes
1answer
29 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
26 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 ...