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)

1
vote
1answer
13 views

Applications of Rouche's theorem

I understand the statement of Rouche's theorem which (very basically) says that if we have a closed anticlockwise contour where functions $f$ and $g$ are holomorphic on a region $R$, and $f$ is ...
0
votes
1answer
87 views

Residue theorem integral calculation

Use the residue theorem to calculate $$\int_{0}^{2\pi} \frac {27} {(5+4\sin\theta)^2} d\theta $$ I know $$ \operatorname{Res}_{z_0} f = \frac 1 {2\pi i} \int_\gamma d\theta f(\theta) $$ My question ...
0
votes
2answers
23 views

Compute $\int_{\Gamma} z^{-2} dz$ where $\Gamma=C_1(0)$ with its usual (counterclockwise) parametrization.

Compute $\int_{\Gamma} z^{-2} dz$ where $\Gamma=C_1(0)$ with its usual (counterclockwise) parametrization. I calculated the integral manually, but I want a little practice with the following theorem ...
1
vote
1answer
18 views

Rouché theorem application

I want to find function's $F(z)=e^{z+1}+z^{2k+1},\ k=2,3,\ldots$ number of zeros inside ball $B(0,1)$ and inside $B\left(0,\frac{k}{k-1}\right) \setminus \overline{B(0,1)}$. I tried using Rouché's ...
1
vote
1answer
24 views

Is $f'(z)$ defined on $A\subseteq \mathbb{C}$, if $f_n'\to f'$ uniformly?

A recall the following theorem: Let $(f_n)$ be a sequence of holomorphic functions defined on $A$ in $\mathbb{C}$. Let $K$ be an compact set in $A$. If $f_n(z)$ converges uniformly to $f(z)$ on $K$, ...
0
votes
2answers
32 views

Complex integration problem via Cauchy's integral formula

I want to integrate the following :$$\int_{|z|=2} \frac{dz}{z^{2}-1}$$ in the positive direction. So my idea is two split the integral into a sum of two integral , something like $$\int_{|z|=2} ...
0
votes
0answers
28 views

What is the explicit expression for the Laurent series?

Is the following an explicit expression for the Laurent series: $$f(z)=\sum_{n=0}^{\infty}(z-z_0)^n\frac{f^{(n)}(z_0)}{n!}?$$ The reason I ask is because this is what I have seen being derived when ...
1
vote
0answers
10 views

Are holomorphic maps that “almost” preserve norm “almost” rotations?

Let's say I have a sequence of injective holomorphic maps $f_n \colon \mathbb{D} \to \mathbb{D}$ such that $f_n(0) = 0$. The main thing is that $f$ "almost preserves norms" in the sense that for all ...
1
vote
0answers
22 views

Trouble evaluating the complex integral $\int_{\Gamma} e^z dz$

Calculate $\int_{\Gamma} e^z dz$, where $\Gamma$ is the graph of $y= \sin x$ parametrized by $x=0$ to $x= \pi$. I tried applying the following theorem to solve the integral: Let $f(z)$ be ...
1
vote
1answer
35 views

Typo in complex Schwarz Lemma analysis problem

Texas A&M U has on their August 2012 qualifying exam this problem: let $f$ be analytic on $\mathbb{D}$ with $|f(z)|\leq 1$. Then we have $$\frac{|f(0)|-|z|}{1-|f(0)||z|}\leq |f(z)| \leq ...
-1
votes
1answer
18 views

limn→∞d∆(1)(an, bn) =? where d∆(1) is the distance with respect to the Poincare metric [on hold]

Let an = 1 −1/n and bn = 1 −1/2n be two sequences in the unit disk ∆(1). Find the limit limn→∞d∆(1)(an, bn) =? where d∆(1) is the distance with respect to the Poincare metric
1
vote
1answer
17 views

Hecke $L$-function of cusp form is entire

Let $f=\sum a(n)q^n\in S_k(N,\chi)$ be a cusp form of integral weight. Can someone give me the proof of the fact that : the Hecke $L$-function $L(f,s)=\sum\frac{a(n)}{n^s}$ is entire. I searched in ...
2
votes
2answers
61 views

Proving a complex inequality in polar form

$$ |e^z -1| \leq e^{|z|} -1 \leq |z|e^{|z|} $$ for all $ z \in C $. My solution for the left inequality: $$ |e^z -1| \leq e^{|z|} -1$$ By triangle inequality $$ |e^z -1| \leq |e^z| -|1|$$ Therefore ...
1
vote
1answer
52 views

calculating contour integral

It would like to find an expression for $ \dfrac{1}{2 \pi i}\int \limits_{c-i \infty}^{c+i \infty} n^s (s-1)^{-e^{iu}}/s^2 ds$ where $0<|u| \leq \pi/6$, so $\Re(e^{iu}) \geq 1/2$ and $\log(s-1) ...
1
vote
0answers
30 views

Relationship between Möbius transformations and flows/vector fields

I've noticed that the pictures illustrating the effect of Möbius transformations on the Riemann sphere (after stereographic projection to the plane) resemble the phase portrait of a vector field. For ...
230
votes
15answers
26k views

Why does $1+2+3+\cdots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
1
vote
1answer
61 views
+50

Holomorphicity of $f(x + iy) = x^2 + iy^2$

By definition: $f: E \rightarrow \mathbb{C}$, where $E$ is an open subset of $\mathbb{C}$ is holomorphic on $E$ if $f$ is $\mathbb{C}$-differentiable at all points of $E$. The key point being ...
3
votes
1answer
62 views

Solutions of $ \tan(z) = \frac{z}{z^{2} + 1}$ in the complexes

In an exam I got this question: Show that if the equation $$ \tan(z) = \frac{z}{z^{2} + 1} $$ has $z_{0}$ as a solution, then $ \Re(z) = 0 $ or $ \Im(z) = 0 $ Writing $z$ as $x + i y$ seems too ...
-1
votes
0answers
26 views

Number theory about lehmers equations [on hold]

1) Find $n$ such that $\phi(n)$ divides $n-2$ 2) Find $n$ such that $\phi(n)$ divides $n+2$ 3) find $n$ such that $\phi(n)$ divides $2n\pm 2$ 4)find $n$ such that $\phi(n)$ divides $8n \pm 2$
1
vote
1answer
42 views

Gaussian integral $\int_{-\infty}^\infty \exp(-(x+\mathrm iY)^2)\,\mathrm d x$ along $[-R,R]+\mathrm i[0,Y]$

Use integration along $\partial Q$ of $Q=[-R,R]+\mathrm i[0,Y]$ to show that for all $Y\geq 0$ it holds that $$\int_{-\infty}^\infty \exp(-(x+\mathrm iY)^2)~\mathrm dx = \int_{-\infty}^\infty ...
2
votes
1answer
29 views

How can I evaluate this contour integral?

Suppose we have the following contour integral, in the complex plane: $$ \int_{\gamma} \frac{e^{\frac{1}{z}}}{z^{2}} \; dz $$ where $\gamma (t) = e^{it}$ for $0 \leq t \leq 2 \pi$. To solve this, I ...
0
votes
0answers
5 views

Integral operator with sub-exponential eigenvalue decay

I am looking for a positive-definite symmetric function $K: \Omega \times \Omega \to \mathbb{R}$ ($\Omega \subset \mathbb{R}$), such that $$ K(x,y) = \sum_{k=0}^\infty \exp(-\sqrt{k}) \phi_k(x) ...
3
votes
1answer
18 views

Find the value of

$\int_{c}\dfrac {1}{\left( z-\alpha \right) ^{3}}dz$ Where C is some closed anti clockwise contour which does not pass through alpha. I was thinking to use Cauchys integral formula, but didnt know ...
0
votes
1answer
20 views

Find the singularity of the polynomial $p(z)=a_0+a_1 z+a_2 z^2+\cdots+ a_n z^n$.

Let $p(z)=a_0+a_1 z+a_2 z^2+\cdots+ a_n z^n$ be a polynomial of degree $n\geq 1$, where $a_0$ and $a_n$ are both non-zero. Then $$f(z)=\frac{1}{p(1/z)}$$, which is meromorphic function on $\mathbb{C}- ...
0
votes
2answers
20 views

Evaluate the following integral

Evaluate $\int_{L}\left( \overline {z}+1\right) dz$ Where L is the line segment from -i to 1+i. On our complex analysis course we have been shown how to evaluate integral using the the FTC and ...
0
votes
1answer
25 views

Solving an integral by Cauchy Formula

I want to solve the integral $$\oint_{|z|=\frac{1}{2}}{\frac{e^{1-z}}{z^3(1-z)}dz}$$ Its a long time ago that I solved such integrals. Is it just by definition of the line integral? Maybe someone can ...
-2
votes
0answers
22 views

What is the order of the zeros of a complex differentiable function? [on hold]

If $f(x) = x^2 + 4x + 4$ is complex differentiable, meaning with cauchy equations fulfilled, what is the order of the zeros of the function? How to compute it?
0
votes
2answers
721 views

how to find integral of z conjugate

please >>how to find the integral of z conjugate where the path is determined as given here>> I saw that we can write z=x+iy >>but how to continue?
2
votes
1answer
537 views

Using a keyhole contour

I've noticed that some complex analysis textbooks discuss evaluating real-valued integrals like $\int_{0}^{\infty} \frac{\sqrt{x}}{1+x^{2}} \, dx $ using a keyhole contour before they have defined ...
0
votes
0answers
49 views

Does $f(\bar{z}) = \overline{f(z)}$ generally? [duplicate]

I have been playing around with this and noticed that it worked for a few functions. Does it work generally? And does that mean that if $f(z)$ is entire or analytic, it's conjugate won't be since ...
9
votes
1answer
292 views
+100

Approximate spectral decomposition

See attempt below I am interested in effective and constructive computations for finding approximate spectral decompositions in some suitable format. Namely, let $A: H \rightarrow H$ be a Hermitian ...
2
votes
1answer
21 views

Finding the limit at infinity of $f(z) = \frac{\overline{z}}{|z|^2}$

I would like to make sure I'm doing everything right and not missing anything, since I know that some familiar functions do crazy things in the complex setting. Since $|z|^2 = \overline{z}z$ I ...
0
votes
1answer
13 views

When is the radius of convergence of the product of two complex power series twice the radius of convergence of the product of the radii

Proving that the product has a larger radius then the product isn't too bad using the nth root test, but another practice question I have asks for examples of power series $\sum a_kz^k$ with radius of ...
0
votes
0answers
37 views

Find all possible values of the integral

Find all possible values of $\displaystyle I= \int_C \frac{dz}{1+z^2}$, where $C$ is a curve with initial point $0$ and final point $1$ that does not meet the poles of $\dfrac{1} {1+z^2}$. It looks ...
1
vote
0answers
32 views

Finding residues at a point $a$ where $a$ is a pole.

I am faced with the following problem: (a) Find $\displaystyle res_{a} \frac{\varphi(z)}{(z-a)^{n}}$ where $\varphi$ is a given function analytic at $a$, $\varphi(a) \neq 0$, and $n$ is a positive ...
2
votes
0answers
34 views

Evaluating Improper Integrals with Residues - don't think I'm calculating the residues properly

I have to evaluate the integrals $\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+p^{2}}$, for $p > 0$, and $\displaystyle \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+p^{2})^{2}}$, for $p > 0$ ...
1
vote
2answers
32 views

Is this function entire on the complex plane?

That is, $f(z) = \overline{z} = x - iy$ I know that a function is differentiable at $z = z_0$ iff the partial with respect to $x$ is equal to $-i$ times the partial with respect to $y$ (that is to ...
-1
votes
2answers
18 views

How to show a $C^2$ real-valued harmonic function is $C^\infty$ [on hold]

We let u be a $C^2$ real-valued harmonic function on a domain $U\subset C$, and I want to show u is $C^\infty$ on U.
0
votes
1answer
56 views

Calculate $\sum_{n=1}^\infty \frac{1}{n^4}$. [duplicate]

Calculate $\sum_{n=1}^\infty \frac{1}{n^4}$. Remark: I know that $\sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90}$, but not how to prove that, I totally stalled.
1
vote
1answer
32 views

Verifying the result of this complex integral (not integrable analytically)

Could someone verify that the following result is correct? $$\frac{1}{2i\pi}\int_{C(0,1)} \frac{1}{e^z - 1 - z} dz = -\frac{2}{3}$$ ($C(0,1)$ represents the unit circle) I am attempting to use the ...
0
votes
0answers
13 views

verifying the complex cosine of a sum formula using exponential definition

We know that for complex numbers $z_1$ and $z_2$ we have the identity: $\cos(z_1 + z_2) = \cos(z_1)\cos(z_2) - \sin(z_1)\sin(z_2)$. I want to show that the rhs is the same as the lhs: ...
17
votes
1answer
387 views

Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
0
votes
0answers
21 views

Alternate proof for the Casorati-Weierstrass theroem

I have to use the following theorem to prove the Casorati-Weierstrass theorem: If $f$ is analytic in a closed disk $D$ of radius $R$, centered at $z_0$, with $f(z_0)=0$, and if $|f(z)|\leq M$ on the ...
0
votes
1answer
26 views

Residue of $1/(\sin(1/z))$ defined at $z=0$? Trying to derive Laurent Series of $\csc (1/z)$ to find it.

This question is related to this one. I was able to figure out on my own that the residue of $\displaystyle \sin \left(\frac{1}{z} \right)$ is defined at $z=0$ by finding the Laurent Series of ...
1
vote
2answers
53 views

How to integrate the following? $\int_{0}^{+\infty}\frac{1-\cos x}{x^{\alpha+1}}\,dx$

$$2\alpha\int_{0}^{\infty}\frac{1-\cos{x}}{x^{\alpha+1}}dx=?$$ I know that it should be solved by integrating on a contour of two semicircles with radius $\epsilon$ and $T$, and the real line. ...
4
votes
3answers
116 views

Equation of a Riemann surface?

Intuitively in complex analysis I know what a Riemann surface is. It is a surface such that at every point on it the value of a function $f(z)$ is single-valued. However, how would I go about finding ...
0
votes
1answer
37 views

Classifying singularities and determing orders of complex functions

Here are a few functions for reference purposes: $f(z) = \frac{sin(2z)}{z^3}$, $ \space g(z) = \frac{sin(z)}{tan(z)}$, $ \space h(z) = z^2 sin(\frac{1}{z})$ Suppose I was calculating the ...
0
votes
3answers
75 views

Why does $z\mapsto \exp(-z^2)$ have an antiderivative on $\mathbb C$?

Why does $z\mapsto \exp(-z^2)$ have an antiderivative on $\mathbb C$? So far I have seen the following results: If $f\colon U\to\mathbb C$ has an antiderivative $F$ on $U$ then ...
3
votes
1answer
36 views

Essential singularity of the resolvent operator of an unbounded operator

Is there an unbounded operator with isolated points in the spectrum, not all of which are eigenvalues? For unbounded operators it is known that isolated spectral points are either poles or essential ...
20
votes
3answers
668 views

Maximum of Polynomials in the Unit Circle

Let $z_{1},z_{2},\ldots,z_{n}$ be i.i.d random points in the unit circle ($|z_i|=1$) with uniform distribution of their angles. Consider the random polynomial $P(z)$ given by $$ ...