1
vote
3answers
32 views

Computing $\int_{|z|=2} {dz \over z^2 + 1}$

Goal: To compute $$ \int_{|z|=2} {dz \over z^2 + 1} $$ by decomposition of the integrand in partial fractions. Attempt: Let $\gamma$ be the circle around the origin of radius $2$. Let us ...
1
vote
1answer
25 views

Showing the winding number of the unit circle is $1$

Let $\gamma$ denote the unit circle parameterized on the domain $[0,2\pi]$. I'm trying to compute $n(\gamma, 0)$ as follows: $$ n(\gamma,0) = {1 \over 2\pi i}\int_\gamma {dz \over z} = {1 \over 2 ...
2
votes
1answer
26 views

Inverse of the function $\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$

It can be proved that the function $f:[-1,1]\to \mathbb{C}$ defined by $$f(x)=\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$$ maps the interval $[-1,1]$ one to one onto the lower part of the unit circle. ...
0
votes
0answers
22 views

Deriving that ${d \over dz}\left(\log\ z \right) = {1 \over z}$ in the complex plane

How does one derive that $$ {d \over dz}\left(\log\ z \right) = {1 \over z}\text{?} $$
0
votes
1answer
19 views

Understanding why $\int_\gamma {dz \over z - a} = k 2\pi i$ for $\gamma$ a closed curve not passing through $a$

The following is a paraphrased proof from Ahlfors. I bolded the part that is confusing me and asked a question about it at the bottom of this post. Hypothesis: Let $\gamma$ be a closed curve that ...
0
votes
1answer
10 views

What is the integrand of $\int_\gamma d\ \log(z-a)$?

Suppose $\gamma$ is a piecewise differentiable closed curve that does not pass through the point $a \in \mathbb{C}$. I'm reading a proof in Ahlfors that shows under this condition we will obtain $$ ...
3
votes
2answers
24 views

Analytic $F(z)$ has $f(z)$ as derivative $\implies$ $\int_\gamma f(z)\ dz = 0$ for $\gamma$ a closed curve

Hypothesis: Suppose that $F(z)$ has $f(z)$ as a derivative. Suppose further that $F(z)$ is analytic. Now consider the complex line integral $$ \tag{1} \int_\gamma f(z)\ dz $$ Question: Does this ...
1
vote
2answers
27 views

Determining why $\int_{\partial R} z\ dz = 0$ and $\int_{\partial R} z\ dz = 0$ independently of Cauchy's Theorem for a Rectangle

Let $R$ be a rectangle on the complex plain and $\partial R$ its closed curve. Without making use of Cauchy's Theorem for a Rectangle (or any of the other Cauchy theorems), I'm curious why we know ...
0
votes
0answers
10 views

Computing ${\partial U \over \partial x}$ and ${\partial U \over \partial y}$ for $U(z)= \int_\gamma (z - a)^n\ dz$

Goal: Let $$ U(z)= \int_\gamma (z - a)^n\ dz $$ I'm trying to compute ${\partial U \over \partial x}$ and ${\partial U \over \partial y}$. Attempt: I know that $(z-a)^n$ is the derivative of ...
0
votes
1answer
20 views

How to know that $(z-z_0^3)(z-z_0^5)(z-z_0^7) = \sum_{k=0}^3 z^k z^{3-k}_0$

How to know that $$(z-z_0^3)(z-z_0^5)(z-z_0^7) = \sum_{k=0}^3 z^k z^{3-k}_0$$ with $z_0$ a root of $z^4+1$. I can check that it is true, but is there a way to tell, by seeing the LHS expression, ...
1
vote
2answers
38 views

Deducing Laplace Formulas

I have to compute the followings integrals $\forall\; b\in \mathbb{C},\; \text{Re} \;b \gt0,p\gt 0$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x-ib}$$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x+ib}$$ ...
2
votes
0answers
15 views

Definition of an integrand

General Question: Say we have integral $$ \int f(z)\ dz $$ Is the integrand in this context (i) $f(z)$ or (ii) $f(z)\ dz$? In any case, is $f(z)\ dz$ a formally defined mathematical object in its ...
0
votes
1answer
23 views

Verifying a condition for which $\int_\gamma p\ dx + q\ dy$ depends only on endpoints

Hypothesis: Suppose there exists a function $U(x,y)$ in $\Omega$ with partial derivatives $${\partial U \over \partial x} = p \quad \quad {\partial U \over \partial y} = q$$ Goal: Show that the ...
0
votes
0answers
23 views

What is meant by a “general line integral of form $\int_\gamma p\ dx + q\ dy$”?

In his text on complex analysis, Ahlfors speaks of "general line integrals of form $\int_\gamma p\ dx + q\ dy$". I'm curious exactly what is meant by this. I take it that $p$ and $q$ are not ...
1
vote
1answer
23 views

How to prove that $\pi \frac{e^{it\frac{\pi}{2}}-e^{it\frac{3\pi}{2}}}{1 - e^{2\pi it}} = \frac{\pi}{2\cos\left(\frac{\pi t}{2}\right)}$

How to prove that : $$\pi \frac{e^{it\frac{\pi}{2}}-e^{it\frac{3\pi}{2}}}{1 - e^{2\pi it}} = \frac{\pi}{2\cos\left(\frac{\pi t}{2}\right)}$$ I start with $$e^{it\frac{\pi}{2}}-e^{it\frac{3\pi}{2}} = ...
1
vote
0answers
17 views

Computing a complex line integral $dz$ in terms of line integrals $dx$ and $dy$

Goal: I'm trying to verify the calculation claimed by Ahlfors that $$\int_\gamma f(z)\ dz = \int_\gamma (u\ dx - v\ dy) + i \int_\gamma (u\ dy + v\ dx)$$ Attempt: $$\int_\gamma (u\ dx - v\ dy) + i ...
1
vote
0answers
35 views

Does $\int_a^b f(z)\ \overline{dz} = \int_a^b f(z)\ dz$? [duplicate]

Question: Attempted Answer: Yes, for if $f = u + iv$ where $u$ and $v$ are real-valued functions, then we have that $$ \int_a^b f(z)\ \overline{dz} = \overline{\int_a^b \overline{f(z)}\ dz} ...
2
votes
1answer
34 views

I want to compute $\int_0^\infty \frac{x^t}{1+x^2}dx \; \forall t \in (-1,1)$ using residue theroem.

I want to compute $$\int_0^\infty \frac{x^t}{1+x^2}dx \qquad \forall t \in (-1,1)$$ using residue theroem. I consider $$f(z) = \frac{z^t}{1+z^2}$$ I find two pole of order 1 in $z=i$ and $z=-i$ with ...
0
votes
0answers
17 views

When does $\int_\gamma f(z) \,dz = \int_\gamma f(z)\, \overline{dz}$?

Suppose $f:[a,b]\rightarrow \mathbb{C}$ satisfies $f = u + iv = u$ (i.e., $v = 0$). Then is it correct to assert that $$ \int_\gamma f(z)\ dz = \int_a^b f(\gamma(t)) \gamma'(t)\ dt = \int_a^b ...
0
votes
0answers
24 views

Visual representation of the complex line integral

Wikipedia has a visual depiction of a line integral of a scalar field: http://en.wikipedia.org/wiki/Line_integral#Vector_calculus I'm curious if this graphical representation could be used to ...
2
votes
0answers
18 views

Is there a simple and fast way of computing the residue at an essential singularity?

Is there a simple and fast way of computing the residue at an essential singularity ? I mean if we have a pole of order $n$ at $c$ we can use the formula : $$\mathrm{Res}(f,c) = \frac{1}{(n-1)!} ...
0
votes
1answer
25 views

Plot of a domain in the complex plane

I am trying to plot the following domain in the complex plane: $\lbrace x\in\mathbb{C}|\: |x^{2}-1|<r\rbrace$ for some $r>1$. I know that in general to take a square root of a complex number ...
0
votes
0answers
21 views

Does $\overline{\gamma'(z(t))} = \left(\overline{\gamma(z(t))}\right)'$?

For $\gamma$ a piecewise differentiable arc from $[a,b]$ to $\mathbb{C}$ via $z(t)$, is it the case that $$ \overline{\gamma'(z(t))} = \left(\overline{\gamma(z(t))}\right)' \text{?} $$
0
votes
0answers
18 views

Computing $\int_\gamma \overline{f(z)}\ dz$

Background: My question concerns a calculation involving the integral $\int_\gamma \overline{f(z)}\ dz$. Consider that if we write $f = g + ih$ with real-valued functions $g$ and $h$, and similarly ...
1
vote
1answer
29 views

Does $\int_a^b \overline{f(z)}\ dz = \int_a^b u(t)\ dt - i \int_a^b v(t)\ dt$?

Hypothesis: Let $[a,b] \subseteq \mathbb{R}$ and $f = u + iv$ with domain $[a,b]$. Question: Do we have that $$\int_a^b \overline{f(z)}\ dz = \int_a^b u(t)\ dt + i \int_a^b -v(t)\ dt = \int_a^b ...
1
vote
3answers
24 views

How to prove $ z^n - z^n_0 = (z-z_0) \sum_0^{n-1} z^kz_0^{n-1-k} $ [duplicate]

I want to prove that with $z_0$ a root of $1+z^n$, I have $$ z^n - z^n_0 = (z-z_0)\sum_0^{n-1} z^kz_0^{n-1-k}$$
0
votes
3answers
31 views

How to find the $n$ zeros of $\displaystyle1+z^n$?

How to find the $n$ zeros of $1+z^n$?
0
votes
1answer
24 views

Is $\sum_0^\infty (-1)^k z^{k-1}$ equal to $\sum_{-1}^\infty (-1)^{k+1} z^{k}$

Is $$\sum_0^\infty (-1)^k z^{k-1}$$ equal to $$\sum_{-1}^\infty (-1)^{k+1} z^{k}$$ i.e am I allowed to reindex the beginning of series ?
1
vote
0answers
15 views

Real part of a holomorphic function is bounded by polynomial then the holomorphic function is a polynomial [duplicate]

Let $u$ be a harmonic function on $\mathbb{R}^2\cong \mathbb{C}$ such that $Ref= u$ where $f$ is an entire function. If $|u(z)|\leq |z|^n$ for any $z\in\mathbb{C}$, then $f$ is a polynomial of degree ...
0
votes
2answers
24 views

How to compute $f(z) = \sum_0^{\infty} (1+2i+(2+i)(-1)^k)^{-k}z^k$

How to compute this serie : $$f(z) = \sum_0^{\infty} (1+2i+(2+i)(-1)^k)^{-k}z^k$$ The serie is convergent if $|z| < \sqrt{2} $ I can find that $$f(z) = \sum_0^{\infty} 3^{-2k}(1+i)^{-2k}z^{2k} + ...
2
votes
1answer
30 views

functions of two variables with one variable defined on a compact set uniformly converge to zero

Let $f$ be a holomorphic function on $[0,1]\times \mathbb{R}$. If for each $x\in [0,1]$ fixed, $\lim_{y\to\infty}f(x,y)=0$, prove that $f$ is bounded. My idea: I do not know how to prove and I also ...
0
votes
0answers
30 views

Power series of characteristic function

I read that if $f \in \ell^1 (\mathbb Z)$ and $w$ is the characteristic function of $\{1\}$ then $$ f(m) = \sum_{n \in \mathbb Z} f(n) w^n(m)$$ where $w^n = w \ast \dots \ast w$ is $n$ times the ...
0
votes
1answer
33 views

How to compute this integral : $\oint \bar{z}^n dz$

How to compute this integral : $$\oint_{|z|=a} \; \bar{z}\;^n dz$$ I choose $z = ae^{i \theta}$, and so $\bar{z}\;^n = a^n e^{-i\theta}$ And $$\oint_{|z|=a} \; \bar{z}\;^n dz = ...
1
vote
1answer
44 views

How to justify, $\sum_{n=1}^{\infty} a_{n} x^{n} - \sum_{n=1}^{\infty}a_{n}y^{n}=\sum_{n=1}^{\infty} a_{n} (x^{n}-y^{n})$?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$ and we let $K_{1}$ be a compact subset of ...
0
votes
1answer
38 views

Question relating to the Casorati-Weierstrass Theorem.

The question I am trying to answer is: Suppose $f$ is analytic in the punctured disc $0 < |z| < 1$ except for poles $\{z_n\}$ where: $$\lim_{n \to \infty}z_n = 0$$ Note that $0$ is not an ...
0
votes
0answers
14 views

Continuity of Complex function and restrictions

I am trying the following question but am stuck at finding the restriction: Prove that $f(z)=1/z^2$ is continuous at $z_0= 1+2i$ Solution: I am trying the use epsilon-delta proof and got it down to: ...
0
votes
0answers
18 views

Showing uniqueness of character identity

How would one show that any complex-valued C1 function satisfying the character identity must be of the form exp(cx) for c complex. Given a function f, it is said to satisfy the character identity if ...
5
votes
1answer
53 views

Whats the differences between the real-entire functions on $\mathbb R^{2}$ and complex entire functions on $\mathbb C$?

We note, as set of points, $\mathbb R^{2}= \mathbb C.$ A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point ...
0
votes
0answers
15 views

conflictions of analytic functions to the boundary and Schwarz reflection principle

Let $\Omega$ be an open subset of $\mathbb{C}$ and $f:\Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Then for any $z\in \Omega$ and any $r>0$ such that $D(z,r)\subseteq \Omega$, $f$ ...
1
vote
0answers
18 views

Cauchy integrals over a line

Can we generalize the Cauchy integral formula from a circle to a line? Since for real integrals, the following types of improper integrals do not converge, is it correct or not that for $z\notin ...
1
vote
1answer
30 views

Conformal mapping on two paths

GIven $ f (z) = z^2$ . Let $p = (0, −1)$ and take the curves $γ_1, γ_2$ passing through $p$ as $γ_1 = $arc of the unit circle through $(0, −1) $ counterclockwise and $γ_2 $ = a straight line ...
1
vote
2answers
32 views

Geometrical Meaning of derivative of complex function

What's the geometrical meaning of f'(z) in complex analysis, as we know in real analysis f'(x) has meaning ie. Slope of curve or gives max/ min. But what does derivative f'(z) has geometrical meaning ...
0
votes
1answer
37 views

Prove that a pseudo-hyperbolic ball is a Euclidean ball. Find the radius and center of the Euclidean ball.

We have that the pseudo-hyperbolic metric in the open unit disk $\mathbb D$ is defined by $$ \rho(z,w) = |\phi_w(z)|, \qquad \phi_w(z) = \frac{w - z}{1 - \overline w z}$$ where $z,w \in \mathbb D.$ ...
0
votes
1answer
19 views

How do i analyze this complex diagram?

I'm asking how to analyze diagrams like this : http://upload.wikimedia.org/wikipedia/commons/thumb/9/96/Complex_LogGamma.jpg/600px-Complex_LogGamma.jpg What do distinct colors here mean? What do the ...
0
votes
0answers
15 views

understanding topological argument in rado-kneser theorem

Rado-kneser choquet theorem states that Poisson integral of a homeomorphism of unit circle is a homeomorphism. It's proof goes like proving it local homeomorphism by proving non vanishing of jacobian ...
1
vote
1answer
22 views

Show uniform convergence of a series of complex function on every compact subset

Let $f:B(0,1)\rightarrow \mathbb{C}$ be an analytic function. Suppose $\sum^\infty_{n=0}f^{(n)}(0)$ converges absolutely. Show that there exists an entire function $g(z)$ such that $g(z)=f(z)$ for ...
0
votes
2answers
53 views

Understanding the Definition of $\int_\gamma f\ \overline{dz}$

Definitionally, we have that $$ \int_\gamma f\ \overline{dz} = \overline{\int_\gamma \overline{f}\ dz} $$ Now let $\int_\gamma f\ dz = w = x +yi$. Question 1: Is it not the case that $\int_\gamma ...
0
votes
1answer
16 views

Defining the Complex Line Integral w.r.t. $x$ and $y$

Ahlfors defines line integrals with respect $x$ as follows: $$ \int_\gamma f\ dx = {1 \over 2} \left( \int_\gamma f\ dz + \int_\gamma f\ \overline{dz} \right) $$ From this I take it as obvious that ...
1
vote
1answer
28 views

$v$ is Conjugate Harmonic to $u$ $\implies$ $f = u + iv$ is Analytic (Proof Verification from Ahlfors)

Hypothesis: Let $u$ and $v$ be two functions from $\mathbb{R}^2$ to $\mathbb{R}$ s.t. $$ \Delta u = {\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} = 0 $$ and $$ \Delta v = ...
1
vote
1answer
49 views

consequences of Schwarz lemma of holomorphic functions of unit disk

Let $D$ be the open unit disk centered at $0$ in the complex plane. Let $f:D\longrightarrow D$ be holomorphic such that $f(0)=0$. Use the Schwarz lemma to prove that $|f(z)+f(-z)|\leq 2|z|^2$ for any ...