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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42 views

Calculating a complex integral by rewriting as a contour integral on |z|=1.

I need to show that $\int_0^{2\pi}\frac{d\theta}{2+i\:sin\theta}=\frac{2\pi}{\sqrt{5}}$ I used $sin\theta=\frac{1}{2i}(e^{i\theta}-e^{-i\theta})$ and substituted $z=e^{i\theta}$. I ended up with ...
1
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1answer
82 views

holomorphic functions, such that $|f|$ depends only on $Re \ z$ and $arg \ f$ depends only on $Im \ z$.

I need to describe all holomorphic functions, such that $|f|$ depends only on $\text{Re}\, z$ and $\text{arg}\, f$ depends only on $\text{Im}\, z$. My thoughts: Let $f=u+vi, z=x+iy$, then $0=d ...
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0answers
26 views

Does a lattice in $PSL(2,\mathbb{R})$ stabilizing $\infty$ have a domain with vertex at $\infty$?

Suppose $\Gamma$ is a lattice in $PSL(2, \mathbb{R})$ acting on the upper half plane. Suppose that the stabilizer in $\Gamma$ of the point at infinity is nontrivial. Does it then follow that the ...
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1answer
47 views

Show that $\ c_X(p,q) \le d_X(p,q)$, for $ p, q \in X$

Update I'm trying to show the Corollary, but I have stuck...That is: For any complex space $X$, we have: $$\begin{align} (1).\ c_X(p,q) &\le d_X(p,q),\ \text{for}\ p, q \in X \\ (2).\ ...
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1answer
17 views

Convergence of a sequence with nonincreasing real terms on the unit disk

Let $a_0 \geq a_1 \geq ... \geq a_n \geq ...$ Then $\sum_{n=0}^{\infty} a_n z^n$ converges for all $|z|=1, z \neq 1$. My take was this: let $z \in \delta D(0,1)- \{1\}$. Then ...
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1answer
54 views

Nth root of complex number (z)

I have to prove that: Prove that $\displaystyle z^{\frac{1}{n}}=e ^{\frac{1}{n}(\text{Log }z+2k\pi i)}$ gives the $n$th root of $z$, taking $k=0,1,2, \ldots$ Well, with the suggestion of Gerry ...
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1answer
93 views

Is this analytic continuation possible?

I'n new to complex analysis and am a little flustered by the following function. I would like help understanding whether or not it is possible to analytically continue it outside of the unit circle. ...
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0answers
58 views

Interpretation of the Argument Principle

Recall that the argument principle states that given a meromorphic function $f$ and a compact region $K \subseteq \mathbb{C}$ whose boundary determines a simple contour and on which $f$ has no ...
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1answer
33 views

Soft Question about Mobius Transformations

Very soft question and I may be completely wrong about this, but does it make any sense to think about the Mobious transformation matrix as a change of basis for C?
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4answers
89 views

On the subject of holomorphic functions on an open disc, D.

The question I am pondering over is an interesting one: If $f(z) = u + iv$ is holomorphic on an open disc $D$, and the range of $f$ lies in either a straight line or a circle, prove that $f$ is ...
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1answer
50 views

MVT doesn't extend to complex derivatives

Let $f(z)=z^3$. For $z_1=1$ and $z_2=i$ prove that there doesn't exist any complex number $c$ on the line segment joining $z_1,z_2$, such that $$\frac{f(z_1)-f(z_2)}{z_1-z_2}=f'(c).$$ A general point ...
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1answer
159 views

Cauchy Integral Formula on Boundary

Suppose that I'm trying to evaluate the following integral: $$\frac{1}{2\pi i}\ \int_{C} \frac{cos(\pi z)}{z^2-1}dz $$ And further suppose that C is a rectangle going over $ 2+i,2-i,-2+i,and -2-i$. ...
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1answer
92 views

Showing that a mobius transformation exists

I'm trying to show that a specific Mobius transf. exists, where I have some points that map to some other points. (I don't wanna be too specific here about what goes where, as I don't wanna run the ...
2
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2answers
79 views

Prove that the entire function $f$ is linear.

Suppose $f=u+iv$ be an entire function such that $u(x,y)=\phi(x)$ and $v(x,y)=\psi(y)$ for all $x,y\in\mathbb{R}$. Prove that $f(x)=az+b$ for some $a\in\mathbb{C},b\in\mathbb{C}$. My approach was: ...
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1answer
47 views

Holomorphic function interpolating $e^{-n}$

Consider the question: does there exist a holomorphic function $f$ on the unit disk in the complex plane such that $f\left({1 \over n}\right) = e^{-n}$ ? I came up with an answer but I'd like to know ...
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1answer
61 views

Complex Conjugation question

I had a complex analysis exam yesterday, and one of the questions is bothering me. Suppose $f(z)$ is an entire function. Show that $g(z) = (f(z^*))^*$ is also entire. Here $^*$ indicates complex ...
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3answers
118 views

If $f$ is holomorphic and $\lvert f\rvert$ is constant, then $f$ is constant.

Let $\Omega$ be a connected open set and $f:\Omega\rightarrow\mathbb{C}$ is holomorphic. If $\lvert f\rvert$ is a constant function then we need to show, $f$ is also a constant function. I tried to ...
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2answers
97 views

Stereographic projection continuous at $\infty$

I have trouble with showing that the stereographic projection is continuous on $\infty$. I was given those two functions: $$\pi:S^2\to\mathbb{C}\cup \{\infty\}$$ $$\pi(x_1,x_2,x_3) =\begin{cases} ...
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3answers
90 views

Evaluating improper integrals of odd functions with hyperbolic and circular elements

The integral $$ \int_{0}^{\infty} {\sin\left(\omega t\right) \over \cosh^{2}\left(t/\sqrt{2\,}\,\right)}\,{\rm d}t $$ with $ \omega >0 $ is an odd function in variable t. This precludes any ...
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1answer
93 views

Minimize norm of a polynomial on a circle

Let $P=\sum_{k=0}^n a_kX^k$ ba a polynomial of degree $n \gt 0$, and let $r\gt 0$. Suppose that $P$ is not the monomial $a_nX^n$, in other words there is at least an $i<n$ such that $a_i\neq 0$. ...
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1answer
50 views

Complex Analysis Integration & Branch Cuts

Suppose that the curve $C$ is any path between $z=0$ and $z=1$ which does not go through any singularities of the function below. I'm trying to show the following: $$\int_{C} \frac{1}{1+z^{2}}\,dz = ...
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1answer
39 views

A proof in $\mathbb{R}^2$ regarding the Cauchy-Riemann equations

Let $u,v$ be a pair of smooth, real valued functions on $\mathbb{R}^2$. Let $(x,y)$ be a point on $\mathbb{R^2}$. Show that the mapping $(x,y)\to(u,v)$ is conformal at the points where the Jacobian ...
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1answer
53 views

Why does $\sin(\operatorname e^i)$ in complex variables have the following solution?

If possible I would like to know the definitions to look at so I can master this material. According to my professor $$ \sin(\operatorname e^i) = \sin(\cos1)\cosh(\sin1)+i\cos(\cos1)\sinh(\sin1) $$ ...
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votes
3answers
141 views

Show, a holomorphic function with constraint on real and imaginary part is constant

Let $f: G\rightarrow \mathbb{C}$ be a holomorphic function on a domain. Let $\left[\Re{(f)}\right]⁴+\left[\Im{(f)}\right]⁴$ have a local maximum in $G$. Why is $f$ than already constant? If I could ...
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2answers
44 views

Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
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0answers
36 views

The Kobayashi pseudo - distance $d_X$ and the Carathéodory pseudo - distance $c_X$

I'm studying the Kobayashi pseudo - distance $d_X$ and the Carathéodory pseudo - distance $c_X$. And I have trouble when I try to show $4$ properties of $c_X$ in my textbook. {It doesn't have ...
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1answer
50 views

Show that all the zeros of $g$

Let us consider the function: $$g(α,β)=\sum_{n=1}^{\infty}(-1)^{n-1}((n^{2α-1}-1)/n^{α}) n^{iβ}$$ My question is: Show that all the zeros of $g$ in $0<α<1$ have the form $(1/2,β)$ where $β$ is ...
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4answers
138 views

Is $f(z)=\bar{z}$ continuous?

I have $z\in \mathbb{C}$, is $f(z)=\bar{z}$ continuous on the whole complex plane? Note that $\bar{z}$ is the conjugate of $z\in \mathbb{C}$ I was thinking that if $z$ is on the real line, then ...
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1answer
181 views

Prove this integral is zero

I'm trying to prove that $\lim_{R\to\infty}\int_{C_R} dz \exp\left(iaz^2\right) = 0$, where $a$ has a positive imaginary part and $C_R$ is an arc from $R$ to $\frac{1+i}{\sqrt{2}}R$ along the circle ...
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1answer
36 views

Cauchy principal value of two integrals

I want to calculate $P.V. \int_{-\infty}^{\infty} \frac{e^{ix}}{x}dx$ and $P.V. \int_{-\infty}^{\infty} \frac{f(x)}{x(x-i)}dx$ I stat by using the definiton to calculate the first one: ...
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1answer
51 views

Solve $\cos \pi z = 0$ for $z \in \mathbb{C}$ [closed]

$\cos \pi z = 0$, so $\cos \pi x \cosh \pi y - i \sin \pi x \sinh \pi y = 0$, $\cosh \pi y$ never be $0$, so $\cos \pi x = 0, \pi x=\pm \pi/2+2k\pi, x = \pm1/2+2k.$ Is this the right way to do?
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1answer
45 views

Can we deduce that the zeros of $g$ are also isolated?

Let $f:Ω→ℂ$ be a non-zero holomorphic function and $g:Ω→ℂ$ be a non-zero non-holomorphic function. We know that all the zeros of $f$ are isolated. Assume that $$f(s)=0⇒g(s)=0$$ Can we deduce that the ...
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1answer
71 views

Explain that the complex sine function is not bounded.

That is, for any positive constant $M$, there exists a $z$ such that $|\sin z|>M$. Given $|\sin z|^2=(\sin x)^2+(\sinh y)^2$.
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1answer
42 views

Closed sets and sequences in Metric spaces

Suppose $x \subset X$ is a closed set, the sequence {$ {x_j}$}${ } \subset F$ and $x \in X$. Show that if $x_j \to x$ as $j \to \infty$, then $x \in F$ Okay so I really don't know where to start with ...
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1answer
103 views

Why is the set of points where a complex polynomial does not vanish is connected?

Let $p$ be a complex multivariate polynomial. Let $C$ be the set of those complex tuples where $p$ is nonzero. Then, $C$ is connected.
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3answers
154 views

Solve the equation $\log(z^2-1)=i\pi/2$

I set $z=x+yi$, so: $$ \log[(x+yi)^2-1]=\log(x^2+2xyi-y^2-1)=\log (r+iθ)=i\pi/2$$ than I get $x^2-y^2=1$ and I have no idea how to continue.
3
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1answer
81 views

complex analysis (Univalent function )

The Distortion Theorem tells us that if $f$ is a univalent function on $\mathbb{D}:=\{z:|z|<1\}$, then $|f'(z)|\leq 12\,|f'(0)|$ for $|z|\leq\frac12$. By iterating this, prove that if ...
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3answers
100 views

Are the Cauchy Riemann conditions sufficient for analyticity

In reading a book All the mathematics you missed I came across this line: (The Cauchy Riemann equations coupled with the condition that the partial derivatives are continuous are sufficient to prove ...
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1answer
87 views

Inverse of an integral transform

Suppose that in a certain domain of analyticity we're given a function $A(s)$ in terms of the integral : $$A(s)=\int_{0}^{\infty}\frac{a(t)}{t(t^{2}+s^{2})}dt$$ How can we recover $a(t)$?
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1answer
191 views

Finding two analytic functions

I want to find two analytic functions (the first one is analytic in the upper half plane the second one in the lower half plane) $f_+(z)$ and $f_-(z)$ which satisfy $f_+(x)-f_-(x)=\frac{1-\cos x}{x}$ ...
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1answer
48 views

Essential singularity question

I'm asked to classify the singularity at the indicated poing and to find the residue at that point for $$f(z)=z^ne^{1/z}$$ for $$z_0=0$$ Here's what I have: ...
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2answers
144 views

Continuity of the derivative

As we all know, all the basic properties of holomorphic functions (i.e. functions which are differentiable in the complex sense) can be deduced from Cauchy's formula. Moreover, Cauchy's formula itself ...
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0answers
39 views

Determining the disc of convergence in two series and determining at which points on the boundary of the disc the series converges.

The two series are as follows: $f(z) = \sum\limits_{n = 1}^\infty n(z+1-i)^{2n}$ and $f(z) = \sum\limits_{n = 1}^\infty n^{-1}z^{n}$ I have worked out that the discs of convergence are, ...
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1answer
52 views

Show, a holomorphic function is constant

I am given that $\left| \frac{g'(z)}{g(z)}\right|\leq \frac{1}{\left|z\right|^2} \hspace{0.3cm}(*)$. I want to show that if $g$ is holomorphic in $\mathbb{C}$, it is constant. I am not sure if ...
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1answer
27 views

Show that the composition of the two functions is the identity.

I have to check that the composition of the following functions gives the identity (or that one function is the inverse of the other): $$\pi:S^2\backslash \{N\}\to\mathbb{C}$$ $$(x_1,x_2,x_3) ...
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2answers
86 views

Can we deduce that $h=f+g≠0$

Let us consider three complex functions $f,g,h$. Let $A$ be a set such that $f≠0$ and $g≠0$ in $A$. Can we deduce that $h=f+g≠0$ in $A$. If not can we add some conditions $f,g$ such that the property ...
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votes
5answers
80 views

Identity $\int_{-\infty}^{\infty}\frac{e^{uz}}{1+e^u} \mathrm{d}u=\frac{\pi}{\sin(\pi z)}$

I want to prove the identity $$F(z)=\int_{-\infty}^{\infty}\frac{e^{uz}}{1+e^u} \mathrm{d}u=\frac{\pi}{\sin(\pi z)}$$ First of all $F(z)$ defines an analytic function for $0<z<1$. I am little ...
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0answers
33 views

Integration of a complex function having square rooted denominator

How to evaluate this integral ? $$\int_{-\infty}^{\infty}\frac{1}{(w-ia)\sqrt{w^2-ibw+b^2c}}dw$$ where $a$, $b$ and $c$ are real and greater than zero. Does the residue theorem work on it? Please ...
0
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0answers
49 views

Prove $f(x,y,z)=e^{iy+z}$ is continuous on $\mathbb R^3$.

Prove $f(x,y,z)=e^{iy+z}$ is continuous on $\mathbb R^3$. I have already proved that other functions are continuous by using that $f, g$ are continuous implies $f+g$ and $fg$ are continuous. ...
1
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1answer
31 views

holomorphic function with integral coefficients

I'm trying to prove that an holomorphic function on $\{Z, |Z|<1\}$ and continuous on $\{Z, |Z|\leq 1\}$ with coefficients in $\mathbb Z$ is polynomial. I have tried to establish some partial ...