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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3
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19 views

Is there any significance to complex function “monotone in norm?”

So, I was reading a question earlier where someone asked if something would be strictly monotone in the complex plane, and the comment was that this would be meaningless, since the complex numbers ...
-4
votes
1answer
34 views

What is the constant term of the Laurent Series for $\cos(z)/z^2$? [on hold]

What is the constant term of the Laurent Series for $\cos(z)/z^2$? I want to prove that the constant term from this series is $-1/2$.
11
votes
2answers
107 views

Is there a good way to solve for z the equation $e^{i\pi} = e^{z\ln2} + e^{z\ln3}$?

$e^{i\pi} = e^{z\ln2} + e^{z\ln3}$ How can I deal with this? I want to solve for z. Does this help? $e^{z\ln2} + e^{z\ln3} = e^{z\ln2}(1 + e^{z(ln3-ln2)})$ If I write out z=x+iy then the ...
0
votes
1answer
16 views

List all the elements of the subgroup of Möbius transformations preserving the set $\{0, 1 + i, \infty\}$

List all the elements of the the subgroup $M_{\{0, 1, \infty\}}$ of the group of Möbius transformations, preserving the set $\{0, 1, \infty\}$ and give an explicit isomorphism $M_{\{0, 1, \infty\}} = ...
1
vote
1answer
18 views

Finding the residue of a function with a surd variable

$$f(z) = \frac{1}{z-2\sqrt{z}+2}$$ Is this the correct way of doing this, please advice - Thanks, what i did was try to rationalise the expression first as follows: $$f(z) = \frac{1}{z-2\sqrt{z}+2} ...
1
vote
1answer
14 views

Elliptic functions $f(z+\lambda_1)=af(z) \; , \; f(z+\lambda_2)=bf(z) $

Let $\lambda_1$ and $\lambda_2$ be complex numbers with nonreal ratio. Let $f(z)$ be an entire function and assume there are constants $a$ and $b$ such that $$f(z+\lambda_1)=af(z) \;\;\;\;,\;\;\;\; ...
5
votes
2answers
55 views

Evaluating sums using residues $(-1)^n/n^2$

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
3
votes
1answer
34 views

Simply connected and connected in complex analysis

Can some one please help me with this, why is third set in the picture not simply connected. The definition of simply connected (in space of complex numbers) is: A set is said to be simply ...
1
vote
2answers
25 views

How is $ \lim_{z \to z_o} (z-z_o)\frac{f(z)}{g(z)} = \lim_{z \to z_o} \frac{f(z)}{g(z)-g(z_o)/(z-z_o)}= \frac{f(z_o)}{g'(z_o)}$?

I was reading this proof in Gamelin Complex Analysis (page 196): If $ f(z) $ and $ g(z) $ are analytic at $ z_o $ and if $ g(z) $ has a simple zero at $ z_o $ $$ Res[ \frac{f(z)}{g(z)},z_o ] = ...
1
vote
2answers
32 views

Comparison of the consequences of uniform convergence between the real and complex variable cases,

In the real variable case, I think that uniform convergence preserves continuity and integrability, i.e., for an integral of a sequence of continuous (or integrable) functions, which converge ...
1
vote
0answers
21 views

Convergence implies Abel summability, and we only need to consider when $s=0$?

Suppose $\displaystyle c_n\in\mathbb{C}\textrm{ and}\sum_{n=1}^{\infty}c_n=s$. Then, prove $\displaystyle\lim_{r\to 1^{-}}\sum_{n=1}^{\infty}r^{n}c_n=s$. In my text, the author hinted that: we only ...
-3
votes
0answers
27 views

inequalities related to linear factional transformation and schwarz`s lemma

In both questions, it is said that I should use schwarzs lemma and linear factional transformation. But I don`t have any ideas how to use it. please give me some more hint
0
votes
0answers
24 views

How to prove that $f$ can be expressed as a ratio of polynomials, given that $|f(z)|=1$ when $|z|=1$? [duplicate]

Given: $f$ is analytic in $| z|\leq1$ and $|f(z)|=1$ when $|z|=1$. Prove that $f(z)=P(z)/Q(z)$ where $P$, $Q$ are polynomials.
3
votes
1answer
78 views

how to compute this complex integral with high order polynomial?

Compute $$ \lim_{R \to \infty} \frac{1}{2\pi i} \int_{|z|=R} \frac{(2z^2+z-1)P'(z)}{P(z)+3}dz $$ where $P(z)=z^{10}+2z^9+z^5+1$. It seems like I may use residue but it contains too high ...
0
votes
0answers
17 views

how to prove that holomorphic function mapping complex onto complex is linear? [duplicate]

How to prove that any one to one holomorphic function mapping complex plane onto itself is linear?
1
vote
1answer
25 views

Help with simplification of an expression

I was solving the residues of $f(z)e^{zt} = e^{zt}\frac{\ln(z)}{z^2+1}$ as follows: $$\operatorname{Res}(f(z)e^{zt}, i) = \lim_{z\to i} (z-i)\frac{e^{zt}\ln(z)}{(z-i)(z+i)} = ...
7
votes
0answers
73 views

Function's analytic continuation is its own derivative

This is the question we were asked at the university by our professor for complex analysis. Not as an exam, but as a challenge. I don't think he knew the answer himself. Find a nontrivial example of ...
0
votes
2answers
43 views

Taylor expansion of the Error function

The error function $\operatorname{erf}(z)$ is defined by the integral $$ \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2}\,dt,\quad t\in\mathbb R$$ Find the Taylor expansion of ...
7
votes
2answers
122 views

How can I prove $\sum_{n=1}^{\infty }\frac{1}{n^3(n+1)^3}=10-\pi ^2$

Can the residue theorem prove this? $$\sum_{n=1}^{\infty }\frac{1}{n^3(n+1)^3}=10-\pi ^2$$
2
votes
0answers
34 views

Need help with holomorphic functions on a domain interval removed.

I want to prove that for a region $\Omega$ with interval $I=[a,b]\subset\Omega$, if $f$ is continuous in $\Omega$ and $f\in H(\Omega-I)$, then actually $f\in H(\Omega)$. Is this problem related to ...
6
votes
1answer
103 views

Show that $\{e^{in}: n\in\Bbb N\}$ is Dense in the Unit Circle

This problem gave me fits when I was in grad school. Looking back at it now, it still escapes me. The problem is from Conway's Functions of One Complex Variable. I'm looking for a proof from basic ...
0
votes
1answer
24 views

Fourier transform, quadratic function

I'm trying to compute this convolution: $\frac{2 \alpha}{\alpha ^2 + 4 \pi ^2 x^2} * \frac{2 \beta}{\beta ^2 + 4 \pi ^2 x^2}$ I know that the Fourier transform of a convolution of two functions is ...
0
votes
2answers
26 views

In dual numbers, what number is represented by the following matrix?

In dual numbers, what number is represented by the following matrix? \begin{pmatrix}0 & 0 \\1 & 0 \end{pmatrix}
0
votes
1answer
28 views

Two different results with contour integration

This is probably going to be a stupid question ( I don't feel great today) but I can't get around this problem. $$I = \int_\mathbb R \frac 1 {(3x-2i)^2} dx $$ I thought that using contour ...
0
votes
1answer
35 views

$|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$

Let $p(z)= \sum_{k=0}^n a_k z^k$ , $a_n \neq 0$ , be a polynomial of degree $n$ such that $|p(z)| \leq M$ for $|z| \leq 1$. Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$ This was an exam ...
1
vote
1answer
48 views

$ \int_\gamma \frac{1}{z\sin z}dz$ where $\gamma$ is the circle $|z| = 5$

My understanding is that if this integral exists in the real sense, i.e. real Riemann-wise, then I can apply the residue theorem. If not, I may use the Cauchy Principal Value, to obtain a value. To ...
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0answers
22 views

an example of functions which is essentialy bounded but not continuous in circle

Can you give me an example of a function which is essentially bounded but not continuous in the unit circle and bounded in the open unit ball?
0
votes
1answer
37 views

How can I prove that the zeroes of $f(z)=1+1/2^z$ have no real part?

I want to prove that the zeroes of the function $f(z)=1+1/2^{z}$ have no real part. Is the following correct? $f(z)=0$ so $2^{z} = -1$ and $-1=e^{i\pi}$ so $e^{i\pi} = e^{z\ln2}$ therefore $z= ...
0
votes
1answer
19 views

Bounded functions composed with Möbius maps

Hopefully easy question here: What is the most succinct method/technique to prove the following statement?: Let $u \in L^{\infty}(\Bbb D)$. Show that $||u(\varphi_{z})||_{\infty}$ is ...
0
votes
0answers
40 views

What does complex square root as defined on Wikipedia look like: two questions

If you look at the third picture here, the surface representing the complex square root intersects the negative real axis at $0$. Later in the article the definition of the complex square root is ...
1
vote
3answers
28 views

How to show that a complex function have a branch in a domain

I've given as homework to show that the function $$f(z)=\sqrt{\frac{z+1}{z-1}} $$ has a branch on $G = \mathbb C \backslash [-1,1] $. I'm having a hard time in finding the way to approach this kind ...
0
votes
0answers
24 views

Searching for a constant transformation in $ \mathbb C$

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with a set $B \subseteq \mathbb C $, which is bounded. Now I want to proove that $ A = f^{-1} (B)$ is NOT bounded! I know it ...
2
votes
1answer
77 views

An entire and one-to-one function must be of the form AZ+B, A non-zero. How to rule out higher degree polynomials in z? [duplicate]

Show that if f is entire and one-to-one, then it must be of the form AZ+B, with A not equal to zero. I am editing my question, since there are duplicates on this forum to the question of why an ...
0
votes
0answers
28 views

Is restriction of $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection?

I have this question: Is the restriction of exp function $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection? Here's what I tried: ...
0
votes
1answer
17 views

Biholomorphic mapping of $\tan(z)$

I'm supposed to solve this question: Show that the function $\tan$ maps the vertical strip $-\frac{\pi}{4}<x<\frac{\pi}{4}$ biholomorphically to $\dot B(0,1)$ It is obvious that ...
8
votes
1answer
50 views

Is this contour continuously deformable into a circle?

As an exam question, we had to solve the integral of $\frac{1}{z}$ over the following contour: (The contour is a sequence of straights arcs joining -1, -$\frac{i}{2}$, $\frac{1}{2}$, i, ...
2
votes
2answers
47 views

How to compute the residue of $(z^2+2z+1)\sin\left(\frac{1}{1+z}\right)$

This was an example given in my notes but all it concluded was with something about an infinite principal part. How do we compute it? we have it equal to $ \left( z + 1 \right)^2 \cdot \sin \left( ...
0
votes
1answer
16 views

The Laurent series expansion of 1/((z^2)(z+1))

In an example in our notes it says: Compute the Laurent series for f(z)= 1/z^2(z+1) and determine the annulus of convergence. No more information was provided. So I did it on my own by factoring it ...
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vote
2answers
46 views

Integral of rational function in the complex plane

Let $P$, and $Q$ be complex polynomials such that $\deg Q \ge \deg(P) + 2$ Prove that there exists $r > 0$ such that if $\gamma$ is a closed curve outside $\{z : |z| \le r\}$, then $$\int ...
8
votes
1answer
84 views

Prove that an analytic function, real-valued on radii $[0, 1)$ and $[0, e^{i\pi\sqrt 2})$, is constant on the open unit disk

Suppose $f$ is analytic in the open unit disc and real valued on the radii $[0, 1)$ and $[0, e^{i \pi \sqrt 2})$. Prove that $f$ is constant. I'm not sure how to solve this.
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votes
1answer
23 views

Limit a complex contour integral

Let $z_{0}$ be a simple pole of $f$ and $\gamma_{r}$ an arc of circle centered on $z_{0}$, of the radius $r$ and angle $\alpha$, i.e., $\gamma_{r}=z_{0}+re^{it}$, with $t\in ...
0
votes
1answer
47 views

find the possible values of z

given two complex number $z,w$ such number that $|z|\le1,|w|\le1$ and $|z+iw|=|z-i\overline{w}|=2$, then find the possible values of $z$ i tryed to use triangular inequality and got that ...
4
votes
1answer
46 views

How to show $\int_0^1\frac{e^{e^{2\pi it}}}{e^{2\pi it}}dt=1$

I was trying to integrate the contour integral $$\int_\gamma \frac{\vert z \vert e^z}{z^2}$$ where $\gamma$ parametrizes the unit circle counterclockwise. I cannot use the Generalized Cauchy Integral ...
2
votes
1answer
54 views

Constructing an entire function

This is a question from my complex analysis final exam: Does there exists an entire function $f$ such that $f(\log k)=1/\log k$ for all $k\geq 2$, integer. My answer is a no. What do you guys think?
0
votes
2answers
59 views

How to find the residues of $\frac{1}{(z^4+4)^2}$?

How to find the residues of this function? $$\frac{1}{(z^4+4)^2}$$ So far, I found the poles: $z_1=-1-i$, $z_2 = -1+i$, $z_3=1-i$, $z_4=1+i$. I know they are of the second order. But I have ...
0
votes
1answer
26 views

Find the transformation that maps real axis to itself and imaginary axis to the circle $|w-\frac{1}{2}|=\frac{1}{2}$

Find the transformation that maps real axis to itself and imaginary axis to the circle $|w-\frac{1}{2}|=\frac{1}{2}$ What I did: $$z_{1}=0,z_{2}=i,z_{3}=\infty ...
0
votes
1answer
19 views

$\left|(1+R^2e^{2i\theta})^2\right| \geqslant (R^2-1)^2$ in complex integration

I need to prove: $$\lim_{R\to +\infty} \left|\int_0^\pi \frac{e^{iaR(\cos\theta+i\sin\theta)}}{(1+R^2e^{2i\theta})^2}iRe^{i\theta} d\theta\right| =0$$ Could someone give me some pointers? A ...
1
vote
3answers
70 views

How to use complex analysis to find the integral $\int^\pi_{−\pi} \frac 1 {1+\sin^2(\theta)} d\theta$?

How can I use complex analysis to solve the following: $$\int^\pi_{−\pi} \frac 1 {1+\sin^2(\theta)} d\theta$$
2
votes
1answer
44 views

Can two analytic functions that agree on the boundary of a domain, each from a different direction, can be extending into one function?

Let $D=\{z:|z|\leq 1\}$ be the unit disc in $\mathbb{C}$. Say $f$ is analytic on $D$ and $g$ is analytic on $\overline{D^c}$, and that $f|_{\partial D}=g|_{\partial D}$. Is there necessarily an ...
3
votes
1answer
27 views

Is i holomorphic over the whole complex plane?

That is, is i entire? I know that it's derivative with respect to z bar is 0, so I would think that the answer is yes, although I'm not sure.