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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38 views

If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant

If $f:\mathbb{C}\longrightarrow\mathbb{C}$ is an entire function such that $f(z)\neq w$ for all $z\in \mathbb{C}$ and for all $w\in [0,1]\subset \mathbb{R}$, how to prove that $f$ is constant (without ...
1
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2answers
35 views

If $f[\mathbb{T}]\subset \mathbb{R}$ then $f$ is constant

If $f:\overline{\mathbb{D}}\longrightarrow\mathbb{C}$ is a holomorphic function over $\mathbb{D}$ and $f(\mathbb{T})\subset \mathbb{R}$ then is $f$ constant? Consider: ...
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1answer
24 views

Examples of divergent series summed by means of the analytic continuation of the corresponding

For my Bachelor's thesis, I am investigating divergent series. This is (yet another) question on this topic. Apparently, a divergent series $$ S = \sum_{n=1}^{\infty} a_{n} $$ can be summed by means ...
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1answer
17 views

Suppose that $f(z) = g(z)/h(z)$ is analytic on the annulus $\{1 < |z| < 2\}.$ Show that $f$ can be written as $f = G(z)/H(z)$

Suppose that $g, h$ are continuous, nowhere vanishing functions on $\{|z| < 2\},$ $\{{|z| > 1} ∪ ∞\}$ respectively. Suppose that $f(z) = g(z)/h(z)$ is analytic on the annulus $\{1 < |z| < ...
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0answers
23 views

If $\sum_{n=0}^{\infty}f^{(n)}(z_0)$ converges then $\sum_{n=0}^{\infty}f^{(n)}(z)$ converges [duplicate]

If $f:\mathbb{C}\longrightarrow \mathbb{C}$ is an entire function such that $\displaystyle \sum_{n=0}^{\infty}f^{(n)}(z_0)$ converges for some $z_0$, how to prove that ...
0
votes
0answers
6 views

An entire divison of two p-order entire fucntion is also p-order at most

Let $f_1$ and $f_2$ be two entire functions such $g=f_1/f_2$ is also entire. It is given that $f_1,f_2$ are of finite order $p$. I need to show show that $g$ is also of order $p$ at most. I've been ...
0
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1answer
75 views

Can a finite value for $\int_1^\infty \exp(x^2)\,dx$ be defined?

Why should $$\int_1^{\infty}\exp(ix^2)dx,\int_1^{\infty}\exp(-ix^2)dx,\int_1^{\infty}\exp(-x^2)dx$$ converges but not: $$\int_1^{\infty}\exp(x^2)dx$$ Is there any way that assigns a value to ...
1
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0answers
37 views

How many zero's does a general real entire function $f(z)$ have?

Let $f(z)$ be a real entire function. How do we find the number of solutions for $f(w)=0$ ? Can we express the number of zero's of $f$ in terms of its Taylor coëfficiënts ? Im not looking for the ...
0
votes
1answer
14 views

Principal Part of Laurent series' expansion of $f(z)=\frac{\sin(z^3)}{(1-\cos z)^3}$

I need to calculate principal part of the Laurent series expansion of $f$ at $z_0=0$ with $$ f(z)=\frac{\sin(z^3)}{(1-\cos z)^3} $$ I can see that $f$ has a pole of order 3 at $z_0=0$ , and also ...
1
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0answers
23 views

Jensen's Formula and the measure of $\{\theta \in [0, 2\pi]: f(e^{i\theta}) = 0\}$

Let $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. Suppose $f$ is continuous on $\overline{\mathbb{D}}$ and analytic on $\mathbb{D}$ with $f(0) \neq 0$. Then if $r$ is such that $0 < r < 1$ ...
0
votes
0answers
23 views

Power series with complex variables inequality

I am struggling to prove the following inequality: For $z \in \mathbb{C}, r \in \mathbb{R}, n \in \mathbb{N}$, if $|z| \leq r$ and $1 \leq r < n$ then ...
0
votes
2answers
46 views

geometric description of set of complex number

A set of complex number: $$S=\{ z\in \Bbb C : |z|=\lambda |z-1|\}$$ what's the geometric description? I try to draw it ... which seems like a circle but cannot find the equation to describe it..
2
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3answers
27 views

Factorization of $z^4 +1 = (z^2 - \sqrt 2z+1)(z^2 + \sqrt 2 z+1)$ for complex z

How can I get this equation from LHS to RHS by using the four roots of $z^4 +1 = 0$ are $z=\pm\sqrt{\pm i}$ $$z^4 +1 = (z^2 - \sqrt2 z+1)(z^2 + \sqrt2 z+1)$$
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0answers
17 views

Validity of Laurent series's principal part calculation

I need to calculate the principal part of the Laurent expansion of $f$ around a given $z_0$ in an annulus of the form $\{z\in \mathbb{C}:0<|z-z_0|<r$} and then use this to find $Res(f,z_0)$ ...
0
votes
1answer
53 views

Show that an entire function that is real only on the real axis has at most one zero, without the argument principle

Could someone advise me on how to approach this problem: Suppose an entire function $f$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero. without the use of argument principle ...
2
votes
2answers
46 views

If $f,g$ are entire functions and$\ fg\equiv 0$ then either $f \equiv 0$ or $g\equiv0. $

Let $f,g$ be entire functions such that $g \not\equiv 0.$ If $fg\equiv0$ in $\mathbb{C},$ could anyone advise me how to show $f \equiv0$ in $\mathbb{C} \ ?$ Thank you.
0
votes
1answer
27 views

Smooth parametrisation in the complex plane.

My book defines a complex smooth parametrisation like this. First a parametrisation is a complex funtion z of a real variable t. Where t is defined on $[a,b]$. It is smooth if $z'(t)$ exists and is ...
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1answer
25 views

Verification of Laurent Series calculation

I tried to calculate the Laurent series of these functions but I have no way to verify my answers. i) $$ \begin{align} f(z)=\frac{e^{z^2 }-1}{z^4}, \mathbb{D}=\mathbb{C} \backslash \{0\} ...
1
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5answers
71 views

If $p(z)$ is an injective polynomial $\Longrightarrow$ $p(z)=az+b$ [on hold]

If $p(z)$ is an injective polynomial, how to prove that $p(z)=az+b$ with $a\neq 0$. $p(z)\in\mathbb{C}[z]$. Any hint would be appreciated.
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votes
1answer
20 views

Supremum of the set $\{\operatorname{Re}(iz^3+1) : |z|<2\}$

I need to find supremum of the set of all real numbers of the form $\operatorname{Re}(iz^3+1)$ such that $|z|<2$. By the inequality $-|w|\le \operatorname{Re}(w)\le |w|$ we have ...
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2answers
34 views

If $p(z)$ is a monic polynomial then $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$

I need some help with this problem: If $p(z)$ is a monic polynomial of degree $n$ then there is a $b\in\mathbb{C}$ such that $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$ where $z_1,z_2,\dots,z_n$ are simple ...
2
votes
1answer
66 views

There is no nonconstant entire function $f$ such that $f(z+1)=f(z)$ and $f(z+i)=f(z)$ [duplicate]

Claim: there is no entire non-constant function $f$ such that $f(z+1)=f(z)$ and $f(z+i)=f(z), \forall z\in \mathbb{C}.$ May I verify if my proof is valid? Or is there a better way to approach this ...
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0answers
29 views

Prove: $|f(z)| ≤ \frac{2A|z|}{1−|z|}$.

Let $f$ be a holomorphic function on the unit disc $\{z : |z| < 1\}$ satisfying $f(0) = 0$ and $Ref(z) ≤ A$ for some positive number $A > 0.$ Prove: $|f(z)| ≤ \frac{2A|z|}{1−|z|}$. Not sure how ...
10
votes
6answers
373 views

Reference request for undergraduate complex analysis.

I am a second year student studying electrical engineering. I self-study pure mathematics and want to pursue a career as a mathematician. What are some prerequisites for studying complex analysis? ...
0
votes
2answers
51 views

Complex Analysis Problem and Advice

Let $f$ be an odd function that is holomorphic in $\mathbb{C}- \{0\}$ such that $|f(z)| \leq \dfrac{1}{|z|}+ |z|^2, $ where $z \neq 0.$ Could someone advise on how to show $f(z) = \dfrac{a_{-1}}{z} + ...
2
votes
1answer
14 views

Guidance / Help with Laurent series expansion in a certain annulus

I am trying to study complex analysis and I've come across this $$ \begin{align} f(z)= \frac{1}{1+z^2} \end{align} $$ I need to determine the Laurent series expansion for the annulus ...
0
votes
1answer
20 views

A question about the relation between divergence and absolute divergence.

Princeton Lectures in Complex Analysis by Stein and Shakarchi says the following: If $|z| > R$, then a similar argument proves that there exists a sequence of terms in the series whose ...
2
votes
2answers
24 views

how to calculate |exp(-ia)+exp(-ia')|^2

What is the correct way to calculate something like $|\exp(-ia)+\exp(-ia')|^2$ ? I have tried simply multiplying the term inside the absolute value by its complex conjugate, ...
3
votes
2answers
32 views

Characterization of entire functions to be a polynomial

I need some help with this proposition: If $f:\mathbb{C}\longrightarrow \mathbb{C}$ is an entire function such that $\{z\in \mathbb{C}:f(z)=w\}$ is finite for all $w\in \mathrm{Im} (f)$ then $f$ is a ...
0
votes
2answers
58 views

Direct evaluation of a series from Euler's identity.

Is there a direct way to evaluate: $$ \sum_{k=0}^{\infty} (-1)^k \dfrac{\pi^{2k}}{(2k)!}=-1 $$ Note that this follows from Euler's identity.
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1answer
31 views

Complex roots (review) (advise)

I have to find the complex roots and want a review of my procedure to see if is correct A. $$\sqrt{3i}$$ $$\left |z \right |=3 $$ $$phase= 90^{\circ}=\displaystyle\frac{\pi}{2}$$ ...
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vote
1answer
15 views

The number of zeros of a function regular on the closure on $U$ and the max of $\operatorname{Re}\frac{zf'(z)}{f(z)}$

Suppose that $f(z)$ is holomorphic on the closed unit disk $\bar U$ and never vanishes on the boundary $\partial \bar U$. Prove that the maximum of $\displaystyle ...
0
votes
0answers
23 views

Guidance or advice with the determination of the type of singularities

I need to determine the type of singularity in $f$ at $z_0=0$ and calculate the Residue at that point. $$ \begin{align} f(z)=(z^2+z) \cos\left(\frac{1}{z}\right) \end{align} $$ I know that ...
1
vote
2answers
45 views

What are the entire functions $f$ such that $|f'(z)| \leq |f(z)| \ ? $

Could someone advise me on how to determine all entire functions $f$ such that $|f'(z)| \leq |f(z)|, \forall z\ ?$ Hints will suffice, thank you.
0
votes
1answer
30 views

how do we find out maximum value of $|f(z)|$:complex analysis [duplicate]

Let $f(z)=2z^2-1$.Then what is the maximum value of $|f(z)|$ on the unit disc $D=\{z\in C:|z|\le1\} $ equals $2$ $3$ $1$ $3$ more than minimum value This question can have more than 1 answer? I ...
2
votes
0answers
65 views

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration$$ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$$ I am not sure as to how to work with the branch cuts of both $\ln{x}$ and $\sqrt{1-x^2}$ ...
0
votes
1answer
54 views

Evaluation of $\begin{align} \int^{\infty}_{0}\end{align} \dfrac{1}{1+x^n}dx$ with the use of Residue theorem [duplicate]

Could anyone advise me on how to show$\begin{align} \int^{\infty}_{0}\end{align} \dfrac{1}{1+x^n}dx=\dfrac{\pi}{n\text{sin}\dfrac{\pi}{2}} ,\ $ for all integers $n \geq 2 \ ?$ Thank you. Here is my ...
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votes
0answers
18 views

stereographic projection-how to solve the problem [on hold]

How do we calculate the point for the following question? by stereographic projection with south pole at origin z=i is projected as ? how do we consider the coordinates of the south pole? ...
3
votes
2answers
34 views

If $a_n\to0$, there exists $\pm$ such that $\sum\limits_n\pm a_n$ converges [duplicate]

Our Analysis I lecturer in his last lecture for the course gave us a problem to think about. I've been thinking about it for a while and has been bothering me for some time. It looks like a ...
0
votes
1answer
31 views

the absolute value of $\frac{1}{e^{i\omega t}-1}$

I am told to get the absolute value of $$\frac{1}{e^{i\omega t}-1}$$ I sense that there's something ridiculously simple about this, but I tried working from the fact that if I square it, the absolute ...
1
vote
1answer
42 views

Entire functions of order zero

I came across this question: If $f:\mathbb{C}\longrightarrow \mathbb{C}$ is an entire function of order zero then $f$ is a polynomial? Note that the converse is true. Any hint would be ...
1
vote
1answer
33 views

Proving that $\int_0^{\pi/2} |\exp(ire^{it})|dt < \pi /2r$.

Let $r >0$. I want to prove that $\int_0^{\pi/2} |\exp(ire^{it})|dt < \pi /2r$ for $t \in [0, \pi/2]$. The hint is to use that $\sin t \ge 2t/\pi$ for $t \in [0, \pi/2]$. I really don't know ...
0
votes
1answer
17 views

Analytic region problem

I am asked to express the function in terms of $u(x,y) +iv(x,y), z=x+iy$ and then determine the region where f is analytic. The function is: $f(z)=e^{z^2}$ I found $f(z)$ expressed in terms of ...
1
vote
2answers
46 views

Sum of Complex series

Let $\theta\in\mathbb{R}$ and $\theta \neq k\pi$ for $k\in\Bbb Z$. By summing a geometric progression show that $$1 + e^{2i\theta} + e^{4i\theta}+e^{6i\theta} + e^{8i\theta}= ...
-1
votes
2answers
59 views

complex analysis exercise

I have this exercise: The only thing I need help to is explaining why I can assume that z is real? I've tried assuming z is complex, and trying to rewrite it to a equivalent expression, but it didn't ...
0
votes
2answers
52 views

Can I conjugate a complex number: $\sqrt{a+ib}$?

Can I find the conjugate of the complex number: $\sqrt{a+ib}$? Actually my maths school teacher says and argues with each and every student that we can't conjugate "square root of $a+ib$" to "square ...
0
votes
0answers
26 views

Joukowski transformation of streamlines around cilinder in mathematica

I have problem transforming streamlines around cilinder, which is in fact simple circle that rotates, to a airfoil. It is done using Joukowski transformation $z = z+\frac{c}{z}$. The circle transforms ...
1
vote
1answer
15 views

An inequality related to the maximum of an analytic function in the disk

Let $f(z)$ be an analytic function in the disk $D$, continuous on the closure $\bar D$. Let $L$ be $\{\operatorname{Re}z=\frac{1}{2}\}\cap \bar D$. Let $M$ be the maximum of $\mid f(z)\mid$ in $\bar ...
4
votes
2answers
103 views

How to prove $\int^{\pi/2}_0 \log{\cos{x}} \, \mathrm{d}x = \pi/2 \log{1/2}$

ALREADY ANSWERED I was trying to prove the result that the OP of this question is given as a hint. That is to say: imagine that you are not given the hint and you need to evaluate: $$I = ...
1
vote
1answer
32 views

Differentiability of non-analytic complex functions

Any complex function that is analytic on an open set is differentiable on that set. But can a function fail to be analytic on an open set but still be differentiable? For example, the function ...