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

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 ...
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20 views

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

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

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

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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0answers
17 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? ...
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2answers
31 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 ...
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1answer
26 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 ...
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1answer
35 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 ...
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1answer
30 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 ...
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1answer
11 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 ...
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2answers
44 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}= ...
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2answers
51 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 ...
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2answers
49 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 ...
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0answers
21 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 ...
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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 ...
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2answers
87 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 = ...
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1answer
31 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 ...
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27 views

Tractable indefinite integral of the exponentiation of some function

Consider the function $z(s)\in\mathbb{C}$ defined as $z(s)=\int_0^s \exp\left[i(q u+\lambda(u))\right]du$ for some $q\in\mathbb{Q}-\mathbb{Z}$ and $\lambda(s)$ a $2\pi$-periodic real differentiable ...
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2answers
34 views

When can I and when can I not use complex replacement?

If I want to calculate: $$(2 cos(t))^3$$ Can I not replace cos(t) with $Re(e^{it})$ and calculate $(2e^{it})^3$ to be $8e^{3it}$ and thus the real part of this becomes 8cos(3t)? But that answer is ...
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1answer
17 views

Question about constructing complex entire function

How to construct an entire function for infinitely many prescribed values? i.e. I hope to find an entire function $f$ given $f(z_k) = w_k$ ($w_k$ might not be zero) for a sequence of $\{z_k\}$ with ...
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1answer
34 views

That for a polynomial of degree $n$ , $\displaystyle\frac{M(r)}{r^n}$ is non-increasing function of r .

Note: My previous question was wrong. This is the opposite of it, and I hope it is more sensible. Where the $\displaystyle M(r)=\operatorname{Max}_{∣z∣=r} \mid f(z) \mid$ , where $f(z)=p_n (z)$ , a ...
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0answers
54 views

Residue with half order pole?

I'm having issues evaluating the following integral using Cauchy's residue theorem. $$\int_{-\infty}^{\infty} \frac{e^{ix}}{\sqrt{x^2 - 1}} dx $$ Here's what I have tried. We have to make a ...
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1answer
26 views

Changing Cauchy integral along triangle to an integral along a circle

I don't want to tease the whole question. So basically I have a line integral which has a pole in a closed triangular path. I was just wondering if I can draw a circle enclosing this triangular path ...
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2answers
35 views

Evaluation of the type of singularity and Residue at that point

$$ \begin{align} f(z)=z^2e^{\frac{1}{z^3}} \end{align} $$ I need to determine the type of singularity and evaluate the Residue at $z_0=0$ We know that $e^{\frac{1}{z}}$ has an essential singularity ...
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0answers
33 views

Weird conformal map problem

Construct a conformal map from the region $\omega$ = open disk of radius 1 centered at 0 minus the closed disk of radius 0.5 centered at 0.5 to $\mathbb{D}$ = disk radius 1 centered at 0. I really ...
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1answer
23 views

Proof that $\int_{C}\frac{f(s)}{s-z}ds-\int_C \frac{f(s)}{s-\frac{1}{\bar{z}}}ds=2\pi i f(z)$.

Let $f$ be analytic inside and on the unit circle $C$. Show that, for $0<\mid{z}\mid<1$ $$ 2\pi i f(z) = \int_C \frac{f(s)}{s-z}ds - \int_c \frac{f(s)}{s-\frac{1}{\overline{z}}} $$ Using ...
3
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1answer
44 views

Let $f(z) = z + z^2$ and let $V = \displaystyle \{z \in \mathbb{C} : |z| < \frac{1}{2}, \frac{3\pi}{4} < arg\{z\} < \frac{5\pi}{4}\}$.

Let $f(z) = z + z^2$ and let $V = \displaystyle \{z \in \mathbb{C} : |z| < \frac{1}{2}, \frac{3\pi}{4} < arg\{z\} < \frac{5\pi}{4}\}$. $(a)$ Show that $f(V) \subset V.$ $(b)$ Let $f_n$ be ...
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20 views

Cauchy Goursat problem

Let k be the rectangle with corners $-2-2i,2-2i,2+i,-2+i$. Evaluate the integral: $$ \int_k \frac{\cos(z)}{z^4}dz $$ Would the best way to do this problem be to integrate along each contour line using ...
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0answers
32 views

Find a Harmonic conjugate $v(x,y)$ to $u(x,y)$.

Show that $u(x,y) = \frac{y^2}{x^3+y^3}$ in some domain and find the harmonic conjugate $v(x,y)$ to $u(x,y)$.
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1answer
34 views

Complex Analysis -Proving convergence

Suppose that $$z_n,z\in G:=\mathbb{C}-\{z\,:\,z\leq 0\}$$ and $$z_n=a_n e^{i\theta_n},z=ae^{i\theta}$$ where $-\pi<\theta,\theta_n<\pi$. Prove that if $z_n\to z$ then $\theta_n\to\theta$ and ...
4
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1answer
36 views

That for a polynomial of degree $n$, $\frac{M(r)}{r^n}$ is a non-decreasing function of $r$.

Where the $M(r)=\operatorname{Max}_{\mid z\mid=r}f(z)$, where $f(z)=p_n(x)$, a polynomial of degree $n$. My first attempt: maybe this is related to the Cauchy's inequality of estimating derivatives. ...
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1answer
17 views

Differentiation under the integral sign (one complex variable)

Let $u(z), u'(z)$ be complex-analytic functions on an open neighborhood $\Omega \subseteq \mathbb{C}$ of the origin. Also, let $f(X)$ be a complex-analytic function. For $s \in [0,1],$ define $$g(s,z) ...
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1answer
26 views

Estimate of complex integral

Prove that $$ \left|\int_c (2-\frac{e^z}{z-\log 2}) dz \right| <\frac{2}{3} $$ when C is the part of circle $\left| \frac{z}{\pi} -1 \right|^2 =2$ where $Re(z)\geq 0$. ($\log$ means natural ...
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2answers
33 views

How to find all values of $z$ at which $\begin{align} \sum^{\infty}_{n=1} \dfrac{1}{n^2} \end{align}\text{exp}\left(\dfrac{nz}{z-2}\right)$ converges

Could anyone advise me on how to find all $z$ such that $\begin{align} \sum^{\infty}_{n=1} \dfrac{1}{n^2} \end{align}\text{exp}\left(\dfrac{nz}{z-2}\right)$ converges ? Does it suffice to find all $z$ ...
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1answer
22 views

What is the proper way to determine the order of a root?

Find the multiplicity of the root at $z_0$ for these functions i) $$ \begin{align} f(z)= e^{zcos(z)-z}-1, z_0=0 \end{align} $$ Let $z_0$ be a root of a holomorphic function $f$ , and let n be the ...
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3answers
73 views

Show that $\displaystyle \int_\gamma \frac{f'(z)}{f(z)}=0$ for every closed curve $\gamma$ in $\Omega$

I have just started taking complex analysis course,The following problem is given in my class.Please help me solving it.Thnx in advance. Suppose $f(z)$ is anlytic and satis fies the relation ...
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1answer
27 views

Existence of a holomorphic map

Does there exist a holomorphic map $f:\mathbb C^2 \to \mathbb C^2$ whose rank at (0,0) is 0 but at all other points is 2? ($f$ could also be defined on a domain in $\mathbb C^2$)
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75 views

Can these holomorhic functions $f:D(0,1)\to \mathbb{C}$ exist

i) $$ \begin{align} \text{Let }f:D(0,1)\to \mathbb{C} \text{ holomorphic ,$\\$ Show that } f(\frac{1}{n})\ne \frac{1}{n+1} \end{align} $$ for all natural numbers,except maybe for some finite cases. ...
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1answer
46 views

Can there be an entire function such that $f(\frac{1}{2n})=\frac{1}{2n},$ and $f(\frac{1}{3})=\frac{1}{9}$

$$ \begin{align} f(\frac{1}{2n})=\frac{1}{2n}, \text{ and } f(\frac{1}{3})=\frac{1}{9} \end{align} $$ Prove that there can't be an entire function $f$ such that the upper conditions are fulfilled. ...
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46 views

Understanding proof for $\sum_{k=1}^{n-1}{\sin{\frac{2\pi k}{n}}} = 0$

(English is not my native language, so I apologize if I fail to use the right technical terms) I am stuck in proving the following. I'll explain how far I got and maybe someone can help me out by ...
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1answer
33 views

What is the generalization for a convolution in $\mathbb C$?

Since the integration range of "the" convolution is $\mathbb R$, what is a sensible generalization in complex numbers? Would one still integrate over $\mathbb R$, or some other path, or over the ...
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4answers
86 views

If $f,g$ are entire functions such that $f(g(z))=0, \forall z, $ then $g$ is constant or $f(z) =0, \forall z \ ?$

Let $f,g$ be entire functions such that $f(g(z))=0, \forall z.$ Could anyone advise me on how to prove/disprove: either $g(z)$ is constant or $f(z) =0, \forall z \ ?$ Hints will suffice, thank you.
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1answer
36 views

Can use residue theorem for this integral

I need to compute $$I=\int_C \dfrac{e^{\sqrt{1+u}}\cdot\sqrt[4]{1+u}}{\sqrt{u}} \,\mathrm {d}u$$ where $C$ is the unit circle. I am confused about whether I can use the residue theorem to compute it? ...
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1answer
17 views

Can any meromorphic function be represented as a product of zeroes and poles?

Given any meromorphic function, can it be represented as $$c\prod_i (z-z_i)^{n_i} $$ where $ n_i\in\mathbb Z$ and $n_i> 0$ denotes the multiplicity of the zero $ z_i $ and $ n_i <0$ for the ...
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2answers
35 views

Show that $f(x+iy)$ is holomorphic if and only if it can be expressed as a polynomial in the single variable $z$

I am very much new in complex analysis.The following question is given in class. Please help me to solve it. I have tried something myself which is also mentioned. Please help me. Thnx in advance. ...
9
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1answer
137 views

$f: \Omega \rightarrow \Omega$ holomorphic, $f(0) = 0$, $f'(0) = 1$ implies $f(z) = z$

Let $\Omega$ be a bounded connected open subset of $\mathbb{C}$ containing $0$. Let $f: \Omega \rightarrow \Omega$ be holomorphic and $f(0) = 0$, $f'(0) = 1$. The problem I am working on is to show ...
3
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2answers
81 views

Radius of Convergence of $ \sum\limits_{n=2}^{\infty} \pi(n) z^{n}$

I came across this question: If $\pi(n)$ denote the Euler function. What's the radius of convergence of this power series? $$ \sum\limits_{n=2}^{\infty} \pi(n) z^{n}$$ Any hint would be ...
0
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1answer
49 views

Absolute convergence of series $\sum_{n=1}^\infty \frac{1}{\frac{1}{z^n}-1}$

$$ \begin{align} \sum_{n=1}^\infty \frac{1}{\frac{1}{z^n}-1} \end{align} $$ Determine the values of $z,z\in\mathbb{C}$ so that the series converges absolutely I know that the series converges for ...
4
votes
1answer
51 views

Polydisc is not biholomorphic to any strictly pseudoconvex domain

I want to prove the poly disc $P=\left\{z\in \mathbb{C}^2 : |z_1|<1,|z_2|<1\right\}$ is not biholomorphic to any strictly pseudo convex domain in $\mathbb{C}^2.$ Can any one provide a hint?
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41 views

How to show that if a complex function is analytic then it is infinitely many times differentiable geometrically? [duplicate]

I am going through the theorem which proves that if a complex valued function is analytic than it is infinitely many times differentiable. But I am not sure how to explain this geometrically without ...
5
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2answers
57 views

Physical or geometric meaning of complex derivative?

As in, the real derivative of a function at a point is a slope of a function at that point. What is the physical or geometric meaning of complex derivative of a function at a point? Any help is ...