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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1answer
10 views

How do you deal with a multivalued antiderivative for complex integration?

I'm having to integrate functions of the kind $\frac{1}{z}$ on some path. For example the integral: $\int_{1}^{-1}\frac{dx}{x-i1}$ which corresponds to integrating on a straight line that connects ...
1
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0answers
29 views

Fast formula for gamma function

I am trying to find an efficient way of evaluating the gamma function gamma(t) where t is a complex number. The Wolfram Mathworld page http://mathworld.wolfram.com/GammaFunction.html gives a number ...
1
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0answers
24 views

Definite Integral: $\int_0^{2 \pi} \frac{d\phi}{z + b cos(\phi)}$

During my work, I stumbled upon this definite integral $$\int_0^{2 \pi} \frac{d\phi}{z + b cos(\phi)} = sgn(\Re(z))\frac{2\pi}{\sqrt{z^2-b^2}} \qquad z \in \mathbb{C}$$ which result I cannot really ...
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1answer
19 views

Sketching the Points in the Complex Plane

I am asked to sketch the points in the complex plane satisfying the given inequality: -pi < arg(z) < pi/2 If arg(z) = pi/4, what exactly is there to sketch?
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0answers
18 views

Contour Integral solution to differential equations, Euler transformation?

In Spain's book, Functions of mathematical physics he introduces the contour integral method of solving ODEs. The baseic idea is: given an ODE $\sum_0^m a_r(t) \frac {d^rf}{dt^r} = 0$, a solution may ...
0
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1answer
22 views

Find a conformal mapping which…

Find a conformal mapping which reflects the $|z|=1$ and $|z+2|=1$ circles in concentric circles with a center at the $z=2$ point.
0
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1answer
25 views

Verification of attempt Related to a question on Loiville's theorem(complex analysis)

I have to prove or disprove :There exists an entire(analytic on whole of the complex plane)function F such that F($0$)=$0$ ; F(1)=i ; F(z)==>$0$as z approaches $\infty$. My attempt: Since F(z)==>$0$ ...
2
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1answer
20 views

Show that if $T \in \mathcal{H}$ then $(T(z_1),T(z_2),T(z_3),T(z_4))=(z_1,z_2,z_3,z_4)$

Let $_1,z_2,z_3,z_4$ be disctint points of $\hat{\Bbb{C}}$, and let $$(z_1,z_2,z_3,z_4)=\frac{z_{1}-z_{2}}{z_{1}-z_{4}} \cdot \frac{z_{3}-z_{4}}{z_{3}-z_{2}}$$ Be the cross ratio (I ...
0
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0answers
26 views

How can I extract value of s from Xi function?

How can I extract value of s from Xi function? $$\xi(s)=(s/2)\Gamma(s/2)(s-1) \pi^{s/2}\zeta(s)$$ Exmple $y=x^2-1$ $x=\sqrt {y-1}$ Yes, I mean inverse function.
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0answers
14 views

Can a harmonic function exist which doesn't have a harmonic conjugate. [on hold]

Will partial derivatives of u(x,y) always be harmonic if u(x,y) is harmonic?
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1answer
15 views

Complex Analysis analytic function problem

Suppose that $f$ ang $g$ are two analytic functions on the set $\Bbb C$ of all complex numbers with $f(1/n) = g(1/n)$ for $n= 1,2,3,\ldots$, then show that $f(z) = g(z)$ for all $z\in\Bbb C$.
3
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4answers
58 views

Find $z$ when $z^4=-i$?

Consider $z^4=-i$, find $z$. I'd recall the fact that $z^n=r^n(\cos(n\theta)+(i\sin(n\theta))$ $\implies z^4=|z^4|(\cos(4\theta)+(i\sin(4\theta))$ $|z^4|=\sqrt{(-1)^2}=1$ $\implies ...
1
vote
1answer
35 views

Derivation of Perron's formula

I tried to derive Perron's formula, but got really screwed up. I know of other ways to derive it, but I'm not quite sure why this way isn't working. I would appreciate some pointers on where I'm going ...
1
vote
1answer
35 views

Radius of convergence of a complex power series?

Let's say I have a series which converges in a radius $R$ about point $a$. Let's say I expand the same about point $b$. How does the radius of convergence change? My progress so far :- I obtain it ...
0
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0answers
15 views

Can I determine complex differentiability by differentiating wrt to z?

If I have a function in terms of $z\in C$ and need to determine the points where it is differentiable, can I simply find the derivative wrt z and see where it is defined? I know that one solution is ...
4
votes
0answers
21 views

Complex Chebyshev Polynomials

Chebyshev Polynomials can be used to compute a very nearly minimax polynomial approximation of an analytic function on $[-1,1]$. Is there a complex analog that can compute a nearly minimax polynomial ...
5
votes
1answer
87 views

Proving $(1-x)\cdot (1-x^2)\cdots(1-x^{n-1})=n$ if $x^n=1$ and $x\neq 1$ [on hold]

If we have a equation $x^n=1$, then how can we prove $$(1-x)\cdot (1-x^2)\cdots (1-x^{n-1})=n $$ when $x$ is not $1$? I know that $x= e^{(2\pi + 2k\pi)/n}$ and we can get different value of $x$ when ...
6
votes
2answers
139 views

The meaning of the Imaginary value of the Residue while Evaluating a Real Improper Integral

When evaluating the improper integral $$\int_{0}^{\infty}\frac{x^{3}\sin\left(2x\right)}{\left(x^{2}+1\right)^{2}}\,dx$$ (which is an even function, so half of the $(-\infty,\infty)$ integral), I used ...
2
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0answers
25 views

Assume that if in the power series expansion(around $0$) of $p(z)/q(z)$ all the coefficients of power series are integer then $q(z) \in \mathbb Z[z]$

I'm reading a paper in which following result is left by saying that its an easy exercise but I'm finding it a bit hard.Can someone give some ideas to complete this problem? Let $p(z),q(z) \in ...
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votes
4answers
42 views

How to solve the equation $a\cdot e^x-b=x \cdot e^x$ with $a,b\in\mathbb{C}$ [on hold]

$$a\cdot e^x-b=x \cdot e^x$$ How to find all the solutions through Lambert function ?
0
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0answers
37 views

How to calculate $\arg(e^{-1+7i})$

I'm trying to calculate $\arg(e^{-1+7i})$. I have: $$ \arg(e^{-1+7i})=\arg(e^{-1}e^{7i})=\arg(e^{-1})+\arg(e^{7i})=\arg(e^{7i})=\arctan\left(\frac{\sin(7)}{\cos(7)}\right). $$ According to my ...
6
votes
5answers
349 views

Power series problem in complex analysis

Suppose that $f(z)= (e^z)/(1-z)$ How can I find out Power series expansion of f about $z=0$?? Is the use of cauchy product must here ? Can it be done without using cauchy product?Please help.. Many ...
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0answers
36 views

Possible proof strategy for Sendov conjecture?

Sendov's conjecture says that if all roots of a polynomial lie within the unit disk, then for every root, there exists a critical point at a distance at most one from the root. I read that Sendov ...
0
votes
0answers
35 views

Real integral done by complex methods [duplicate]

$\int_{-\infty}^{\infty} \frac{cosx}{x^2+25} dx $ = $ \frac{\pi}{5e^5}$ Any ideas?
0
votes
1answer
31 views

Show that if $f$ is analytic on a domain $D$, and if $|f|$ is constant, then $f$ is constant. [duplicate]

If $f(z)=0$ for some $z\in D$ then since $0$ is a constant, $f'(z)=0$ on $D$. Also since $f$ is analytic, then by theorem $f(z)$ is constant. Here is where I get stock! If $f(z)\not=0$. I want to ...
1
vote
1answer
17 views

Continuity of the following complex function

I am working through the text Complex Analysis by George Cain and I have a question stemming from the problem #9 in section 2.2. Consider the function $f(z)$ given by $\frac{\overline{z}^2}{z}$ when ...
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2answers
35 views

Is my explanation correct regarding Maximum value of Sine function over $\Bbb C$?

Question: What is the maximum value of sine function taking domain as $\Bbb C$? My answer is: The maximum value is not defined. Explanation: Since the range of sine function is $\Bbb C$ and $\Bbb C$ ...
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1answer
23 views

Isolated singularities of this function?

I have a function $\sin (\frac 1 z) \sin(wz)$ where $w=\exp (\frac {i\pi} {4} )$. I need to find out the singularities of this function. My progress :- This function will have singularities on ...
4
votes
1answer
59 views

Is the set open?

Define a complex polynomial $p:\mathbb{C}\longrightarrow\mathbb{C}$ where $\deg p=n\in\mathbb{N}$. \begin{equation} p(z) = \alpha_{n}z^{n}+\alpha_{n-1}z^{n-1}+\dots+\alpha_{1}z+\alpha_{0},\quad ...
1
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0answers
33 views

conjectured generalization of euler's formula.

Given the elliptic modulus $k$ ,such that the complementary modulus is defined by $$k'\equiv \sqrt{1-k^2}$$,the jacobi amplitude $$\phi\equiv am(u|k)$$ and $K(k)$,is the complete elliptic integral of ...
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votes
1answer
26 views

Regarding Power series in complex analysis [on hold]

Suppose that I have a series $\sum_n^{\infty} \frac{z^n}{n}$.It is convergent for $|z|<1$. I want to know why the above series converges for $|z|=1$ except at $z=1$.
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votes
2answers
74 views

In $\Bbb C$, are polynomials open maps?

If $p$ is a polynomial, is it true that for every open $A\subseteq\Bbb C$, $p(A)$ is open? I really don't know how to approach this. I'm fairly certain that they're closed maps, though.
3
votes
1answer
20 views

Residue of $g(z)g'(z)$

I know how to use the residue theorem and the winding number to find the residue of $f$, but I have no idea how to relate the residue of $g$ to that of $g\cdot g'$, especially without knowing ...
5
votes
1answer
58 views

Proof of Vandermonde's Identity using a “different approach” using complex integration

Hi I'd like to know if the following proof of Vandermonde's Identity is correct (is really easy): Let $m,n,r$ be natural numbers such that $r\le \min \{m,n\}$. The Vandermonde's Identity gives us ...
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0answers
18 views

What are the loci of points z which satisfy the following relations?

a) |Z-Z1|=|Z-Z2| b) 0< Re(iZ)<1 c) |z|=ReZ+1 d) Im((Z-Z1)/(Z-Z2))=0 The professor did not explain loci in class and the text does not have any examples, so I am completely lost.
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0answers
18 views

Prove that $ \left|\sum_{k=1}^{n}z_kw_k\right|^2 z_k,w_k \in \mathbb{C}$ [on hold]

Prove that $$\left|\sum_{k=1}^{n}z_kw_k\right|^2=\sum_{k=1}^{n}|z_k|^2 \cdot \sum_{k=1}^{n}|w_k|^2-\sum_{1 \leq k , l \leq n}^{}|z_k\bar{w_l}-z_l\bar{w_k}|^2$$
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2answers
52 views

(complex analysis) Prove that: $\arg ((z_3-z_2)/(z_3-z_1)) = 1/2 \arg z_2/z_1$

If $|z_1|=|z_2|=|z_3|$ Urgent help needed. I have used: $z_1=x_1+\mathrm iy_1,z_2=x_2+\mathrm iy_2,z_3=x_3+\mathrm iy_3$ and obtained $$\arg\frac{z_3-z_2}{z_3-z_1} = \arctan ...
5
votes
2answers
46 views

Showing that $\{z\in\mathbb{C}:|z-1|<|z+i|\}$ is an open set

Got stuck on some homework (from H. A. Priestley, Complex Analysis). My topology ain't quite up to speed yet. So, I want to show that $S=\{z\in\mathbb{C}:|z-1|<|z+i|\}$ is open. Geometrically it's ...
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1answer
46 views

Problems with understanding analyticity

I have a problem understanding the idea behind Analytic functions. (Please correct me on my terminologies while I state my problem). An analytic function, is a function that has a power series that ...
0
votes
1answer
29 views

Calculating the residue of a complex funciton with ln(z) at z=0

How can I calculate this residue: $$Res\left(\frac {z\ln(z)}{(z^2 +1)^3} , 0\right) $$ if it's possible at all. I know $0$ is a branch point for $\ln(z)$ and therefore isn't a pole, but when i plug ...
0
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2answers
49 views

Finding an Arg(w)

The question is: Describe (in words) and sketch the set of all $z \in \mathbb{C}$ such that $$\displaystyle 0<\arg\left(\frac{i-z}{i+z}\right)<\frac{\pi}{2}$$ I believe that I am supposed to ...
1
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1answer
22 views

Applying Cauchy theorem to the integral of $\overline{z}^2$ over two different curves.

I have solved the following exercise, in which I had to compute $$\int _\gamma \! \overline z ^2 \, \mathrm{d}z,$$ where $\gamma$ is The circumference $\left| z\right| =1$ The circumference ...
1
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1answer
32 views

Find the analytic function

$f(z)=1 $ satisfies the condition Using Identity Theorem $f(z)=1$ can be only function that satisfies this. so option (b) is NOT true. Am I on correct path?
2
votes
2answers
49 views

Sketch the complex function: $z\overline{z}+(1+2i)z+(1-2i)+1=0$

Tried sketching the complex function: $z\overline{z}+(1+2i)z+(1-2i)+1=0$ I first simplified it by converting $z=x+yi$ I got: $(x+yi)(x-yi)+(1+2i)(x+yi)+(1-2i)+1=0$ Which gave me this implicit ...
0
votes
1answer
21 views

Show that the imaginary part of $\frac{z^2}{z-z_p}$ is harmonic

Let $z\in\Omega \subset\mathbb{C}$ and $z_p\notin \Omega$. Show that $\text{Im}(\frac{z^2}{z-z_p})$ is harmonic in $\Omega$, where $\text{Im}(z)$ is the imaginary part of $z$. So far: For $z = \alpha ...
0
votes
1answer
23 views

why $f(z) = z^{(3/2)}$ does not have derivative at z = 0 in complex plane.

it seems that the $f'(z) = z^{(1/2)}$ means that this function has derivative for every complex value. But why $f(z) = z^{(3/2)}$ does not have derivative at z = 0
2
votes
0answers
19 views

Classify entire functions satisfying $|f(z)|\leq (1+|z|)^2$

I have to classify entire functions satisfying $|f(z)|\leq (1+|z|)^2$ for all $z\in \mathbb{C}$. Using Cauchy integral's formula, I've shown that $f^{(3)}=0$. Thus $f(z)=a+bz+cz^2$ for some $a,b,c ...
0
votes
1answer
27 views

Harmonic functions on $\{1<|z|<2\}$

I have to find all complex-valued harmonic functions on $\{1<|z|<2\}$ that extend continuously to $\{|z|=2\}$ and take value $0$ on that circle. My first idea was to map $\{1<|z|<2\}$ to ...
1
vote
3answers
40 views

Proving uniform convergence on disk within radius of convergence

Needham's Visual Complex Analysis 2.III.2 states that a power series $S_k=\sum{C_k z^k}$ with RoC $R$ converges uniformly on any disk $r<R$. He leaves the proof as an exercise to the reader. But ...
-4
votes
0answers
37 views

Entire function satisfying $f(z + 1) = f(z)$ and $f(z + i) = f(z)$ is to be proven constant [on hold]

Show that an entire function satisfying $f(z + 1) = f(z)$ and $f(z + i) = f(z)$ is a constant.