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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190 views

the real part of a holomorphic function on C \ {0, 1}

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω = \text{C \ {0, 1}}$. Show that there exist unique real numbers $a_0$, $a_1$ such that $u(z) = h(z) − a_0 \log |z| − a_1 \log ...
3
votes
1answer
114 views

Let f be analytic on ∆

The problem is: let $f$ be an analytic function on $\Delta$ and satisfy $|f|<1$. Prove that if $f(1/2)=f(−1/2)=0$, then $|f'(0)|\le 1/4$. I tried to expand $f$ at $0$ and then plug in $1/2$ and $-...
2
votes
1answer
27 views

How to find the singularities of the function $z(1-e^{\frac{1}{z}})$ and classify them

Find the singularities of the function $z(1-e^{\frac{1}{z}})$ and classify them. I'm fairly sure that due to the exponential term overpowering the factor $z$, there will be an essential singularity ...
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1answer
61 views

Find $\int_{|z|=R} \frac{1}{(z-b)(z-a)^m} dz$

I have to find $\int_{|z|=R} \frac{1}{(z-b)(z-a)^m} dz$ for $|a| <R < |b|$ I would use Cauchy formula but first what can I do with $\frac{1}{(z-b)(z-a)^m}$? I dont remember it.
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2answers
387 views

If |f| is constant, f is constant.

I am confused as to how they got from the two equations being equal to 0 to the derivative being 0. I could be really tired right now but this isn't really making sense to me. I was thinking of doing ...
0
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1answer
97 views

Distance on riemann sphere [duplicate]

Let we have $C$ the set of complex numbers and $z_1 , z_2 \in C $ we have $Z_1 , Z_2 \in S$ correspond on riemann sphere and we will define : $$ d(Z_1,Z_2)=\frac{2|z_1-z_2|}{\sqrt{1+|z_1|^2} \sqrt{1+|...
2
votes
2answers
219 views

Why does the function $f(z) = 1/\sin(\pi/z)$ have isolated singular points?

In the complex analysis text book "Complex Variables and Applications 8th edition", it states the function $1/sin (\pi/z)$ has singular points $z = 0$ and $z = 1/m; (m = \pm 1,2,3,4,\dots.) $ I sort ...
5
votes
2answers
146 views

How to choose a contour in order to use the residue theorem to sum up a series from Ryzhik?

I would like to know how to sum up to following series (from the Gradshteyn-Ryzhik tables): $$\sum_{n=-\infty}^\infty\frac{e^{in\alpha}}{(n-\beta)^2+\gamma^2}=\frac{\pi}{\gamma}\frac{e^{i\beta(\alpha-...
2
votes
2answers
173 views

Using Complex Analysis to Compute $\int_0 ^\infty \frac{dx}{x^{1/2}(x^2+1)}$

I am aware that there is a theorem which states that for $0<a<2$ we have $$\int_0^\infty\frac{x^{a-1}}{x^2+1}dx=\frac{\pi \cos\big(\frac{a\pi }{2}\big)}{\sin (a\pi) }$$ but I prefer to evaluate ...
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0answers
21 views

Switching between Cartesian coordinate and polar coordinates

Under what assumption, every non-zero complex number represented in Cartesian coordinate system admits unique polar representation and vice versa ?
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1answer
49 views

What are 'Regular Products'?

When looking at the functional equation for the Riemann zeta function, I came across the statement: For $s$ an even positive integer, the product $\sin{(\frac{\pi s}{2})}\Gamma({1-s})$ is regular. ...
2
votes
2answers
97 views

When the argument of complex numbers is a well defined real valued function?

I know that the argument $\arg:\Bbb C\setminus\{0\}\to\Bbb R$ is multivalued function and also that if we consider $\arg:\Bbb C\setminus\{0\}\to{\Bbb R}/{2\pi \Bbb Z}$, then it is a well defined ...
0
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0answers
115 views

Holomorphic functions (continuity of partial derivatives)

Let $f:\Omega\rightarrow \mathbb{C}$ be an holomorphic function i.e. for any $z_0\in \Omega$ there exists the limit: $$f^{'}(z_0) = \lim_{z\mapsto z_0}\frac{f(z)-f(z_0)}{z-z_0}.$$ Let us write $f(z) =...
1
vote
1answer
126 views

Residue theorem

Let us say we need to perform the classic integral $$ I=\int_{-\infty}^{+\infty}dz \,\frac{e^{itz}}{z^2+1}~, $$ where $t>0$. What is normally done is the following. We consider the integral $$ K=\...
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1answer
18 views

Unsure about expansion

Hello, can someone tell me how this expression is expanded in this proof. Does it follow from some other theorem?
0
votes
1answer
91 views

Show that $\sum r^n cos(nx)=rcos(x)-r^2/(1-2rcos(x)+r^2)$

I'm a little unsure about how to approach this. I've been told that we have to use the relationship that $\sum r^n=1/1-r$. However, I'm not too sure what to do with the $\cos(nx)$. Can someone give ...
0
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1answer
74 views

Taylor Series expansion and radius of convergence for $e^z+e^{-z}+2cosz/4$

So I did this by taking apart bits of that long equation: $e^z=\sum z^n/n!$, $e^{-z}=\sum(-z)^n/n!$ $2\cos z=e^{iz}+e^{-iz}$ So when we put these together as a Taylor Series, do we just add them ...
1
vote
1answer
48 views

Constructing an entire function with a given isolated zero set

We know that a nonzero entire function on $\mathbb C$ has an isolated set of zeroes. Is it the case that, given an isolated set, there is an entire function which vanishes precisely on that set? If ...
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0answers
123 views

Real Cross Ratio Example

Theorem The cross ratio $(z_1, z_2, z_3, z_4) $ is real if and only if the four points lie on a circle or on a straight line. I know by geometry I can obtain $arg(z_1, z_2, z_3, z_4) = arg(\frac{...
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0answers
39 views

Counterexample to Line Integral depending on end points

Theorem The line integral $$\int_\gamma p\, dx + q\,dy$$ defined in $\Omega$, depends only on the end points of $\gamma$ if and only if there exists a function $ U(x,y) $ in $\Omega$ with the ...
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2answers
59 views

Analyticity of complex derivative and conjugate

Given a complex function $\phi(z)$ that is analytic, can I say that the following are analytic? 1) $\phi '(z)$ 2) $\overline{\phi '(z)}$ 3) $z\overline{\phi '(z)}$ My end goal is to compute the ...
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0answers
65 views

Application of Reflection Principle

Let $f(z)$ be an entire function whose modulus is constant on some circle. Show that $f(z)=c(z-z_0)$ for some $n\geq 0$ and some constant $c$, where $z_0$ is the center of the circle. So far I have ...
3
votes
1answer
70 views

Therem of Residue application

I want to determinate the following integral: $$\int_{\gamma} \frac{e^z}{\cos{(z)}} dz$$ Where $\gamma (t)=\frac{\pi \cos t}{1 +\sin^2 t}(1+i\sin t)$, $0\leq t \leq 2\pi$ So I see that $\...
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vote
1answer
59 views

Find and show that the residues of the meromorphic differential $dx$ for Fermat's curve $C:$ $\xi_1^4+\xi_2^4=\xi_0^4$ is zero

Find the residues of the meromorphic differential $dx$ of Fermat's curve $C:$ $\xi_1^4+\xi_2^4=\xi_0^4$ at its poles. Check that their sum is zero, as it must be. Attempt: Let $\xi_2\not=0$. Then ...
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2answers
137 views

How many distinct elements are there in $C=\{zw\mid z∈A$,$w∈B\}, z^{24}=1$ and $w^{54}=1$.

Let $A$ be a set of all complex numbers $z$ such that $z^{24}=1$ and let $B$ be the set of all complex numbers $w$ such that $w^{54}=1$. That is: $A$={$z$|$z^{24}=1$} and $B$={$w$|$w^{54}=1$} Finally,...
0
votes
0answers
67 views

Integral Evaluation Trouble

I have the following indefinite integral: $$\int\frac{2e^x}{e^{2x}+1+2x}dx$$ I was thinking substitution, so let $u=1+2x$. then $du=2dx$ and thus $$\int\frac{e^{\frac{u-1}{2}}}{e^{u-1}+u}du$$ I've ...
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2answers
66 views

Derivatives of $f$ doesn't match accordingly to the Cauchy-Riemann equations?

For the function $f(x+iy) = 2xy+i(x+\frac2 3y^3)$, I've decided that $f$ is differentiable at the points $-1/2$ and $-1/2 + i$ by using the Cauchy-Riemann equations: $\frac {\partial u}{\partial x} = ...
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2answers
32 views

Establishing a bound in the complex plane

so my function is $$f(z)= \frac{e^{iz}}{z^2+a^2} $$ What is getting to me and probably I should've been comfortable with this fact is how they establish this upper bound: $$\bigg|\int_{0}^{\pi} f(Re^...
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vote
1answer
32 views

Algebraic vs. analytic definition of the multiplicity of a polynomial's root

Let $f(x) = a(x - c_1)^{d_1}(x - c_2)^{d_2} \dots (x - c_n)^{d_n}$ be a polynomial over the complex numbers ($n, d_i \in \{1, 2, \dots\}$, $a \in \mathbb{C}\setminus \{0\}$), where the roots $c_1, c_2,...
-1
votes
1answer
52 views

Determine all points where $f(x+iy) = 2xy + i(x+\frac 2 3 y^3)$ is differentiable in $\mathbb C$.

Consider the function $f : \mathbb C \rightarrow \mathbb C$ given by $f(x+iy) = 2xy + i(x+\frac 2 3 y^3)$. I want to determine all points at which $f$ is differentiable as a complex function. To do ...
5
votes
1answer
87 views

How to prove $\oint_\Gamma \nabla\theta\cdot\vec{dr}=\pm2\pi $ around a phase singularity/over a cut

How would you prove that $$\oint_\Gamma \nabla\theta\cdot\vec{dr}=\pm2\pi $$ We know that $\theta\in(-\pi,\pi)$, suppose that $\theta$ is continuous in the region bounded by and along $\Gamma$ apart ...
2
votes
1answer
64 views

Show that the meromorphic differential of the homogeneous polynomial is holomorphic and not isomorphic to $\mathbb{P_1}$

Consider the elliptic curve i.e. non-singular cubic, $X$ given by the equation $\xi_0\xi_2^2=\xi_1^3-\xi_0^2\xi_1$ in projective coordinates $(\xi_0:\xi_1:\xi_2)$, or, equivalently, by the equation $y^...
2
votes
1answer
39 views

Evaluate the countour integral $\int _\Gamma z dz$

Can someone please help me setup a) $\int _\Gamma z dz$ b) $\int _\Gamma \bar z dz$ and given the admissible parametrization of $\Gamma$ $\Gamma_1 : z_1: 2 + i(t - 1) ; 1 \leq t \leq 2$ and $...
2
votes
1answer
82 views

Divisor of the meromorphic differential $\omega=\frac{dx}{y^3}$ on C: $\xi_1^4+\xi_2^4=\xi_0^4$

Consider Fermat's curve of degree 4 defined by C : $\xi_1^4+\xi_2^4=\xi_0^4$ in projective coordinates $(\xi_0 :\xi_1 :\xi_2)$ or, equivalently, by the affine equation $x^4 + y^4 = 1$ in the affine ...
0
votes
2answers
38 views

Show $\\Log z_1z_2 \neq Log z_1 + Log z_2$. given $z_1 = i$ and $z_2 = -\sqrt 3 + i$.

Show by evaluating both sides that for $z_1 = i$ and $z_2 = -\sqrt 3 + i$, $\\Log z_1z_2 \neq Log z_1 + Log z_2$. Recall the definition: $\\Log z = Log |z| + iArg z$ Attempt: left side: $\\Log ...
2
votes
1answer
67 views

Proving that $\phi_a(z) = (z-a)/(1-\overline{a}z)$ maps $B(0,1)$ onto itself.

I want to prove that if $\phi_a: B(0,1) \to \Bbb C$ is given by $\phi_a(z) = (z-a)/(1-\overline{a}z)$ with $|a| < 1$, then $|\phi_a(z)| < 1$. Resist the itch on your finger urging you to close ...
3
votes
3answers
126 views

Why does the residue method not work straight out of the box here?

I'm trying to evaluate the integral $$I = \int_0^{\infty} \frac{\cos(x)-1}{x^2}\,\mathrm{d}x $$ The way I've done this is by rewriting $\frac{\cos(x)-1}{x^2}$ as $\Re\left[\frac{e^{iz}-1}{z^2}\right]$,...
2
votes
1answer
63 views

Laurent series of $\frac{1}{\sin^2z}$ around $0$?

I tried to expand $\frac{z^2}{\sin(z)^{2}}$ using Taylor expansion, but the coefficient involved some limit of $\frac{0}{0}$ and was really difficult to calculate. (I tried to convince myself the ...
1
vote
2answers
46 views

complex integral using cauchy's formula

I have to find the value of: $\int_{|z-2|=\frac{3}{2}}\frac{\cos{z}}{z^2(z^2-\pi^2)}$ If there is no singularity in the closed disk $D=|z-2|\leq\frac{3}{2}$ then the integral should be 0. For the ...
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vote
2answers
58 views

A dense subspace of L^2

Let $\mathcal{H}$ be the Hilbert space of holomorphic functions defined on the unit disc $D\subset\mathbb{C}$ which is the clousure of the complex polynomial functions on the disc with respect to the ...
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2answers
71 views

Trig function integral

I'm trying to solve $$\int_{0}^{\pi}\frac{dx}{cos^2(x)-a^2}, \hspace{5mm} 0<a<1$$ There are numerous examples of similar integrals but non with the condiction that $0<a<1$, say $a = 0.5$....
0
votes
1answer
35 views

Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1.

Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1. Then, (a) the set {z : |h(z)| = 5} is unbounded by the Maximum Principle; (b) the set {z : |h'...
1
vote
1answer
44 views

Suppose $|\alpha + \beta| \le |\alpha' +\beta'|$. Is it then possible that $|\alpha'| + |\beta'| \le |\alpha| + |\beta|$?

Suppose $\alpha, \beta, \alpha', \beta' \in \mathbb C$ and that $|\alpha + \beta| \le |\alpha' +\beta'|$. By using the triangle inequality is it then possible that $|\alpha'| + |\beta'| \le |\alpha|...
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votes
2answers
46 views

Show that the function $g(z) = f(e ^z )$ is not a polynomial.

Let $f :\mathbb C \rightarrow \mathbb C$ be an entire function. Show that the function $g(z) = f(e ^z )$ is not a polynomial. What is the technique to show a function is a polynomial?
3
votes
2answers
95 views

How do I use residue theorem to evaluate this improper integral to get a good looking solution?

The problem is $\int_{0}^{\infty} \frac{\sqrt{x}}{x^2+2x+5}dx$ I replace x with z, and did some algebra, but the solution was rather nasty. it contains exponential and arctan such and such. However, ...
1
vote
2answers
45 views

Finding the zeroes of this complex polynomial

Now I thought it wouldn't be too much of an issue, but it is becoming hell to find the zeroes of: $$z^4 + 10z^2 +1 $$ Now reason I need them is for the poles of a function I am working on. So with ...
1
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0answers
75 views

simple root of a complex variable function

I have encountered the following question: Let $f_{\epsilon}(z)=\sin(z)+\epsilon e^z$. Prove that $f_{\epsilon}(z)$ has a simple root $z_{\epsilon}$ with the property that $\lim_{\epsilon \to 0} \,\,\...
1
vote
1answer
19 views

How does this inequality of a complex function hold

I cannot figure out how $\Re[g(\lambda)]\leq |\lambda|$ implies $|g(\lambda)|\leq|2 r-g(\lambda)|$ where $\lambda$ is an arbitrary complex number s.t. $|\lambda|\leq r$, and $g$ is an entire function. ...
3
votes
1answer
45 views

Residue of 1/sin^3(z)

What are the residues of $ \frac{1}{sin^{3}z} $? From the residue theorem the residues are at $$\lim_{z \rightarrow z_{0}} \frac{1}{(n-1)!} \frac{d^{n-1}}{dz^{n-1}} (z-z_{0})^{n} f(z)$$ $$\lim_{z ...
1
vote
1answer
165 views

Find the Laurent expansion for $f(z)=\frac{\exp{1/z^2}}{z-1}$ about $z=0$.

Find the Laurent expansion for $f(z)=\frac{\exp{(1/z^2)}}{z-1}$ about $z=0$. I was able to determine the series for each of the factors. We have $$e^{1/z^2}=1+\frac{1}{z^2}+\frac{1}{2!z^4}+\frac{1}{...