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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29 views

Construct holomorphic function from harmonic function

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω=C$\ {0, 1}. Show that there exist unique real numbers $a_0, a_1$ such that $$u(z)=h(z)−a_0log|z|−a_1log|z−1|$$ is the real ...
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1answer
23 views

Using Partial Fraction Decomposition to acquire appropriate form for GCIF

I need to find the PFD so I may continue with a complex integral $\int_C \frac{ze^z}{z^6 - 1}dz$, $z \in \mathbb{C}$. The contour $C = |z-a|=a$, $a>0$ I have found all $6$ roots of $z^6 - 1$, so ...
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1answer
16 views

Further from Cauchy inequality

Let $f$ be entire and$ M(R)=sup_{|z|=R}|f(z)|$ and $A(R)=supn_{≥0}|a_n|R^n$ where $a_n$ = $f^{(n)}(0)/n!$. Prove that $2A(2R) ≥ M(R)$ I tried to approach this question the same way as the Cauchy ...
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22 views

Reference Request for Complex Analysis (with some specificity regarding Ahlfors and Cartan)

I am a self-studier and am making my second pass through Complex Analysis. I have read the reference request posts many times. Yet perhaps I could get some advice as to the relevant benefits of ...
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1answer
32 views

Calculate $\int_\Gamma \frac{2z+i}{z^2(z^2+4)}$ with residue theory. Where $\Gamma:|z-3i|=4$ is positively oriented circle.

Calculate $\int_\Gamma \frac{2z+i}{z^2(z^2+4)}$ with residue theory. Where $\Gamma:|z-3i|=4$ is positively oriented circle. Pls, for check my solution. poles: $z_1=0$ (order 2 pole) $z_2=-2i$ ...
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1answer
25 views

laurent series expansion problem 1

I am trying to find the laurent series for the function $\frac{1}{z+z^2}$ for domain $0<|z+1|<1$. I separated the function into: $f(z)=\frac{1}{z}\frac{1}{z+1}$ I am having trouble with the ...
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1answer
31 views

Find all polynomials such that $P(A)\subset U$ for a countable subset of the unit circle $U$

I recently answered a question, in which I proved that If a polynomial fixes the unit circle then $P$ is a monomial (a classical result),i,e: $$\forall P\in \Bbb C[X]\ \ \ \ (\forall z\in \Bbb C \ \ ...
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1answer
39 views

Simple Residue calculation

$$\int_{\gamma(0;2)}\frac{e^{i\pi z/2}}{z^2-1} \, dz$$ Using the residue calculus i got $$-2\pi$$But the answer is $$=i$$ I must be wrong at this, but shouldn't the answer have $\pi$ at least since ...
2
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1answer
43 views

Trying to understand a proof in Rudin concerning winding number

In the proof of theorem 10.10 in Real and complex analysis Rudin states that if we will differentiate $$\phi(t) = \exp \left\{\int_a^t \frac{\gamma'(s)}{\gamma(s)-z} \,\textrm{d}s\right\}, \textrm{we ...
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32 views

The inverse of the Gamma function at $-\infty$

Let $\Gamma$ be the analytic continuation of the Gamma function $$\Gamma:z\mapsto \int_0^{+\infty} x^{z-1}e^{-x}dx$$ on the complex plane except non-positive integers. We know that $\Gamma$ has no ...
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0answers
37 views

Complex Exponential in Differential Equations.

I am a physics student, but have taken courses in analysis and algebra. My knowledge of differential equations is purely methodical, and I was hoping for a more math oriented insight with regards to ...
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0answers
27 views

What is An Image (of the Riemann Sphere)? [closed]

What's the image of the left half plane of the Riemann Sphere? {z in C; Re(z)<0} How do I find the image??
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1answer
36 views

Show that $g(z) =\bar{z}$ is continuous at every point of $\mathbb{C}$ and that it's not differentiable at any point. [closed]

Show that $g(z) =\bar{z}$ is continuous at every point of $\mathbb{C}$ and that it's not differentiable at any point.
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1answer
22 views

Taylor Series Expansion for ${z^2+4z^4+z^6}/(1-z^2)^3$

So I know for sure that the Taylor Series expansion for $1/(1-z^2)^3$ is $\sum {k(k-1)z^{2k-4}/{2}} $ assuming |x|<1. But what do we do with the top? I think its already in the expanded form, ...
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1answer
37 views

Cauchy inequality

This is supposed to be an upper bound counterpart for the Cauchy inequality. Let $f$ be entire and $M(R) = sup_{|z|=R} |f(z)|$ and $A(R) = sup _{n≥0} |a_n|R^n$ prove that $2A(2R)$ ≥ $M(R)$ I used ...
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43 views

Integrate x^(2m)/(1 + x^(2n)) from 0 to infinity, where 0 <= m < n. [closed]

Integrate $x^{2m}/(1 + x^{2n})$ from $0$ to $\infty$, where $0 \le m < n$.
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1answer
77 views

Rearranging $\sum_{k = 0}^{+\infty} \left(z+\frac{1}{2}\right)^k$.

Consider the complex series: $$\sum_{k = 0}^{+\infty} \left(z+\frac{1}{2}\right)^k.$$ Clearly the series converges for $\left|z+\frac{1}{2}\right| < 1$, by the ratio test. I am supposed to write ...
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0answers
15 views

Map in $C(\mathbb{T})$ with constant modulus 1

I know that a function in A($\mathbb{D})$ (analytic on the open disk and continuous on its boundary $\mathbb{T}$) with constant modulus on $\mathbb{T}$ is a finite Blaschke product. But what about a ...
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1answer
23 views

How to compute the following real integrals using the residue theorem?

How to compute the following real integrals using the residue theorem: $$\int_{-\infty}^{\infty} \frac{1}{(x^2+p^2)(x^2+q^2)} dx$$ $$\int_{0}^{2\pi} \frac{sin^2(\theta)}{5+4cos(\theta)} d\theta$$ ...
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3answers
47 views

Can a non-constant analytic function have infinitely many zeros on a closed disk?

I think not, however my proof is quite sketchy so far.. My attempt: Suppose an analytic function f has infinitely many zeros on some closed disk D. Then there exists a sequence of zeros in D with a ...
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1answer
23 views

How does squaring a function affect it's removable singularities?

This is a simple question. say you have a function, f, with a removable singularity. does f^2 have a removable singularity at the same point? I strongly suspect that squaring the function would only ...
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1answer
72 views

Holomorphic function injective on annulus => injective on unit disk?

Let $f(z)$ be analytic on the unit disc, and suppose that there is an annulus $U =$ {$z ∈ C| r < |z| < 1$} such that $f(z)$ restricted to the annulus U is injective. Show that f is injective on ...
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32 views

Affine curve is union of $d$ lines through point of multiplicity $d$. [closed]

Let $C$ be an affine curve defined by a polynomial of $P(x, y)$ of degree $d$. Show that if $(a, b)$ is a point of multiplicity $d$ in $C$ then $P(x, y)$ is a product of $d$ linear factors, so $C$ is ...
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1answer
42 views

What are the zeros of the j-function?

Recall that, for a complex number $\tau$ with positive imaginary part, the $j$-invariant is given by $j(\tau)=1728 \frac{g_2(\tau)^3}{g_2(\tau)^3-27g_3(\tau)^2}$ where $g_2(\tau)=60 ...
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0answers
70 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
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1answer
102 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
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1answer
9 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
17 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
347 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 ...
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1answer
30 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} ...
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2answers
24 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 ...
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2answers
91 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
15 views

Where exactly is coupling used in this probablistic proof of Liouvelle's theorem? [closed]

In the last section of the following blog post https://blameitontheanalyst.wordpress.com/2012/01/24/probabilistic-coupling/ coupling is supposedly used but it is not clear at which step they ...
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0answers
16 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
40 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. ...
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2answers
35 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 ...
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29 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) ...
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1answer
53 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 $$ ...
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1answer
17 views

Unsure about expansion

Hello, can someone tell me how this expression is expanded in this proof. Does it follow from some other theorem?
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1answer
24 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 ...
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1answer
22 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 ...
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1answer
14 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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27 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) = ...
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31 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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28 views

Is every natural recursive relation necessarily holomorphic?

Define the set of algebraic primitive recursive relations as the set of functions defined by: $$ F(n,a,k) = F(n-1,F(n-1,F(n-1,a,a),a)...,a)_{\text{nested to depth k}}$$ $$ F(0,a,k) = a + k $$ Along ...
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2answers
35 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
23 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
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1answer
57 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 ...
1
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1answer
43 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
39 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 $B$={$w$|$w^{54}=1$} ...