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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89 views

Prove that if integral of f around any closed disk in U is 0 then f is holomorphic in U?

I know goursats theorem says that if integral of f over any triangle in U is 0 then f is locally integrable in U and hence by Moreras theorem is holomorphic in U. But here I need to show that if f is ...
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0answers
29 views

Prove that if $g = r +ip$ is analytic on $C$ and $r(x,y) \leq M$, with $M > 0$, for all $(x,y)\in C$, $g$ is constant.

Let $g = r +ip$ be analytic on $C$. If for some $M > 0$ we have $r(x,y) \leq M$ for all of $C$, then $g$ is constant. The theorem is given without proof in my notes and I can't find any examples ...
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1answer
89 views

$f$ is holomorphic iff $df$ is $\Bbb C$-linear

Let $\Omega\subseteq\Bbb C^n$ open connected, $f:\Omega\to\Bbb C$ differentiable in the real sense. We know that $f$ is holomorphic iff $\partial_{\bar z_j}f=0\;\;\forall j=1,\dots,n$ . We know also ...
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1answer
282 views

Understanding angle-preserving definition

My book (Real and complex analysis, by Rudin) gives the following definition: Let $A(z) = \frac z{|z|}$. Then we say $f$ preserves angles at $z_0$ if $$\lim_{r \to 0}e^{-i\theta} A[f(z_0 + re^{i\...
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1answer
36 views

Finding $a_{-n}$ where $\cot (\pi z)=\sum_{n=-\infty } ^\infty a_nz^n$

The following is problem 5.11.2 of Berkeley Problems in Mathematics. Let $\cot (\pi z)=\sum_{-\infty} ^\infty a_nz^n$ be the Laurent expansion for $\cot (\pi z)$ on the annulus $1<\vert z \vert ...
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1answer
36 views

Rouche's theorem to P(z)/Q(z)

Find that the number of roots of $$\frac{z^2-4}{z^2+4} + \frac{2z^4-1}{z^2+6} = 0$$ within the unit circle is zero. So I have solved for $P(z) = (z^2-4)(z^2+6)$ and $Q(z) = (2z^4-1)(z^2+4)$. I can't ...
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2answers
58 views

set of arbitrary positive measure conformally equivalent to unit disk

Show that for any  $\epsilon$ > 0, there is a dense subset of $\mathbb{C}$ with measure less than $\epsilon$ and which is conformally equivalent to the unit disc. To make a dense set that has ...
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2answers
77 views

Morera's Theorem Proof

Hello, I am having trouble with Morera's Theorem. How does the integral being equal to 0 matter? I can't see why this condition is necessary for this theorem to hold true. Also, if someone could ...
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1answer
66 views

Gluing together holomorphic functions on $\mathbb{P}^n$

The problem Let $U_j$ for $0\leq j\leq n$ denote the standard coordinate charts of the complex manifold $\mathbb{P}^n$. Fix $d\geq 1$ and assume we are given holomorphic functions $f_j:U_j\to \mathbb{...
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1answer
185 views

Mittag-Leffler Expansion

I am attempting to perform what is described in my notes as a "Mittag-Leffler Expansion", but first I must prove that this expansion is valid. Given that $$ f(z) = \frac{1}{\sin{z}} - \frac{1}...
2
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1answer
71 views

Complex modulus Inequality using $|exp(z)-1|$

I think I am almost there: Prove $\left|z\right|/4 < \left|\exp(z)-1\right|<7\left|z\right|/4$ for all $0<|z|<1$. MY ADVANCES First we note that $$ \left|\exp(z)-1\right| = \left|\...
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0answers
35 views

Is a holomorphic function with zero derivative on a close connected set constant?

I know that it is constant if the set is open and connected but I don't know why the condition of openness is necessary.
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1answer
59 views

Integral of a two-valued function with two branch cuts

Is it possible to calculate in closed-form the integral $\int_{-\infty}^{+\infty}\mathrm{sinc}(\sqrt{1+x^4})\,dx$ (sinc being the cardinal sine, $\sin(x)/x$)? The function is everywhere defined (all ...
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1answer
41 views

Procedure for plotting domain $|2z+3|\gt 4$

I want to plot $|2z+3|\gt 4$. Firstly I plotted $|2z|\gt 4$, by taking it to be:$$|z|\gt 2$$ $$|x+iy|\gt 2$$ And this I am comfortable with, it is just everything greater than(in x or y values) than ...
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2answers
43 views

Minimising $|a+bw+cw^2|$ such that a,b,c are consecutive integers?

Suppose we are given a expression $k=|a+bw+cw^2|$ such that $w$ is cube root of unity ($w\neq1$) such that $\{a,b,c\}$ are consecutive integers , then how can we minimise value of expression ? I was ...
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1answer
77 views

Complex Analysis: Schwarz's Lemma

Let $f$ be a one-to-one holomorphic mapping from the unit disk onto itself, $f(0)=0$, $f^{\prime}(0)>0$. Prove that $f(z)=z$. Attempt: Since the hypothesis gives us $f(0)=0$ and $|f(z)<1|$ (...
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1answer
73 views

An entire function is a polynomial iff the Taylor expansion around $0$ converges uniformly

Let $g:\mathbb{C} \to \mathbb{C}$ an entire function. Prove that the Taylor expansion around $0$ converges uniformly in all $\mathbb{C}$ if and only if $g$ is a polynomial. 1/2 PROOF I think I have ...
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0answers
21 views

Real Roots of Complex Variable Equations [duplicate]

Prove that the equation tanz=z has only real roots....I am stuck on this so any help would be nice... Many things do not work for example one cannot just graph like in calculus and look at the ...
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75 views

Is the union of two projective curves in the projective plane a projective curve?

As the title suggests, is the union of two projective curves in the projective plane a projective curve? Any help would be appreciated, thanks.
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1answer
28 views

All the zeroes of analytic $f$ in $A$ are isolated, or $f \equiv 0$ on $A$.

Let $f$ analytic on an open connected domain $A$. I need to prove that all the zeroes of $f$ in $A$ are isolated, or $f \equiv 0$ on $A$. What I did: I let $B=$ the set of limit points of zeroes in $...
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0answers
55 views

Interpreting and understanding the identity $e^{iz} = \cos(z) \pm \sqrt{\cos^2(z) - 1}$

A question in my complex analysis book (Gamelin's "Complex Analysis", question I.8.7) asks me to prove that $e^{iz} = \cos(z) \pm \sqrt{\cos^2(z) - 1}$. Using the identity $\cos(z) = \frac{e^{iz} + e^{...
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1answer
32 views

Pole of elliptic function

Let $f:C→P1$ be such that $f(z+1)=f(z+i)=f(z)$ for all z∈C. Let $Γ=\{m+ni:m,n∈Z\}$. Show that if $f$ is holomorphic on $C∖Γ$, and $z⋅f(z)$ is bounded in a neighbourhood of $z=0$, then $f$ is constant....
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1answer
186 views

Are the integrals of the following function path independent in the following domain?

Are the integrals of the function: $$f(z)=\frac{1}{z+1}+\frac{1}{(z+1)^2}+e^{\frac{1}{z}}$$ path independent in the following domain: $$D= \{Re z >0\}\setminus\{1\}$$ My thoughts on the ...
3
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0answers
114 views

Construct holomorphic function from harmonic function

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω=\Bbb C \setminus \{0, 1\}$. Show that there exist unique real numbers $a_0, a_1$ such that $$u(z)=h(z)−a_0 \log |z|−a_1 \log |...
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1answer
27 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
27 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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1answer
34 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) $z_2=-2i$ (...
2
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1answer
32 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 ...
0
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1answer
41 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 \ \ |...
2
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1answer
58 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
63 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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0answers
65 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
60 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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1answer
54 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
24 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, ...
5
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1answer
61 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 ...
2
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1answer
79 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
25 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
69 views

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

How to compute the following real integrals using the residue theorem: $$\int_{-\infty}^{\infty} \frac{1}{(x^2+p^2)(x^2+q^2)} dx$$ $$\int_{0}^{2\pi} \frac{sin^2(\theta)}{5+4cos(\theta)} d\theta$$ ...
2
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3answers
680 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
34 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 ...
6
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1answer
205 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 $U = \{z ∈ C \mid r < |z| < 1\}$ such that $f(z)$ restricted to the annulus $U$ is injective. Show that $f$ is ...
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0answers
53 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
71 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 \sum_{(m,n)\neq(...
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2answers
53 views

Prove the function to not be continuous at $z = 0$

$$f(3) = \begin{cases} \dfrac{\mathrm{Re}(z)}{|z|} & \text{when $z \neq 0$} \\ 0 & \text{when $z = 0$} \end{cases}$$ Can someone please explain the concept behind solving such a problem? ...
5
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1answer
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 ...
0
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1answer
60 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.
4
votes
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 ...