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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Circular Contour Integration .

Doing some revision for an upcoming exam I have stumbled across the following problem: Evaluate the integral $\int_{C}\log(z)$ where $C=C(2,1)$ the positively oriented circular contour, centre 2, ...
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70 views

Classify the singularities of the function .

Classify the singularities of the function $\frac{1-\cos(z)}{z^2(z-1)}$. I think my answer may be that I have a simple pole at $z=0$ and a removable singularitie at $z=-1$ however i am not too sure. ...
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60 views

Evaluate integral using Residue Theorem

Let $\gamma$ be the semicircle $[-R,R]\cup\{z\in\mathbb{C}:|z|=R\ and\ Im{z}>0\}$, traced in the positive direction, and $R>1$. Evaluate $$\int_\gamma\frac{dz}{z^4+1}.$$ I note that ...
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85 views

Prove a union is a domain

Prove that if S and T are domains that have at least one point in common, then S union T is also a domain I wrote: A domain is a set that is open and connected. The union of open sets is easily open. ...
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1k views

If $f,g$ are both analytic and $f(z) = g(z)$ for uncountably many $z$, is it true that $f = g$?

If two analytical functions of $\mathbb{C}$ f and g are equal on an infinite number of input values, than they are equal. I can't seem to find a counterexample, but I haven't seen this anywhere ...
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88 views

Contour integration with a branch cut. Parameterizing f(z) properly

I have a contour integral of a function of the form $(z^6-P)^\alpha z^\beta$ Here $\alpha\in R$, $\beta\in N$ and $P$ is some constant. I therefore have branch points at the sixth roots of $P$. The ...
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63 views

Entire functions of order 0

Sorry, this may be a stupid question, but I am just beginning to learn about this and cannot find the answer anywhere I have looked so far. Clearly if we have any polynomial $P(z)$, then it is easy to ...
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89 views

path dependence of the integral $f(z)=\frac{1}{(z-4)^2} + \sin z$

Are the integrals of the $$f(z)=\frac{1}{(z-4)^2} + \sin z$$ path independent in the following domain $$D= \{\operatorname{Re} z >0\}\setminus\{4\}$$ My thought is that since ...
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35 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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142 views

On the construction of hyperelliptic Riemann surfaces.

I have seen two ways to construct hyperelliptic curves, and it seems to me that the intuition behind the change of coordinate is not the same. I like better the second construction (which is pretty ...
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56 views

$\tan z=az+b$ has infinitely many solutions

I try to prove following questions. Prove that, for all complex numbers $(a,b)\neq (0,\pm i)$, the equation $\tan z = az + b$ has infinitely many solutions. By assuming $(a,b)=(1,0)$, I tried ...
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90 views

infinite product expansion $\frac{1}{\sin(z)}-\frac{1}{z}$

I have successfully solved that $$\frac{1}{\sin(z)}-\frac{1}{z} = \sum_{n=-\infty, n \neq 0}^\infty (-1)^n\left(\frac{1}{z-n\pi}+\frac{1}{n\pi}\right)$$ and am now attempting to integrate both sides ...
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1answer
30 views

Bounded entire fuctions. [duplicate]

Let $f$ be an entire function and assume f(0) = 1 and $|f(z)| \large \geq\frac{1}{3}\left| {\LARGE e^{z^{3}}}\right|$ for all $z$. Show $f(z) = {\huge e^{z^{3}}}$ for all $z$. Can this be shown ...
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86 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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1answer
87 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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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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35 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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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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74 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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131 views

Find a meromorphic function $f$ with poles at $\dots,-3,-2,-1$

I need help with the following problem: Find a meromorphic function $f:\mathbb{C}\longrightarrow \mathbb{C}$ whose only singularities are simple poles at $\dots,-3,-2,-1$ with residues $n$ at $z=-n$. ...
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68 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| = ...
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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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153 views

Key differences between almost complex manifolds and complex manifolds

I know the technical difference between an almost complex manifold and a complex manifold, namely in the former the almost complex structure $J$ may not be integrable while in the later it is. ...
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72 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 ...
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61 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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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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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} ...
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131 views

Help with an inequality in Cazenave's book “Semilinear Schrodinger equations”

I'm reading Cazenave's book "Semilinear Schrodinger equations" and I found this inequality at page 84 $$\vert\vert u_1\vert^\alpha u_1-\vert u_2\vert^\alpha u_2\vert\vert\leq C (\vert ...
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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. ...
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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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72 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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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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73 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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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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183 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 ...
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1answer
69 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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100 views

The Poisson Integral is harmonic

We have proved that for $h(e^{\mathcal{i}\theta})$ continuous on the unit circle, the Poisson Integral of $h$ defined by ...
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53 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} + ...
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1answer
30 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 ...
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170 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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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) $z_2=-2i$ ...
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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 ...
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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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39 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
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
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 ...
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 ...
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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
59 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.