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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20 views

Roots of complex polynomial in $S^1$

roots of complex polynomial Let is $ p(z) = a_0 + ... + a_n z^n$ such that $a_n$ is not equal to zero there is $ j \in \{0, ... ,n \}$ with $ |a_j| > ( \sum_{k=0}^n |a_k|) - |a_k| $. Show that the ...
2
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1answer
35 views

Assumptions on functions so that integral is zero

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two arbitrary functions. Assume $g\in L^2(\mathbb{R})$. I'm looking to find out the minimal set of assumptions on $f$ and $g$ such ...
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22 views

How to calculate this residue which has a pole of order n-r?

So I have this complex integral: $$ \oint \frac{dz}{2\pi}\frac{e^{iz(br-(n-r)a)}}{\left(1-(1-q)e^{-ik_{1}-iza}\right)^{n-r}}$$ b,r,q,a,n are all constants in this context. However I'm not entirely ...
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2answers
31 views

What does this condition mean for Cauchy's integral formula?

** Theorem: ** Suppose that $\Omega$ is a simply connected domain in $\mathbb{C}$, that $f \in H(\Omega)$, that $\Gamma$ is a simple closed contour in $\Omega$ and that $w \in \text{Int} ...
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1answer
49 views

What does holomorphic at the cusp infinity means

In the usual theory of classical modular forms, the modular forms defined to be "holomorphic at the cusp infinity". I do not know what this should mean? can anyone explain it for me? Thanks
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1answer
16 views

Solve the moebius transformation $f(z) = \frac {z−1}{iz+i} $For φ ∈ R

I just don't know how the term in the denominator $|e^{iφ} +i|^2$ got there obviously if we factor out the $i$ like $i(e^{iφ} +1)$ and then multiply the numerator and denominator by the conjugate of ...
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1answer
32 views

Integration along a keyhole

(H. Priestley complex Analysis Chapter 7 Exercise 9) Suppose $f$ is holomorphic inside and on $\gamma(0,1)$. By integration around the usual keyhole like this one : Integration of $\ln $ around a ...
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2answers
33 views

Complex analysis: partial fraction decomposition

Suppose $p$ and $q$ are polynomials of degrees $m$ and $n$ respectively where $n \ge m+1$, and suppose q has simple zeros at $b_1$,...,$b_n$. By considering $f(w)=\frac{p(w)}{q(w)(w-z)}$ obtain the ...
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0answers
16 views

Reciprocals of theta functions

I've spent the last few months with partial fraction expansions, and thought to create a function with simple poles over a lattice of zeros, like that of any of the Jacobi theta functions... but I ...
6
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2answers
59 views

Is a meromorphic function satisfying $f(2z)=\frac{f(z)}{1+f(z)^2}$ constant?

Let $f(z)$ be a holomorphic function on the unit disk satisfying $f(0)=0$ and $$f(2z)=\frac{f(z)}{1+f(z)^2}.$$ Extend it to a meromorphic function on the entire complex plane using this recursion. ...
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1answer
36 views

Analytic continuation of a certain Dirichlet series

Is there an elementary way to analytically continue $$f(s)=\sum_{n=1}^\infty \frac{(-1)^n}{(2n+1)^s}$$ to the entire complex plane? It is not hard to see (by grouping terms in pairs and using the ...
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117 views

Subring of $\mathcal O(\mathbb C)$

Let $\mathfrak A \subset \mathcal O(\mathbb C)$ be the subring generated by the nowhere zero analytic functions $f: \mathbb C \to \mathbb C$. Does we have a precise description of $\mathfrak A$ ? Is ...
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1answer
57 views

Ideal in $\mathcal O(\mathbb C)$

Let $\mathfrak {I}$ the ideal generated by all the holomorphic functions which are never zero. Question : is $\mathfrak {I} = \mathcal O(\mathbb C)$ ?
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2answers
49 views

How do I prove that for any two points in $\mathbb{C}$, there exists a $C^1$-curve adjoining them?

Let $G$ be an open-connected subset of $\mathbb{C}$. Let $a,b$ be two distinct points in $G$. How do I prove that there exists a $C^1$-curve $\alpha:[0,1]\rightarrow G$ such that $\alpha(0)=a$ and ...
9
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1answer
54 views

Functions $f$ such that $f(z+1)-f(z)$ is holomorphic

Find all functions $f:\mathbb C\to\mathbb C$ such that $f(z+1)-f(z)$ is entire. I am curious about this, because an algebraic analog states that if $f:\mathbb Z\to\mathbb N$ is such that ...
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1answer
24 views

Calculate complex integral

Let $C$ be a circle $\gamma=\partial B (0,2)$ oriented positively. I have to calculate $$\int_\gamma \frac{-\cos(1/z)}{\sin(1/z)z^2}dz$$ My attempt: Notice that $\sin(1/z)$ is meromorphic inside ...
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1answer
21 views

The Laurent series around $z=0$ of the function $f(z) = \frac{z}{(z-i)(z-2)}$ in the annulus $A(0,1,2)$

What I got so far: $$ \frac{z}{(z-i)(z-2)} = \frac{z}{(2-i)(z-i)} + \frac{z}{(2-i)(z-2)} $$ which is equal to $$ \frac{z}{(2-i)(z-i)} + \frac{z}{(2-i)(z-2)} = \frac{z}{(2-i)z + 1-2i} + ...
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3answers
52 views

How many zeros does $g(z)= z^4+iz^3 +1$ have in the first quadrant?

Let $g(z)= z^4+iz^3 +1$. How many zeros does $g$ have in $\{z\in \Bbb{C}: \text{Re }(z), \text{Im }(z)>0\}$? I tried comparing the number of zeros of $g$ to that of $z\mapsto z^4$ and ...
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1answer
22 views

Find the residue of $e^{\frac{1}{z^2-1}}\sin(\pi z)$ at $z=1$

I'm dealing with the following problem (from an old qualifying exam): Let $\gamma$ be a closed curve in the right half-plane that has index $N$ with respect to the point 1. Find $$ ...
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1answer
28 views

Why are compact complex manifolds Liouville?

I know this is true but strangely can't find references. Also, consider the trivial $n$-bundle over any connected compact manifold, does Liouville imply that all holomorphic sections are constant? ...
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42 views

Using estimation theorem to show integrals are zero.

How would I use the Estimation theorem to show the last two integrals on the right equal zero? \begin{equation} \int_{-R}^{R} e^{-\pi x z^{2}}dz= \int_{-R+i \frac{m}{x}}^{R + i \frac{m}{x}} e^{-\pi x ...
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2answers
82 views

Nontrivial entire $f(z)$ never equal to $0$ [on hold]

I'm looking for nonconstant entire functions $f(z)$ such that $f(z)\neq 0$ for any $z$. More specifically I'm looking for nontrivial cases. So $\exp(z),\exp(z^2),...$ is not what I am looking for. ...
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2answers
36 views

Why aren't these two properties of complex powers the same?

Let $z\in\mathbb{C}$ s.t. $z=u+iv$. As an example, take the square in this trivial manner: $(u+iv)^2=u^2-v^2+2iuv$. On the other hand taking the square using the properties of complex powers, i.e. ...
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55 views

limit of a complex expression

Suppose that $$R(r)=\left[(a_0-r^2a_2)e^{ir\tau}+(b_0-r^2b_2)\right]^{\frac{1}{r}}\;,$$ where $a_0,a_2,b_0,b_2\in\mathbb{R}$ and $\tau>0$. What is $\displaystyle\lim_{r\rightarrow\infty}R(r)$? I ...
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1answer
56 views

Showing a function map to itself

Let $ D = \{ z \in \mathbb{C} : |z| < 1\}$. Fix $ w \in D$ and define $f: \bar{D} \to \mathbb{C}$ by $$f(z) = \frac{w-z}{1-\bar{w}z}$$ Show the following: $f$ maps $D$ to $D$ and $\partial D$ to ...
8
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1answer
98 views

For which complex $a, b, c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
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1answer
130 views

What proof uses both the Riemann Hypothesis and its negation?

Some time ago I happened to see a proof that was remarkable in that it used both the Riemann Hypothesis and its negation. That is, it considered the two cases: RH is true, and RH is false, obtaining, ...
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0answers
19 views

Smooth interpolation for complex variable function. [on hold]

Is there any smooth interpolation function $T(z)$ that could smoothly connect two complex variable rational polynomial function $H_1(z)$ and $H_2(z)$, for example $$ H(z) = \begin{cases} ...
2
votes
2answers
57 views

Weierstrass's M-test example for uniform convergence and switching Sum and Integral.

How would I go about finding $M_n$ in \begin{equation} \sum_{n=1}^{\infty} \int_{0}^\infty x^{\frac{s}{2}-1}e^{-\pi n^{2}x}dx \end{equation} to show that it is uniformly convergent?
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41 views

$\eta(1) = \ln(2)$ proof using Abel's Theorem

Hi I was just wondering how does one justify $\eta(1) = \ln(2)$. Looking at the power series for $\ln(1+x)$ we have \begin{equation} \ln(1+x)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^{n}}{n} ...
4
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1answer
88 views

Find the natural boundary of $\sum_{n=1}^\infty \frac{z^n}{1-z^n}$

I'm asked to prove that the natural boundary of $\sum_{n=1}^\infty \frac{z^n}{1-z^n}$ is the unit circle. My try: First, use root test to show that the series converges for $|z|<1$. Then I have ...
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1answer
27 views

How can I calculate the singularities and residues of…?

$$\frac{e^z}{z^3(z-1)}+\frac{1}{z^3}$$ I have problems specially for $z=0$ Can anyone show me how to do it?
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1answer
35 views

Evaluating $\int_{\gamma} \frac{z}{\cosh (z) -1}dz$

Evaluate $\int_{\gamma} \frac{z}{\cosh (z) -1}dz$ where $\gamma$ is the positively oriented boundary of $\{x+iy \in \Bbb{C} : y^2 < (4\pi^2 -1)(1-x^2)\}$. I just learned the residue theorem, ...
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3answers
50 views

Question about the Fourier Inversion Formula

We have $$\hat{f}(\xi)=\mathcal{F}f(\xi):= \int_{-\infty}^{\infty}f(x)e^{-2\pi i\xi x}dx,$$ with $f\in L^{1}$, and the Fourier inversion formula says that ...
3
votes
3answers
56 views

How do limits work in complex functions?

I don't quite understand one example in my notes it says. My query is this: I don't understand what the significance of $\theta$ is. Why does it matter that $\theta \in (-\pi,\pi]$? I see the ...
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1answer
38 views

Integrating $\sin^2(x)/x^2$, discrepancies with other solutions

I know this is an old problem and it has been answered some times, but I encountered some discrepancies with former solutions and I am not certain what the underlying mathematical reason is. Here ...
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4answers
208 views

Calculate $\int _0^\infty \frac{\ln x}{(x^2+1)^2}dx$

Calculate $$\int _0^\infty \dfrac{\ln x}{(x^2+1)^2}dx.$$ I am having trouble using Jordan's lemma for this kind of integral. Moreover, can I multiply it by half and evaluate $\frac{1}{2}\int_0^\infty ...
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1answer
81 views

Verifying the complex integral: $\int_0^{\infty}\frac{\cos{ax}}{1+x^4}dx$

Verifying the integral: $$\int_0^{\infty}\dfrac{\cos{ax}}{1+x^4}dx$$ I started considering: $$\cos{x}=\dfrac{e^{ix}+e^{-ix}}{2}\implies \cos{ax}=\dfrac{e^{iax}+e^{-iax}}{2}?$$ So: ...
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2answers
25 views

Closedness of a complex $1$-form defined by homogeneous functions

How can I prove this 1-form is a closed one within the specific subset $A\subset \mathbb C$ ? $$\omega=\frac{f(x,y)}{xf(x,y)+yg(x,y)}dx+\frac{g(x,y)}{xf(x,y)+yg(x,y)}dy$$ Where $f, g\in C^1$ are ...
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0answers
46 views

Can the triangle function approximate the Gaussian curve for complex numbers?

I was thinking about approximating the Gaussian curve with a triangular curve. The graphs look like this: their respective functions are: $$ y_1(x) = t(x) = max(0, 1 - |x|)$$ $$ y_2(x) = e^{ - ...
2
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1answer
40 views

Roots of polynomial outside a vertical strip of $\mathbb C$

Let $P(z)$ be an arbitrary polynomial with real coefficients. I'd like to guarantee that all roots of $P$ have real parts outside the interval $(0, 1)$. Is there some simple condition on P that will ...
2
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1answer
160 views

Why are conformal mappings necessarily 1 to 1?

Say, by the Riemann Mapping Theorem, there exists a biholomorphic, conformal mapping from the upper half plane to the (open) unit disk (since the UHP is simply connected and is not the entire complex ...
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2answers
74 views

Find the Laurent series of $f(z)=\frac{1}{z(1-z)}$

I am having difficulties finding Laurent series of the above function, around these two domains: $$0<|z-1|<1$$ and $$|z-1|>1$$ The function $f(z)$ takes the form ...
2
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1answer
42 views

Harmonic complex function

Can anyone help me with this question? Show that a $C^2$ function, $f:U\longrightarrow \mathbb C$, is harmonic iff $\frac{\partial }{\partial z}\frac{\partial f}{\partial \bar z}\equiv0$. Thank you. ...
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4answers
210 views

Help with the contour for this integral using residues

$$ PV \int_0^\infty \frac{dx}{\sqrt{x}(x^2-1)} $$ A keyhole contour can't be used because we have a pole in the real positive axis, isn't it?
5
votes
1answer
118 views

Definite integral with logarithm and arctangent inside of arctangent

How to prove $$\int_0^1 \left[ \frac{2}{\pi }\arctan \left(\frac 2 \pi \arctan \frac{1}{x} + \frac{1}{\pi }\ln \frac{1 + x}{1 - x}\right) - \frac{1}{2} \right]\frac{\mathrm{d}x} x = \frac{1}{2} \ln ...
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1answer
19 views

Determine a meromorphic function satisfying certain conditions

Suppose $f$ is meromorphic on the Riemann sphere, and suppose also that $f(0) = 0$, $f(-1) = 2$, $f(3) = 3$, $f$ has a simple pole at $1$ with residue $1$, and $f$ has a triple pole at $2$ with ...
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1answer
32 views

Solving $\int_{0}^{+ \infty} \frac{x \cos(x)}{x^4 + 4 a^4} dx$ with residues

We also have the condition $a > 0$. My attempt was to, as usual, define $f(z) = \displaystyle\frac{z e^{iz}}{z^4 + 4 a^4}$. Then I tried to integrate $f$ over a curve $\gamma$ which goes from $0$ ...
0
votes
2answers
38 views

Proof composition of analytic functions is analytic

Title says it all I looked for a proof on this site but couldn't find one. Prove if $f$ is analytic on $D$ and $g$ is analytic on $\Omega$ containing the range of $f$ show $g(f(z)$ is analytic. ...
3
votes
2answers
72 views

Complex analysis: Prove a meromorphic function to be rational.

I come across a problem about complex analysis: Show that a meromorphic function on the complex plane, which achieves any complex number no more than fixed given times, must be rational. The only ...